R.R.Puri MATHEMATICAL METHODS OF QUANTUM OPTICS Berlin: Springer, 2001, pp. XIII+285 This book provides an accessible introduction to the mathematical methods of quantum optics. Starting from first principles, it reveals how a given system of atoms and a field is mathematically modelled. The method of eigenfunction expansion and the Lie algebraic method for solving equations are outlined. Analytically exactly solvable classes of equations are identified. The text also discusses consequences of Lie algebraic properties of Hamiltonians, such as the classification of their states as coherent, classical or non-classical based on the generalized uncertainty relation and the concept of quasiprobability distributions. A unified approach is developed for determining the dynamics of two-level and a three- level atom in combinations of quantized fields under certain conditions. Simple methods for solving a variety of linear and nonlinear dissipative master equations are given. Contents 1. Basic Quantum Mechanics 1 1.1 Postulates of Quantum Mechanics 1 1.1.1 Postulate 1 1 1.1.2 Postulate 2 11 1.1.3 Postulate 3 11 1.1.4 Postulate 4 11 1.1.5 Postulate 5 13 1.2 Geometric Phase 16 1.2.1 Geometric Phase of a Harmonic Oscillator 18 1.2.2 Geometric Phase of a Two-Level System 18 1.2.3 Geometric Phase in Adiabatic Evolution 18 1.3 Time-Dependent Approximation Method 19 1.4 Quantum Mechanics of a Composite System 20 1.5 Quantum Mechanics of a Subsystem and Density Operator 21 1.6 Systems of One and Two Spin-l/2s 23 1.7 Wave-Particle Duality 26 1.8 Measurement Postulate and Paradoxes of Quantum Theory 29 1.8.1 The Measurement Problem 3 0 1.8.2 Schrodinger's Cat Paradox 31 1.8.3 EPR Paradox 32 1.9 Local Hidden Variables Theory 34 2. Algebra of the Exponential Operator 37 2.1 Parametric Differentiation of the Exponential 37 2.2 Exponential of a Finite-Dimensional Operator 38 2.3 Lie Algebraic Similarity Transformations 39 2.3.1 Harmonic Oscillator Algebra 41 2.3.2 The SU(2) Algebra 42 2.3.3 The 577(1,1) Algebra 43 2.3.4 The SU{m) Algebra 45 2.3.5 The SU(m, ή) Algebra 45 2.4 Disentangling an Exponential 48 2.4.1 The Harmonic Oscillator Algebra 49 2.4.2 The SU(2) Algebra 50 2.4.3 5(7(1,1) Algebra 51 2.5 Time-Ordered Exponential Integral 52 2.5.1 Harmonic O scillator Algebra 52 2.5.2 SU(2) Algebra 53 2.5.3 The SU(1, 1) Algebra 53 3. Representations of Some Lie Algebras 55 3.1 Representation by Eigenvectors and Group Parameters 55 3.1.1 Bases Constituted by Eigenvectors 55 3.1.2 Bases Labeled by Group Parameters 56 3.2 Representations of Harmonic Oscillator Algebra 60 3.2.1 Orthonormal Bases 60 3.2.2 Minimum Uncertainty Coherent States 61 3.3 Representations of SU(2) 68 3.3.1 Orthonormal Representation 68 3.3.2 Minimum Uncertainty Coherent States 70 3.4 Representations of SU( 1, 1) 76 3.4.1 Orthonormal Bases 76 3.4.2 Minimum Uncertainty Coherent States 77 4. Quasiprobabilities and Non-classical States 81 4.1 Phase Space Distribution Functions 81 4.2 Phase Space Representation of Spins 88 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components 93 4.4 Classical and Non-classical States 95 4.4.1 Non-classical States of Electromagnetic Field 95 4.4.2 Non-classical States of Spin-l/2s 97 5. Theory of Stochastic Processes 99 5.1 Probability Distributions 99 5.2 Markov Processes 102 5.3 Detailed Balance 105 5.4 Liouville and Fokker-Planck Equations 106 5.4.1 Liouville Equation 107 5.4.2 The Fokker-Planck Equation 107 5.5 Stochastic Differential Equations 109 5.6 Linear Equations with Additive Noise 110 5.7 Linear Equations with Multiplicative Noise 112 5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations 113 5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations 114 5.8 The Poisson Process 115 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise 116 6. The Electromagnetic Field 119 6.1 Free Classical Field 119 6.2 Field Quantization 121 6.3 Statistical Properties of Classical Field 123 6.3.1 First-Order Correlation Function 125 6.3.2 Second-Order Correlation Function 126 6.3.3 Higher-Order Correlations 126 6.3.4 Stable and Chaotic Fields 127 6.4 Statistical Properties of Quantized Field 130 6.4.1 First-Order Correlation 131 6.4.2 Second-Order Correlation 132 6.4.3 Quantized Coherent and Thermal Fields 132 6.5 Homodyned Detection 134 6.6 Spectrum 135 7. Atom— Field Interaction Hamiltonians 137 7.1 Dipole Interaction 137 7.2 Rotating Wave and Resonance Approximations 140 7.3 Two-Level Atom 144 7.4 Three-Level Atom 145 7.5 Effective Two-Level Atom 146 7.6 Multi-channel Models 149 7.7 Parametric Processes 150 7.8 Cavity QED 151 7.9 Moving Atom 153 8. Quantum Theory of Damping 155 8.1 The Master Equation 155 8.2 Solving a Master Equation 160 8.3 Multi-Time Average of System Operators 162 8.4 Bath of Harmonic Oscillators 163 8.4.1 Thermal Reservoir 164 8.4.2 Squeezed Reservoir 166 8.4.3 Reservoir of the Electromagnetic Field 167 8.5 Master Equation for a Harmonic Oscillator 168 8.6 Master Equation for Two-Level Atoms 170 8.6.1 Two-Level Atom in a Monochromatic Field 171 8.6.2 Collisional Damping 172 8.7 Master Equation for a Three-Level Atom 173 8.8 Master Equation for Field Interacting with a Reservoir of Atoms 174 9. Linear and Nonlinear Response of a System in an External Field 177 9.1 Steady State of a System in an External Field 177 9.2 Optical Susceptibility 179 9.3 Rate of Absorption of Energy 181 9.4 Response in a Fluctuating Field 183 10. Solution of Linear Equations: Method of Eigenvector Expansion 185 10.1 Eigenvalues and Eigenvectors 186 10.2 Generalized Eigenvalues and Eigenvectors 189 10.3 Solution of Two- Term Difference-Differential Equation 191 10.4 Exactly Solvable Two- and Three-Term Recursion Relations 192 10.4.1 Two- Term Recursion Relations 192 10.4.2 Three- Term Recursion Relations 193 11. Two-Level and Three-Level Hamiltonian Systems 199 11.1 Exactly Solvable Two-Level Systems 199 11.1.1 Time-Independent Detuning and Coupling 202 11.1.2 On- Resonant Real Time-Dependent Coupling 208 11.1.3 Fluctuating Coupling 208 11.2 N Two-Level Atoms in a Quantized Field 210 11.3 Exactly Solvable Three- Level Systems 210 11.4 Effective Two-Level Approximation 212 12. Dissipative Atomic Systems 215 12.1 Two-Level Atom in a Quasimonochromatic Field 215 12.1.1 Time-Dependent Evolution Operator Reducible to SU(2) 217 12.1.2 Time-Independent Evolution Operator 219 12.1.3 Nonlinear Response in a Dichromatic Field 223 12.2 N Two-Level Atoms in a Monochromatic Field 224 12.3 Two-Level Atoms in a Fluctuating Field 236 12.4 Driven Three-Level Atom 237 13. Dissipative Field Dynamics 239 13.1 Down-Conversion in a Damped Cavity 239 13.1.1 Averages and Variances of the Cavity Field Operators 240 13.1.2 Density Matrix 242 13.2 Field Interacting with a Two-Photon Reservoir 245 13.2.1 Two-Photon Absorption 245 13.2.2 Two-Photon Generation and Absorption. 247 13.3 Reservoir in the Lambda Configuration 248 14. Dissipative Cavity QED 251 14.1 Two-Level Atoms in a Single-Mode Cavity 251 14.2 Strong Atom-Field Coupling 252 14.2.1 Single Two-Level Atom. 252 14.3 Response to an External Field 255 14.3.1 Linear Response to a Monochromatic Field 256 14.3.2 Nonlinear Response to a Bichromatic Field 257 14.4 The Micromaser 259 14.4.1 Density Operator of the Field 259 14.4.2 Two-Level Atomic Micromaser 263 14.4.3 Atomic Statistics 266 Appendices 267 A. Some Mathematical Formulae 267 B. Hypergeometric Equation 270 C. Solution of Two-and Three-Dimensional Linear Equations 272 D. Roots of a Polynomial References Index absorption spectrum 182 ac Stark splitting 223 algebra - harmonic oscillator 41 -51/(1,1)43 - SU(2) 42 - SU(m) 44 - SU{m,n) 45 antibunching 132 antinormal ordering 49, 83 Bargmann representation 64 Bell's inequality 35 Bloch-Siegert shift 142 Bloembergen resonances 224 Bom approximation 157 bunching 134 cat paradox 31 cavity QED 151 Chapman-Kolmogorov equation 103 characteristic function 100 coherent multiphoton process 148 coherent states 58 - generalized 57 - Glauber 58,62 - of e.m. field 58, 133 - of harmonic oscillator 62 - of spins 70 - of 5*7(2) 70 - pair 78, 248 - Perelomov 57 coherent states, completeness relation 57 - for harmonic oscillator 62 -for 51/(1,1)78 - for 51/(2) 72 coherent states, minimum uncertainty 59 - of harmonic oscillator 64 - of spins 70, 72 - o f 51/(1,1)77 273 277 283 Index coherent states, uncorrelated equal variance minimum uncertainty 59 - of harmonic oscillator 62 - of spins 70 - o f 51/(1,1)77 collapses and revivals 206 collisional damping 172 complementarity 12,27 cumulants 101 density operator 21 Descarte's rule 277 detailed balance 106, 108, 160, 225 differentiation, parametric 8 - of exponential operator 37 - of operator product 8 disentangling an exponential 48 - harmonic oscillator algebra 49 - algebra 51 - SU(2) algebra 50 down conversion 151,239 dressed states 204 e.m. field - chaotic, classical 127 - chaotic, quantum 133 - coherence time 130 - coherent 127,133 - correlation functions, classical 123 - correlation functions, quantum 130 - quantization 121 effective two-level approximation 212 effective two-level atom 147 eigenvalue 7,186 - generalized 189 eigenvector 7,186 - generalized 189 entangled state 20, 25 EPR Paradox 32 equal variance minimum uncertainty state 13 Fokker-Planck equation 107 four-wave mixing 223, 257 - collision induced resonances 224, 257 - quantum resonances 258 Gaussian process 102 geometric phase 16 - in adiabatic evolution 18 - of a harmonic oscillator 18 - of a two-level system 18 Hamilton-Cayley theorem 190 Heisenberg equation 14 hidden variables theory 34 - local 34 Hilbert space 1 homodyned detection 134 Hurwitz criterion 277 Husimi function 87 incompatibility 12 interaction picture 15 interference 26,27 Jaynes-Cummings model 143, 144, 204 Jordan canonical form 189 Lie algebra 40 Lie group 56 Markov approximation 158 Markov process 103 master equation 104, 105, 158 measurement problem 30 micromaser 259 - trapping condition 264 minimum uncertainty states 12, 59 - of harmonic oscillator 61, 67 - of spins 70 - of 5E7(M) 77 mixed state 22 moments 100 multi time joint probability 15 multi-channel models 149 noise - additive 110 - coloured 109 - delta correlated 109 - Gaussian white 109 - multiplicative 110, 112 - white 109 non-classical states 95 - of e.m. field 95 - ofspin-l/2s 97 normal ordering 49, 83 Omstein-Uhlenbeck process 110-112 P-function 86 - for spins 91 parametric processes 150 phase - dynamic 16 - geometric (see geometric phase) 16 photon 122 Poisson process 115 probability amplitude 11 probability density 11,99 - conditional 103 -joint 99 pure state 22 Q-function 87 - for spins 92 quantum eraser 29 quasiprobability distribution 83 - for spins 89 Rabi frequency 142, 221 random telegraph noise 116 regression theorem 105,163 representations - by eigenvectors 55 - equivalent 56 - labeled by group parameters 56 - of harmonic oscillator algebra 60 - of SU( 1,1) algebra 76 - of SU(2) algebra 68 resonance approximation 144 resonance fluorescence 171,219 - collective 225 rotating wave approximation 143, 181 Rydberg atom 152 s-ordering 83 Schmidt decomposition 25 Schrodinger equation 13 Schwarz inequality 2 - generalized 3 secular approximation 162, 227, 253, 256 semiclassical approximation 139 similarity transformation 39 - harmonic oscillator 41 -5*7(1,1)43 - SU(2) 42 - SU(m) 44 - SU(m,n) 45 Sneddon's formula 37 spectroscopic squeezing 75 spectrum 136 - absorption 223,256 - emission 222 spin operators - collective 69 - lowering 23 - raising 23 squeezed reservoir 166 squeezed states - of harmonic oscillator 67 - of spins 73, 74 squeezed vacuum 166 squeezing operator 65 Stark shift 214 stationary process 100 stochastic differential equation 109 sub-Poissonian distribution 132 superoperator 10 - adjoint 10 superposition, principle of 26 susceptibility 179 - optical 180 symmetric ordering 83 - for spins 94 thermal reservoir 164 three level atom 145 time-ordered exponential integration - harmonic oscillator algebra 52 - SU( 1,1) algebra 53 - SU(2) algebra 53 trace 6 transition probability 103 two-channel Raman-coupled model 150,207 two-level atom 144 two-photon process 146 two-photon reservoir - in ladder configuration 245 - in Lambda configuration 248 uncertainty relation 12 uncorrelated equal variance minimum uncertainty state 13, 62 vacuum field Rabi oscillations 205 vacuum field Rabi splitting 257 vacuum fluctuations 122 wave mixing 149, 181 wave-particle duality 26 welcher weg 28 which path 28 Wiener process 110, 111 Wiener-Khintchine theorem 136 Wigner function 87 - for spins 92 Zeno effect 16 Dedicated to My Inspiration - My Wife Shyama Preface This book is intended to provide a much needed systematic exposition of the mathematical methods of quantum optics, something that is not found in existing books. It is primarily addressed to researchers who are new to the field. The emphasis, therefore, is on a simple and self-contained, yet concise, presentation. It provides a unified view of the concepts and the methods of quantum optics and aims to prepare a reader to handle specific situations. A number of formulae scattered throughout the scientific literature are also brought together in a natural manner. The broad plan of the book is to introduce first the basic physics and mathematical concepts, then to apply them to construct the model hamilto- nians of the atom-field interaction and the master equation for an atom-field system interacting with the environment, and to analyze the equations so obtained. A brief description of the contents of the chapters is as follows. The first chapter introduces the basic postulates of quantum mechanics, brings out their implications and develops the associated operational tech niques. It discusses the measurement problem, the paradoxes of quantum mechanics and the local hidden variables theory, since quantum optics pro vides experimental means of examining these issues. Chapter 2 outlines the algebra of the exponential operator, which plays a prominent role in mathe matical physics. The concept of Lie algebra is introduced and the standard hamiltonians of quantum optics are treated as elements of one or the other finite-dimensional Lie algebra. The question of representations of Lie algebras is addressed in Chap. 3. The notion of coherent states emerges as a continuous representation of a Lie algebra. The concept of quasiprobabilities is developed in Chap. 4. Their usefulness as operational tools and as entities for identify ing purely quantum effects is demonstrated. Chapter 5 presents the essential elements of the theory of stochastic processes. The theory of classical and quantized electromagnetic (e.m.) fields is outlined in Chap. 6. It describes the characterization of the e.m. field in terms of its correlation functions and also their role in identifying the signatures of field quantization. By starting with the hamiltonian for an atom interacting with the e.m. field in the dipole approximation, Chap. 7 describes ways of reducing it to simpler, mathematically tractable forms commensurate with given physical conditions. The standard models of quantum optics are thereby derived. The VIII Preface effects of the environment on an atom-field system are the subject of the quantum theory of damping outlined in Chap. 8. Here the master equation for the evolution of a system in contact with a reservoir is constructed and methods of solving it are discussed. Chapter 9 analyzes the perturbative solution of the master equation of an atomic system in an external field. This leads to the notions of susceptibility, multiwave mixing and the absorption spectrum. The method of solving a set of linear equations with time-independent coefficients in terms of generalized eigenvectors is outlined in Chap. 10. That chapter presents the solution of a two-term recurrence relation and identifies and solves exactly solvable quadratic three-term recurrence relations. These recurrence relations encompass many well-known quantum optical situations. Chapters 11-14 deal with the solution of some standard model systems. Chapter 11 identifies the class of analytically exactly solvable models of an ef fective two-level atom and that of an effective three-level atom in a quantized field. It provides a unified treatment of the exactly solvable hamiltonians of quantum optics. The problem of an externally driven two-level atomic system dissipating into a squeezed reservoir is addressed in Chap. 12. The exactly solvable cases of an arbitrary time-dependent drive are identified. The exact dynamics in a monochromatic drive is investigated and the collective effects in a driven two-level atomic system are highlighted. Chapter 12 also briefly discusses the dynamical behaviour of a three-level atom dissipating into a reservoir at absolute zero temperature and reveals the effects of almost equally spaced pairs of energy levels. The dynamics of a field dissipating into a linear or two-photon non-linear reservoir is the subject of Chap. 13. The evolution of an atomic system in teracting with a single damped quantized cavity mode is investigated in Chap. 14. This chapter also outlines the theory of the micromaser. I am indebted to Girish Agarwal for teaching me the subject of quantum optics. Valuable contributions to my understanding have been gained through my association with Robert (Robin) Bullough, Joseph Eberly, Fritz Haake, Shoukry Hassan, Rajiah Simon, Subhash Chaturvedi, V. Srinivasan, Subha- sish Dattagupta, Surya Tiwari, Dinkar Khandekar and Suresh Lawande. I am grateful to Debabrata Biswas and Aditi Ray for their valuable suggestions and help in preparing the manuscript. I am thankful to Dinesh Sahni for his support and encouragement. Angela Lahee of Springer-Verlag deserves a big thank you for her careful editing. Mumbai, January 2001 Ravinder Puri 1. Basic Quantum Mechanics Quantum optics is the quantum theory of interaction of the electromagnetic field with matter. In this chapter we recapitulate basic concepts and opera tional methods of the quantum theory essential for developing the theory of quantum optics. We delve also in to the controversial issue of interpretation of the quantum theory as a classical statistical theory. Quantum optics provides means for subjecting these conceptually controversial issues to experimental tests. 1.1 Postulates of Quantum Mechanics In this section we state five basic postulates of Quantum Mechanics and discuss some of their important implications. 1.1.1 Postulate 1 An isolated quantum system is described by a vector in a Hilbert space. Two vectors differing only by a multiplying constant represent the same physical state. Following the notation introduced by Dirac [1], we represent a vector by a ket, | ). A Hilbert space is a complex linear vector space equipped with the def inition of a scalar product and spanned by a complete set of vectors [2]. The meaning and implications of these properties of the Hilbert space are explained below. They are crucial for relating the theory with experimental observations. Linear Vector Space. A Hilbert space is a complex linear vector space. We assume familiarity with the notion of a linear vector space over the field of complex numbers (c-numbers) [2], We recall that if \ψι) and ψι ) are vectors in a complex linear vector space then a linear combination ft\ |?/>i) + « 2 1 ^ 2 ) for arbitrary complex numbers αχ, η·ι is also a vector in the same space. A set of vectors \ψ\), · ■ ·, \ψη) is said to be linearly independent if n X J a i |'0 i ) = O, (1.1) i=i 2 1. Basic Quantum Mechanics implies ai = 0 for all i = 1,n. The maximum number of linearly inde pendent vectors in a linear vector space is called its dimension. S c a l a r P r o d u c t. To s ay t h a t t h e Hi l b e r t s pa c e i s a Euclidean or scalar product space means that it is possible to associate with every pair of vectors |φ) and \ψ) in it a complex number, denoted by {φ\ψ), such that 1. {φ\ψ) = {ψ\φ}*, where * denotes the operation of complex conjugation; 2. If \ψ) = + c2\ip2 ) then (φ\ψ) = Ci(0|V>i) + c2( ^#2); 3. (V#> > 0; 4. (ψ\ψ) = 0 if and only if (iff) \φ) = 0. In the following we list some consequences of these axioms. • The scalar product associates with a vector | ) its dual ( | called a bra [1]. • The non-zero positive number |||'0)|| Ξ \/(Ψ\Ψ) is called the norm or the length of the vector. Since two vectors differing only by a multiplication factor represent the same physical state, we can represent a physical state by a vector of a fixed, say unit, norm if the norm is finite. Hence, \ψ) is physically an acceptable vector if its norm is finite i.e. if (ψ\ψ) < oo. (1.2) • The vector \φ){φ\φ) is the projection of a vector |ψ) along the vector \φ). The scalar product {φ\ψ) is a measure of the overlap between the vectors |φ) and |φ). If (φ\ψ) = 0 then |^>) and |φ) are said to be orthogonal to each other. • Two sets of vectors \ψι), · · ·, |φη) and \φι), ■ ■ ■, \φη} are said to be orthonor mal to each other if {Φί\ψ^ = i,j = l,...,n. (1.3) • A set \e\), ■ ■ · \en) of vectors is said to be orthonormal if (e^ \ ej ) Sij, i, j 1,..., n. (1.4) • An important consequence of the axioms defining the scalar product is the Schwarz inequality {.ψ\φ}{·ψ\ψ) > (φ\ψ)(ψ\φ), (1.5) where the equality holds if and only if the two vectors in question are linearly dependent i.e. if \ψ)=μ\φ), (1.6) μ being a complex number. In order to establish this, show that the min imum value of (Ψ(μ)\Ψ(μ)), where 1^) = \ψ) — μ\φ), as a function of μ is (ψ\ψ) — \(φ\φ)\2/(φ\φ). The requirement that this value, due to axiom 3 of the scalar product, be positive leads to the Schwarz inequality in (1.5). Also, according to the axiom 4 above, (Ψ(μ)\Φ(μ)) = 0 iff \Ψ(μ)) = 0 i.e. 1.1 Postulates of Quantum Mechanics 3 iff (1.6) holds. It may be verified easily that (1.5) then holds with equality. In a similar way we can derive the generalized Schwarz inequality where det((^|^>„)) is the determinant of the matrix constituted by the elements ('ψμΐ'ψν), μ, v = 1,n. Invoking the fact that the determinant of a matrix is zero if its rows (or columns) are linearly dependent, it follows that the equality in (1.7) holds iff \ψμ) are linearly dependent. Completeness. In a scalar product vector space of finite dimension n, there always exists a set of n linearly independent vectors (I'i/’i)}, called the basis vectors, such that any vector \ψ) can be expressed as a linear combination [2], The complex numbers {d*} in a scalar product space may be determined by taking the scalar product of (1.8) with the vectors {\φί}} orthonormal to {|·0ΐ)} to give di = (φί\ψ) so that The vector |tp) in an n-dimensional space is thus characterized by n complex numbers {{φί\ψ}}· The column of these numbers constitutes a representation of the vector in the given basis. The dual (ψ\ of |ψ) is then represented by the row constituted by the numbers {{φ\φί)} = {{Φί\Ψ}*}· Thus the representa tion of {ψ\ is obtained by the process of hermitian conjugation (interchanging of rows and columns along with the operation of complex conjugation), de- The expansion (1.9) in a scalar product space is guaranteed if the space is finite-dimensional. However, such an expansion need not exist if the space is infinite-dimensional. In quantum mechanics, we are concerned only with those scalar product linear vector spaces in which every vector is expressible in terms of a basis. Such a space is called a Hilbert space. Now, on invoking the fact that (1.9) is to be valid for an arbitrary \ψ), follows the completeness relation i=l wher e I is the identity operator defined below. If is orthonormal, i.e. if {Iφί)} = {|·0ί)} = (|ei)}, then (1.10) reduces to det((Vv|Vv)) > 0, (1.7) n ( 1.8) i-1 n (1.9) noted by f, of \ψ): (ψ\ = (l^))1. n ( 1.10) i—1 (1.11) 4 1. Basic Quantum Mechanics In our discussion so far we have assumed that the basis vectors are de- numerable. There are, however, occasions which require us to work with a basis labeled by a continuous parameter. Consider an orthonormal set of ba sis vectors |£) labeled by a real continuous parameter ξ. The condition of orthonormality then reads < m = * ( £ - o. (i-i2) δ(χ) being the Dirac delta function. If a < ξ < b then the analog of the expansion of a vector in terms of the basis vectors is \Ψ) = f m 10 άξ· (1-13) J a A vector |φ) in a continuous basis is thus represented by the function c(£) = {ζ\φ) of a real variable ξ. O p e r a t o r s. The action of a force t ransforms t he s t at e of a system. A t r ans formation of a s t a t e of a system may be described by a rule, called an operator, that associates with a vector in the space another vector in the same space. If, for example, an action transforms \φ) to |φ) then we write Α\φ) = \φ) (1.14) where the operator A defines the rule of transformation. We distinguish an operator from a c-number variable by a caret on the former. An operator A is linear if, for any complex numbers ci and c2, Α(θι\φ) + c2|0)) = (θιΑ\φ) + ο2Α\φ) ^. (1.15) We shall be concerned only with linear operators. If Α\φ) = | φ) for all | φ) then A is called the unit or identity operator, often denoted by I. Since I acts like the scalar unity, we do not dress it with a caret and even denote it by 1. In order to obtain a c-number representation of an operator A, consider an orthonormal basis {|ej)}. Rewrite A as I A I where I is the unit operator and express I in terms the completeness relation (1.11) to get n a = Σ (e*i^iej) ie*)<eji· (L i 6 ) *>.7 = 1 The operator A may be represented by an η x n matrix constituted by the complex numbers (ei\A\ej) (i,j = 1 ,...,n ). On operating (1.16) on an ar bitrary vector |φ), it follows that Α\φ) is represented by the product of the matrix {ei\A\ej) representing A with the column {ej\φ) representing \φ). It is straightforward also to show that a product AB is represented by the prod uct of the matrices representing them. Thus, the correspondence between vectors as columns and operators as matrices is not only notational but also operational. 1.1 Postulates of Quantum Mechanics 5 The analog of (1.16) in the continuous representation is evidently The function α(ξ, ξ') = (ξ|-Α|ξ') of two real variables ξ and ξ' now serves as a representation of A. The rules of addition and multiplication of operators and those for the action of an operator on a vector are same as the corresponding ones for the discrete case with summations replaced by integrals. Next, we enumerate some definitions and algebraic operations involving linear operators. The treatment, though may lack rigor at times, is adequate for our purpose. For details, see, e.g., [3]. 1. The product AB denotes the action of B on a state followed by that of A on the resulting state. This result need not be the same as that due to the operation BA. The operator defined by The problem of disentangling the exponential of a sum of non-commuting operators is addressed in Chap. 2. 4. Check that we can write B{AB) m = ( BA) B( AB )m~1 = · · · = ( 'BA) mB. As a consequence of this it follows that if F( AB) is a function expandable as a power series of its argument then [A, b ] = A b - b A (1.18) is called the commutator of A and B. If \A, B\ = 0 then A and B are said to commute. 2. I f m is a positive integer then Am denotes A multiplied with itself m times. 3. A function F(A) of an operator A may be defined by its expansion in terms of the powers of A. Of particular interest is the exponential operator defined by the expansion (1.19) m —0 Using this definition, the reader should verify that if [A, B] = 0 then exp[A + B] = exp(A) exp (B) = exp (B) exp(A). (1. ( 1.20) b f (Ab ) = f (b A) b. ( 1.21) 5. If there exists a nonnegative real number CV<V#> f°r all I1/’) then A is called a bounded operator. The minimum of the numbers C, denoted by JJA||, is called the norm of A. 6. The adjoint of A, denoted by A^, is defined by the relation (φ\Α\·ψ) = (·ψ\Α*\φ)*. (1.22) 6 1. Basic Quantum Mechanics for all |i/j) and |φ) in the given Hilbert space. Combine this with (1.16) to show that the matrix representing the adjoint of an operator is the ' adjoint of the matrix representing that operator. Verify also that ( i B C ) t (1.23) 7. If, corresponding to an operator A, there exists an operator B such that BA = AB = I then B is called the inverse of A. The inverse of A is denoted by A _1: A~1A = AA~1=I. (1.24) An operator is called singular if it does not admit an inverse. It can be shown that AB = I implies BA = I in a finite dimensional Hilbert space but not if the space is infinite dimensional [3]. In an infinite dimensional space, an operator A may be singular but corresponding to it there may still exist an operator A called the left inverse of A such that A ^ 1 A = I or an operator A~^ called its right inverse such that A A ^ = I. Clearly, an operator which is the right as well the left inverse of A is the inverse A ^ 1. It is straightforward to show that (ABC)-1 = 0 - ιΒ~1Α~1, (1.25) provided that A, B. and C are non-singular. Since an operator commutes with itself, it follows from (1.20) that the inverse of exp(A) is exp(—A). Notice also that [ e x p ( i i ) -.- e x p ( i m) = e x p ( - i m) · · · e x p ( - i i ). (1.26) 8. An operator H is called hermitian if H = W . 9. An o p e r a t o r U is called unitary if UW = WU = I. This shows that (V’ilV’j) = {Ψί\ΰ^ΰ\Ψ^)· Hence the set of states {U\ipi)} obtained by a unitary transformation of the set {| )} preserves the scalar product. Hence, the set of vectors obtained as a result of unitary transformation of an orthonormal set of vectors is also orthonormal. If the orthonormal set is complete then so is the set obtained by its unitary transformation. Verify that U can be represented as U = exp (i/7) where H is hermitian. 10. If an operator A commutes with its adjoint i.e. if [A, A^\ = 0 then it is called a normal operator. Note that hermitian and unitary operators are examples of a normal operator. 11. If (φΐΑΐφ) > 0 for all \ψ) then A is said to be positive. Let |φ) = Β\ψ). Then (·φ\Β^Β\φ) = (φ\φ) > 0. Hence B^ B > 0 for all B. 12. T h e s um of t h e di agonal e l eme nt s of a ma t r i x r e pr e s e nt i ng a n o p e r a t o r A is called the trace of A. It is often denoted by Tr(/1). If {|e*)} is an orthonormal basis then, by definition, Tr(A) = (1.27) i =1 1.1 Postulates of Quantum Mechanics 7 The value of the trace of an operator is independent of the basis. Some consequences of the definition of trace are: • The complex conjugate of (1-27), read with (1.22), shows that i= 1 • Verify that the trace of a product of operators possesses the cyclic property: Ti (ABC) =T i (CBA). _ _ (1.31) • If JJ is unitary then, due to (1.31), Tr[i/tΑϋ) = Ττ[ϋϋ^Α] = Tr[/i]. This shows that the trace of A and that of U^AU are equal. 13. If \ψ) is such that where Λ is a constant then \ψ) is called an eigenvector or an eigenstate and Λ the corresponding eigenvalue of A. Expand F{A) in powers of A and use (1-32) repeatedly to show that The problem of solving an eigenvalue equation is addressed in Chap. 10. We recall from that chapter that • The eigenvalues of a hermitian operator are real. • Any normal operator in an n-dimensional space possesses n eigenvec- n n Α\ψ) = Χ\ψ) (1.32) (1.33) tors which are orthonormal. Hence, if A is a normal operator and {|α*)} the set of its orthonormal eigenvectors then {aj\A\aj) = On com bining this with (1.16) follows the expansion n of a normal operator in its eigenbasis. Apply (1.33) to show that n Now, let |a, b) be a simultaneous eigenvector of A and B such that A\a,b) = a\a,b), B\a,b) = b\a,b). (1.36) Operate first (second) of the equations above by B (A) and subtract the resulting equations to obtain [A, B]\a,b}=0. ( 1.37) 8 1. Basic Quantum Mechanics This equation is trivially solved if [A, B] = 0 showing that non commuting operators may possess common eigenvectors. If A and B do n o t commut e, t h e n no gene r al conc l us i on c a n be dr a wn a b o u t t h e s ol va bi l i t y of ( 1.37) exc ept f or s pec i a l cases. For exa mpl e, i f [A, B] is a non-zero constant then (1.37) evidently does not admit a non-trivial solution. This is the case for the pair of position and momentum op erators q and p. Hence, q and p do not have any common eigenvector. Same result can be shown to hold for angular momentum operators. It is generally accepted that non-commuting observables of common inter est do not possess common eigenvectors. However, it can be proved that if A and B do not commute then they do not admit a common set of eigenvectors [3]. 14. If a state \ψ(ί)} or an operator A(t) is a function of a scalar t then its derivative with respect to t has the usual meaning of the calculus of c- number functions. The rules for differentiation of a product of states or of operators with respect to a parameter are also same as for the c-number functions provided that in applying those rules the order of states and operators is retained. Thus, for example, ^\ψ(ί)){Φ(ή\ = (Φ(ί) I + ^(Φ(ί)\ψ(ί)} = 1^(0) + (Φ(ι) I^IV’W); i 1·38) i t A l'"An = ( i t Al)'"An + '" + A l""i t An· (1'39) Hence, verify that 15. Consider the differential equation (1.41) Its solution may be written in the form where the time-ordered exponential integration is defined by (1.42) + / dτη / drn_i · · · / driA(rn) · · ■ A ( n ) H . (1-43) Jto Jto Jto 1.1 Postulates of Quantum Mechanics Here *T is the so called time-ordering operator. It arranges operators in a chronological order with time increasing from right to left. Verify by term-by-term differentiation of (1-43) that rt . rt d d t T exp Γ f άτΑ(τ) = Α ( ί γ τ exp f άτΑ(τ) Jt ο Jto (1.44) On using t hi s it follows t h a t (1-42) indeed satisfies (1.41). In t he following we list some properties of t he time-ordered exponential operation. • Take t he her mi t i an conjugate of (1-43) t o show t h a t j ^ Te x p f άτΑ{τ) | = Ί*exp f dr^4^(r) . (1-45) *- °-+ 0 The operator T in the equation above arranges operator in a chrono logical order with time increasing from left to right. Verify that T* exp f άτΑ(τ ) = T* exp ί f άτΑ{τ) A(t). (1-46) d t L JtQ J L Jto J • If |^4(ij), A(^)j = 0 for all U,tj then the operators under the integral in (1.43) may be shuffled at will like the c-numbers. This property leads to the relation rt f t pTn pT 2 / dτη drn_ i''' / d r i i ( T n) · · · A( n) J ίο J to J ίο 1 n! f ά τ Α ( τ) '-Jto Substitution of this in (1.43) yields ^ e: f dr^4(r)l = exp f dt A( t ) Jto Jto I n p a r t i c u l a r, i f A is independent of t then exp f dr^4| = exp[.4(i — i0)]· LJt0 ( 1.47) ( 1.48) ( 1.49) I f A(t) does not commute at different times then the commutator of A(ti) and A(t j ) would contribute to the integral if A(ti) and A(t j) are interchanged. Hence, on shuffling the operators as in (1.47), we can express the nth term in the time-ordered integral in the form /*ί /*Tn /*T2 / dτη drn_i ·' · / dri^4(rn) ■ J10 Jto Jto ■ M n ) 1_ n\ Jt dr^4(r)j Cn(t), (1.50) the Cn(t) being the contribution from commutators of A(t) at defferent times. This reduces to (1.47) if the commutator of A(t) at different times vanishes. If A is time-independent but B(t ) a function of time then 10 1. Basic Quantum Mechanics = exp[yl(t — ί0)] Ϋ exp J dt B ( t ) , (1.51) - B(t) = exp[—A(t — t0)]B exp[A(t — t 0)]. (1.52) The relation (1.51) may be established by showing that the terms on its two sides have the same derivative with respect to t. The problem of evaluating a time-ordered exponential integral is ad dressed in Chap. 2. Superoperators. We have so far considered the operation of linear trans formation of vectors in a Hilbert space. Another important class of operations consists of transformation of operators. A superoperator defines a rule that as sociates an operator with another operator acting in the same Hilbert space. The operation of transformation of an operator / to another operator g may be expressed as I f = g. (1.53) Here the superoperator L, distinguished from operators by a double caret, defines the rule of the transformation. We restrict our attention to linear superoperators acting on linear operators in a given Hilbert space. Recall from the theory of vector spaces that linear operators acting in a vector space constitute a vector space. The relation (1.53) may then be viewed as defining a transformation in a vector space and a superoperator may be identified as an operator in the vector space of the operators. Furthermore, the vector space of operators may be made a scalar product space by defining a suitable scalar product. A useful definition of scalar product is ( i, S ) = Tr & B . (1.54) It may be verified that this definition is in accordance with the axioms of the scalar product. In analogy with operators, the definition (1-54) of the scalar i t i product leads to the following definition of the adjoint L of L: T r [ p L g ] = T I [ g l i f y. (1.55) In order to obtain a c-number representation of superoperators, let {|e*)} be a complete set of orthonormal vectors spanning an TV-dimensional Hilbert space. The transformation (1.53) in this basis may be written as N gij = ^ (1.56) k,l =1 The superoperator L is thus represented by a tensor {Lij ki} acting on a matrix. If N x N matrices {fi j } and {gij} are represented as column Vec tors having N 2 elements then Lij ^i is represented as an N 2 x N 2 matrix. This provides a means of converting equations involving superoperators in to matrix equations. 1.1 Postulates of Quantum Mechanics 11 To each dynamical variable there corresponds a unique hermitian operator. The reason for associating a hermitian operator with a dynamical variable will become clear after the statement of the postulate 4. 1.1.2 Postulate 2 1.1.3 Postulate 3 If A and B are hermitian operators corresponding to classical dynamical vari ables a and b then the commutator of A and B is given by [A, B] = AB — BA = ih{a, b}, (1.57) where {a, b} is the classical Poisson bracket of a and b and h = h/2π where h is the Planck’s constant. See [1] for the rationale behind this postulate. 1.1.4 Postulate 4 Each act of measurement of an observable A of a system in state \ψ) collapses the system to an eigenstate \ai) of A with probability \(α,ί\ψ)\2. The average or the expectation value of A is given by (i) = Σ α·ί\{α·ί\Ψ)\2 = (^|^|^), (1-58) the ai being the eigenvalue of A corresponding to the eigenstate \ αΐ). The complex number (αι\φ) is called the probability amplitude. The last equality in (1.58) can be derived by (i) rewriting |(a;|^ )| 2 as (φ\αΐ) (a^ip), and (ii) by invoking (1.34) to write the summation over i of ai\a,i)(ai\ as A. An operationally useful way of evaluating t he probability | (α^|·0) | 2 is t o use t he expression (10.11) t o write \ai) (^1 = [Π?# (a, ~ α*)] ~ΧΠ^ [a, - i). (1.59) This assumes that αχ,... ,an are distinct. As a consequence, the probability of observing the eigenvalue a; as an outcome of a measurement of A on |ip) is given by \(αί\Ψ)\2 = [n&i fa -a*)]”1 (^| n^i (aj ~ Α)\φ) . (1.60) In the discussion above, it is tacitly assumed that the eigenvalues axe denu- merable. Let the eigenvalues ξ of the operator associated with the observable be continuous with |£) as the corresponding eigenvectors. Then, | (ξ|'(/>') 12 is identified as the probability density so that |{ξ|'(/;)|2 d£ is the probability that the act of measurement results in a value in the range d£ around ξ. Invoke (A.l) to show that i f °° KilV’)!2 = W(£ - Α)\Ψ) = 2 - J dx(?l>\ exp[i(£ - Α)χ]\ψ). (1.61) 12 1. Basic Quantum Mechanics Since, according to the postulate 2, the observables are represented by linear hermitian operators whose eigenvalues are necessarily real, it follows that the act of measurement would give a real number as its result as any act of measurement does. This also explains the rationale behind associating a hermitian operator with an observable. In the following we list some implications of this postulate. • As a consequence of this postulate, quantum theory predicts only the av erage of the results of measurements on a large number of identically pre pared systems. The results of all such measurements are identical only if the observed state is an eigenstate of the measured observable. Only in that case does the observable possess a definite value, that value being the corresponding eigenvalue. Also, as mentioned before, non-commuting observables do not admit common eigenvectors. Hence, non-commuting ob servables can not have definite values simultaneously. Simultaneous mea surement of non-commuting observables to an arbitrary degree of accuracy is thus incompatible. Non-commuting observables are complementary in the sense that precise knowledge of one excludes that of the other. • Since quantum predictions are probabilistic, it is important to know the extent of the spread in the outcomes of measurements. A measure of the spread in the values of the results of measurements of an observables A of a system in state |ip) is the variance defined by Δ A2 = (ip\[A - (A)]2\ip) = (ip\A2\ip) - (ip\A\ip)2. (1.62) In order to determine the relationship between variances of two observables A and B due to measurements on a system in state |ip), let \ip1) = [A-{A)]\iP), \iP2) = [B-(B)]\iP). (1.63) Invoke the Schwarz inequality (1.5) to arrive at the relation (F)2 + (C)2] (1.64) called the uncertainty relation. Here [A, B\ = iC, F = Ab + BA-2{A){B). (1.65) Note that, as a result of assumed hermiticity of A and B, the operators C and F are also hermitian. The operator F is a measure of correlations between A and B. The uncertainty product is minimum i.e. the equality in (1.64) holds iff, following (1.6), | ^ i ) = — ϊλ | -02), where Λ is a complex number, i.e. iff [A + i\B]\iP) = [(A) + iA<B>] \ip) = z\iP). (1.66) The state | ip) satisfying (1.66) is called a minimum uncertainty state. We may deri ve usef ul general resul t s about t he sol vabi l i t y o f ( 1.66) and t he expressi ons for ΔΑ2 and Δ Β 2 in the minimum uncertainty state. To δ Α2δ β 2 > ] 4 1.1 Postulates of Quantum Mechanics 13 that end, verify that A + i\B is a normal operator if Re(A) = 0. Recall that the eigenstates of only a normal operator are orthonormal. Hence it follows that the minimum uncertainty states (1.66) for a given A are non-orthogonal if Re(A) φ 0. Next, rewrite (1.66) in the form {Α-{Α)]\φ) = -ίΧ[Β-{Β)]\ψ). (1.67) Operate this on the left with A — (A) and take the scalar product of the resulting vector with \ip). Repeat this procedure by operating (1.67) now with B — (B ). On using (1.65), the two equations so obtained read AA2 = ~ [ { F ) + i ( C ) }, A B 2 = ^ [ { F ) - i { C ) ]. (1.68) Set λ = Ar + iAi. Compare the real and imaginary parts of each of the equations above to get δ Α 2 = - 2 \i(F) + \ t(C) Xi{C)-XT(F)=0. (1.69) These relations imply that 1. If | A| = 1 then Λ A2 = A B 2 . The corresponding states may then be referred to as equal variance minimum uncertainty states . 2. I f | A| = 1 al ong wi t h A; = 0 t h e n Δ A2 = A B 2 and (F ) = 0. Since F i s a me a s u r e of c or r e l a t i ons bet ween A and B, the corresponding states may be referred to as uncorrelated equal variance minimum uncertainty states. 3. I f λ Γ φ 0 then (F) = ±(C), ΔΑ2 = ^ φ ), Δ B2 = ±-(C). (1.70) The variances and the correlations in the measurement of two observables in this case is expressible in terms of the average of their commutator C. Clearly, those quantities are completely determined without an explicit construction of the said state if C is a constant multiple of the identity operator. Also, since A A 2,A B 2 are positive, it follows that Ar should have the same sign as (C). Hence, if C is a positive operator then admis sible solutions of (1.66) are obtained only if Ar > 0. As an example, recall that C = h i > 0 for the pair (q,p). Hence, the minimum uncertainty states for the pair (q,p) exist only if Ar > 0. This is borne out also by an explicit solution of (1.66) carried in Sect. 3.2. 1.1.5 Postulate 5 The time evolution of a state |ip) is governed by the Schrddinger equation . d l dt ( i.7 i ) where H(t) is the Hamiltonian which is a hermitian operator associated with the total energy of the system. This postulate provides the means for determining the dynamical evolu tion of a system. The Schrodinger equation (1.71) is of the form (1.41). Its solution is, therefore, given by \ip(t)) = Ψ exp £ dτ Η( τ ) ^ |^(0)) ξ Us (t, t 0)\ip(t0)). (1.72) Invoke (1.45) to show that (ip(t )| = (^(0)| 2* exp ^ άτ Η( τ ) ^ = (ψ(0)\ϋ1(ί, t0). (1.73) If H is time-independent then, on using (1.49), (1-72) reads |xp(t)) = exp i(i - t 0)H/h}j \ip(t0)). (1.74) Expand |^(io)) in the basis of the eigenstates l ^ ) of H and apply (1.33) to reduce (1.74) to n = '^2^v{-'lE i ( t - t 0)t/K){Ei\%jj(t0))\Ei). (1.75) i—l Now, the quantities of physical interest are the expectation values of opera tors. Using (1.72), the expectation value (^(i)|^4|^(i)) of A at time t may be expressed as {■ψ(ή\Α\ψ(ή) = {i(j{t0)\A(t)\ip(t0)), (1.76) where A(t) = tfs ( t,t o ^ s {t,to), (1.77) now carries the time dependence. It is straightforward to see that A(t) evolves according to the Heisenberg equation i f t ^ i ( i ) = A, H(t ) . (1.78) The time evolution of the expectation value of an operator can thus be pic tured either in the framework of the time evolution of the state, called the Schrodinger picture or in that of the operator, called the Heisenberg picture. Ne xt, c ons i de r a s ys t e m des c r i be d by \ip{t)) evolving under the action of a hamiltonian H decomposable as H = H0 + Hi(t) (1.79) where Ho is time-independent. Define |ψι(ί)) = exp (i H0t/h ) |ip(t)). (1.80) It is then straightforward to see that \ipi(t)) evolves according to 14 1. Basic Quantum Mechanics 1.1 Postulates of Quantum Mechanics 15 ihd = (1.81) Hi(t ) = exp ^iH0t/hj Hi (t )exp iH0t/hJ . (1.82) Th e e v o l u t i o n ( 1.81) i s s a i d t o b e i n t h e interaction picture generated by Ho- In many situations, it is required to know the multi time joint probability p({|0i), ti}) that a system in a state |0o(io)) at to is found in the state \φΐ) at t = ti (i = 1,..., n). To find that probability, note from (1.72) that the state of the system at time ii is given by Us(ti, ίο)|0ο(ίο)) and its projection on \φι) is |0i(ii)) = |0i)(0i|i7s(ii,io)|0o(io))· The state |0i(ii)) then evolves till time i 2 to Us(t 2,t\) ^ i ( t i ) ) whose projection along |02) is |<?i>2 (^2 )) = 102) (021 (^27 ί ι ) | 0 ι ( ί ι ) ) · On continuing this argument till time tn it follows that (1.83) Consider now a time-independent hamiltonian so that, by virtue of (1.74), Us(ti, t j ) = exp(—iH( t i —t j)/h). Also, let the observations be spaced at equal time intervals ti — ti - 1 = t/n. The probability that at each time ti the system is observed in its initial state |0o) then reads Ρ({\Φο), ti}) = (0o| exp ^ — i Ht/nf i j \ φ0) ( 1.84) Let t/n < 1. Expand the exponential in (1.84) and retain terms up to t2 to arrive at (0oj exp ( — i Ht/nhj |0O) « 1 - Δ Η 2, ( 1.85) wher e Δ Η 2 = (φ0\Η2\φ0) — (φο\Η\φο)2 ■ The joint probability that the sys tem is observed in its initial state at the time of each of n equally spaced observations on it in time t is then given, on substituting (1.85) in (1.84), by (4] P({l0o),*i}) = 1 f - V \hnJ Δ H2 ( 1.86) Compare this with the probability p(|0o),i) that the system is found in the initial state at time t if it is left unobserved in between. That probability is evidently (1.86) corresponding to n = 1, i.e. p(|0o), t) = 1 - t 2A H 2/h 2 (1.87) This shows that the probability of finding the system in its initial state at a given time is increased if it is observed repeatedly at intermediate times com pared with that probability when the system is left unobserved during its evo lution to that time. In fact, for n » 1, (1.86) approaches exp(—t 2A H 2/n h 2). Hence, the probability of finding the system in its initial state tends to unity 16 1. Basic Quantum Mechanics as the number n of observations tends to infinity. In other words, a system continuously under observation does not evolve! This is the quantum Zeno effect or the watchdog effect. This effect was invoked to predict the inhibi tion of decay of an unstable system [5]. An experimental demonstration of this effect in the context of quantum optics has been reported in [6]. For a discussion of various interpretations of the experimental results, see [7]. 1.2 Geometric Phase Yet another characteristic which exhibits distinctly quantum nature of evo lution is the phase of a state. The observable effects of phase obviously can not be exhibited by the expectation value (·ψ\Α\ψ) of an observable A as it is independent of the phase of \ψ). Effects of phase may be manifested, as we will see in Sect. 1.7, in interference between two non-orthogonal states. Now, consider a state |^(i)) evolving under the action of a Hamiltonian H(t ) according to the Schrodinger equation. The state is assumed to be normalized to unity. The phase difference between the states at two times t\ and t 2, called the total phase, is given by <t>t = arg(^(ii)|^(*2))· (1.88) The phase is defined modulo 2π. It turns out to be useful to introduce another phase, called dynamic phase , defined by Ψά = ~ ^ £ (ψ(τ)\Η(τ)\ψ(τ)) dr = Im {ψ(τ)\ψ(τ))άτ. (1.89) The last equality above is the result of the assumption that |ip(t)) evolves according to the Schrodinger equation and that {ψ\Η\ψ) is real. If H is time- independent, and if the initial state is an eigenstate of H with eigenvalue En then, invoke (1-72) to show that φ% = Φά = En(t x — t 2)/h i.e. the dynamic phase in this case is the same as the total phase. In general, the difference Φ& = Φι~Φά= arg('0(i1)|,0(i2)) - Im f άτ(ψ(τ)\ψ(τ)) (1.90) Jti is called, for the reason elaborated below, the geometric phase. We note first that the definition (1.90) holds even for the states which do not satisfy the Schrodinger equation. In order to see that, consider the Gauge transformation 14>(t)) = exp(i f (t))\4>(t)) (1.91) where f ( t ) is a real smooth function of t. On substituting this for |ψ(ί)) in (1.90) we obtain άτ{ψ(τ)\ψ(τ)). (1.92) Φ& = arg(^(ii)|^(i2)) - Im j 1.2 Geometric Phase 17 This shows that φκ remains invariant under the transformation (1.91). Note that, if |ψ) satisfies the Schrodinger equation, |φ) need not. Now, the trans formation (1.91) defines an equivalence class of vectors. Let P be the space obtained by projecting to the same point the states related by (1.91). With the passage of time, any state |^(i)) traverses a curve C under the action of H. Let Co be the image of that curve in the projective space P. Let C be another trajectory traversed by the state vector under the action of H'. If C' projects on to the same Co in the projective space then its geometric phase is clearly the same as that for the motion on C. It can also be verified that 0g is reparameterization invariant, i.e. it is unchanged under the transformation t —» t' where t' is a smooth function of t [8]. The phase 0g is thus a geometric property of unparameterized Co in the projective space. The freedom in the choice of f (t ) may be used to cast 0g in different instructive forms. To that end, use (1.91) in the first term on the right hand side of (1.92) so that </>g = arg(^(i1) | ^ ( i 2)) + f ( t 2) ~ f {t i ) ~ Im f άτ(ψ(τ)\ψ(τ)). (1.93) Jti If f i t ) is chosen such that f ( h ) — f { t 2) = ar g ( ( ^ ( ii ) |^ ( i 2)) then the geometric phase is the same as the dynamic phase: </>g = - I m f άτ(ψ(τ)\ψ(τ)) = - φ ά· (1.94) Jti Alternatively, use (1.91) in the second term on the right hand side of (1.92) to get Φζ = a r g ( ^ ( i i ) |^ ( i 2)) + f { h ) ~ f ( t 2) - I m ί άτ(·ψ(τ)\ψ(τ)). (1.95) Jti If f (t ) is such that f(ti) - ffo) = Im ί άτ{ψ(τ)\ψ(τ)) (1.96) J t j then the geometric phase is the same as the total phase: </>g = ar g ( ^ ( i i ) | ^ ( i 2)). (1.97) The geometric phase signifies interesting differential-geometric properties of evolution. For a detailed discussion of its differential geometric interpretation, its generalization to non-hamiltonian evolution and references to experiments on its observation, see [8, 9], For the sake of illustration, we evaluate next the geometric phase for the evolution generated by the hamiltonian of a harmonic oscillator and that of a two-level system. 18 1. Basic Quantum Mechanics 1.2.1 Geometric Phase of a Harmonic Oscillator Recall that the eigenstates of the hamiltonian of a harmonic oscillator of fre quency ω are |n) with En = (n + 1/2)Τιω, (n = 0,1,...) as the corresponding eigenvalues. On invoking (1.75), its state at time t is given by °c , , v I^W) = Σ ^ θχρ f - iωί ( πι + J \m). (1.98) Consider the evolution over the period 2π/ω. On substituting (1.98) in (1.90) and on using the orthonormality of the states {|ίττ.)} follows the well-known result OO φΕ = 2π m\Cm\2 = 2π{πι). (1.99) m—0 1.2.2 Geometric Phase of a Two-Level System Consider a system having two states |±) in an external field. The state of such a system at any time is a linear combination of |±) and hence is expressible as |ip(t)) = c o s ( ^ ^ j | + ) + e x p ( i </> ( i ) ) s i n ^ ^ 0 | - ) ( 1.100) where the functional form of θ(ί),φ(ί) is unimportant for the present. For the sake of simplicity, assume that Θ is independent of time. On substituting (1.100) in (1.90) and after a little algebra it may be shown that [8] tan cos (Θ) tan ( ( 1.101) with Αφ = φ(ί 2 ) — φ(ί ι). 1.2.3 Geometric Phase in Adiabatic Evolution The recent surge in interest in the geometric phase owes its origin to the paper by Berry on the geometric phase in adiabatic evolution [10]. Consider a system evolving under the action of a time-dependent hamiltonian H(t ). Let the system be initially in an eigenstate |n(0)) of H( 0). The assumption of adiabadicity means that the state of the system at any time t is \Ψ(ί)) expB i drE”(T) !"(!)) (1-102) where \n(t)) is the eigenstate of H(t ) and En(t) the corresponding eigenvalue. Assume also that after a time T the hamiltonian returns to its form at t = 0. On substituting (1.102) in (1.90) with ti = 0, t 2 = T and |n(0)) = |n(t)} it follows that 1.3 Time-Dependent Approximation Method 19 = —Im f (n(t)\n(t))dt. (1.103) Jo Now, if it is assumed that the time-dependence of the hamiltonian, and con sequently of the states \n(t)), is due to that of the parameters {/?,} then (1.103) reduces to the form 4") = - I m Γ\n(t )\V R n ( R m A t ) (1-104) Jo which is familiar since its introduction in [10]. 1.3 Time-Dependent Approximation Method We have seen that the problem of studying the time-evolution of a quantum hamiltonian system reduces to solving the Schrodinger equation. However, more often than not, the Schrodinger equation is not exactly solvable. That necessitates use of approximation methods for its solution. The approxima tion methods are many a times useful to unveil the salient features of even an exactly solvable problem. The approximation method to be employed de pends, of course, on the nature of the problem. Here we outline a method of frequent use in quantum optics. Consider a system whose hamiltonian is expressible as in (1-79) where Ho is time-independent whereas H\ (t) is time-dependent. Assume that no part of Hi (t ) commutes with Ho. In the interaction picture generated by H0, the system is described by the state vector |^/(i)) related to the state vector j^(i)) in the Schrodinger picture by (1.80). The formal solution of (1.81) governing its evolution is \ipi(t)) = i7/(i)|^(0)), f//(i) = exp J άτ Ηι ( τ )/Η^ , (1.105) where Hj (t ) is defined in (1.82). Let Ho and H\{t) be such that N Hi(t ) = h'^ 2 ^ Fk exp (—ii?fc t ) + exp (iJ?fc i ) J . (1.106) k = 1 Subst i t ut e t hi s in Ui(t) of (1.105). Expand Uj(t) as in (1.43) and express its 7ith term as in (1.50). It follows that if ||Ffc||/rtfc « 1 (1.107) then the time-ordered expansion of Ui(t) is a perturbative expansion in the smallness parameter ||Ffc||/i?fc. It may be terminated at a desired order. A more instructive form of perturbation expansion is obtained by sep arating the contribution from commutators of Hi(t ) at different times. To that end, (i) express the time-ordered expansion of {//(<) in the form 20 1. Basic Quantum Mechanics Uj(t) = 1 + x = exp(ln(l + £)), (ii) expand (1 + x) in powers of x, (iii) group together the terms having the same number of Hj (t )'s to get Ui(t ) = exp Lfc=i (1.108) M2(t) = ( - - s i “ τ"'( τ ) · o’jM dTlJff/(T2)Jf f/( r 1) - - M x2(i) (1.109) and so on. Comparison of (1.108) and (1.50) shows that if Hj (t ) commutes at any two times then Mk(t) = 0 for k > 2. Hence, Mfc(i) for k > 2 contain contribution from the commutators of Hj{t) at different times. 1.4 Quantum Mechanics of a Composite System The state vector provides a quantum theoretic description of an isolated system. At times we need to describe the state of a system in terms of its constituents known as its subsystems. Here we outline the approach for the quantum theoretic description of a system in terms of its subsystems. For the sake of simplicity, consider a system made up of two subsystems: A and B. The state vector \Ψa +ii) of the combined system may then be expressed in terms of orthonormal basis vectors (|a*)} and (|&i)} respectively for the subsystems A and B as |Ψα+b) = '^2,OLij\ai,bj), (Xij = {a,i, bj\^A+B)· (1.110) where {jcii,bj)} ξ (|α*)} ® (|&j)} is the direct product of the sets of vectors (Ιαΐ)} and {IM}· |Ψα +β ) be normalized to unity so that Σ M 2 = L (L m ) *i.7 = l Now, if aij = a\A^a^B^ then (1.110) shows that | ΦΑ + Β ) = [ Σ α *( β ) Μ Ξ \Φα )\Φ β ), (1.1 1 2 ) * 3 where \Φα ) (IΦβ )) is the state vector only of the subsystem A (B ). In this case the state vector of the combined system factorizes in to those of its subsystems. The state of a composite system which can not be factorized' in to a product of the states of its subsystems is called an entangled state. The entangled states play a crucial role in understanding purely quantum effects. 1.5 Quantum Mechanics of a Subsystem and Density Operator 21 1.5 Quantum Mechanics of a Subsystem and Density Operator Consider a system composed of subsystems A and B. Let it be that we are interested in the behaviour of only one of the subsystems, say, the subsystem A. That behaviour is determined by the expectation values of the operators { Xa } which act on the states of the subsystem A alone. On using (1.110) for the state vector of the composite system, the expectation value of X a is seen to be given by (Xa) = (Ί/α+β\Χα\Ί/α+β) = ΣΣ«« aij(ak\XA\ai)(bi\bj) k,l i,j = ^ ^ ^kj ^i j \Xa |flj) ~ ^ ^ Cjk (flfc \ X a \ &i) i (1.113) i,j,k i,k cik = Y ^ a ija*kj = c*ki. (1.114) j In writing the first line in (1.113) we have made use of the fact that X a does not act on the vectors representing the subsystem B, and in writing the second line we have invoked the orthonormality of the basis {| )} - Use of (1.111) in (1.114) yields Σ i= 1 Cii = 1. (1.115) Now, on invoking (1.30), (1.113) may be written as (Χ α ) = Ύ ϊ\Χ αρ ^\ (1.116) where Pa — Cik\ai)(ak\, (1.117) i,k is called the density operator of the system A. On taking the matrix element of (1.117) it follows that (kk = (ai\pA\ak)· (1.118) The elements {(a,\pA\aj)} constitute a matrix representation of the density operator p a ■ Since it is the expectation values of operators which are the quantities of physical interest and since all such expectation values for a subsystem can be found by using the density operator by means of the relation (1.116), it follows that a density operator describes the state of a system interacting with other systems in the same way as a state vector describes the state of an isolated system. In the following we enumerate some properties of the density operator. 1. The expectation value of an operator X of an isolated system in state \Ψ) may be written, using (1.30), as (^\Χ\Φ) ξ Tr[X|i')(i>’|]. On comparing this with (1.116) we see that the density operator of a system in the state |Φ) is given by p = \Ψ)(Ψ\. If the density operator p of a system is expressible as p = then it is said to be in a pure state. Else it is said to be in a mixed state. 2. Le t Trij denote the operation of trace only over the subsystem B. Check, using (1.29), that 22 1. Basic Quantum Mechanics Tr b \ai,bj)(ak,bi\ =\ai )(ak\TTB \bj)(bi\ = |a*>(afclfy· (1.119) Using this to carry the operation of trace over B in \Ί/α +β )Ψ/α +β\ with Wa +b ) given by (1.110) and on invoking (1.114) it follows that Trf \&Α+ β ){Ί/Α+ β\ = T r B [ PA+B\ = PA- (1.120) If the evolution of the system A is governed by the hamiltonian HA(t) then the kets and bras of A evolve according to (1.72) and (1.73). Hence, the density operator (1.117) of A at time t is given by PA(t) = exp [ dt H a (t) pA{0)3*exp ^ f dt H a {t) n Jto J β Jto .( 1.121) I t i s s t r a i g h t f o r wa r d t o ver i f y t h a t t h e e q u a t i o n of e vol ut i on o f pA (t ) is 4. On combining (1.114) and (1.117) we infer that pA = p\ i.e. p is tian. 5. The operation of trace over (1.117) combined with (1.115) leads to ( 1.122) hermi- M pa ) = Σ C?'?' 1. (1.123) 6. The probability that a system is in state |ψΑ) is given by the expecta tion value of \ψΑ)(ψΑ\ which, by virtue of (1.116), is (ψΑ\ρ\ψΑ). Since measurable probability should be a positive number, it follows that {Ψα Ιρ α ΙΨα ) > 0. Hence pA is a positive operator. 7. As a consequence of (4), we note that the eigenvalues of pa are real. If {I A*)} are the eigenstates of pA corresponding to eigenvalues {A*} then, on applying (1.34), pa can be represented as 1.6 Systems of One and Two Spin-l/2s 23 Since pA > 0, it follows that A* > 0. The equation (1.124) also shows that the trace of Pa is sum of its eigenvalues. This, along with the condition (1.123) imply that A, < 1. If one of the A[s, say, Ai = 1 then A* = 0 for all i Φ 1. Then, pA = |Ai)(Ai| which is the density operator for the system in pure state |Ai). On squaring (1.124) and by using the orthonormality of the states {|Aj)}, it follows that Ρ2α = Σ\Ϊ\\) ( Μ < Ρ α · (1.125) i= 1 The equality in this holds, as discussed above, only if pA describes a pure state. Since Tr[p^] = 1, we note that Tr[p^] = 1 if the state is pure and that Tr[p^] < 1 if the state is mixed. Recall from the list of properties of the trace that if U is unitary then Tr[^4] = Ύτ\ϋAU^) for any A. Hence, if p obeys the inequalty Tr[/52] < 1 then so does UpU^ if U is unitary. A mixed state thus remains mixed and a pure state remains pure under a unitary transformation. Note in particular that the transformation generated by the Schrodinger evolution is unitary. Hence, under the Schrodinger evolution, a mixed state evolves to a mixed state and a pure state to a pure one. Next we discuss the quantum mechanics of one and two two-state systems which are of interest in the discussion to follow. 1.6 Systems of One and Two S p i n - l/2 s Assuming familiarity with the concept of spin in quantum mechanics, we let the vector operator S represent spin in the ordinary three-dimensional space. Its components Sx,S y,S z in three orthogonal directions, say, the directions x,y,z, obey the commutation relation (letting Ti = 1 for convenience) SX, Sy i Sz, SZ, Sx | = i Sy, Sy, Sz = iSx. (1.126) I t woul d t u r n o u t t o be usef ul t o i n t r o d u c e t h e spin raising and lowering operators S± = Sx ± i S y, S+ = S f_. (1.127) On applying (1.126), it is straightforward to verify that S+, = 2Sz, sz, s± \ = ±s±. (1.128) I t i s a s p i n - 1/2 i f me a s u r e me n t on any of i t s c ompone nt s yi el ds one of t h e t wo val ues, ± 1/2. I n a d d i t i o n t o t h e c o mmu t a t i o n r e l a t i o n s ( 1.126), t h e o r t h o g o n a l c o mp o n e n t s of a s p i n - 1/2 obe y al s o t h e a n t i c o mmu t a t i o n r e l a t i o n s -S'm'S'i/ + βνβμ = 0, μ φ v = x, y, z. (1.129) 24 1. Basic Quantum Mechanics The anticommutation relation between the components of spins in two arbi trary directions is obtained by expressing them in terms of the x, y, z com ponents. Verify that SaSb + SbSa = ~, (1.130) the Sa and Sb being the spin components along the directions a and b. By c ombi ni ng t h e c o mmu t a t i o n a n d a n t i - c o mmu t a t i o n r e l a t i ons, a ny p r o d u c t of s p i n - 1/2 o p e r a t o r s ma y be expr e s s e d i n t e r ms of a si ngl e s pi n- 1/2 o p e r a t o r. Th u s, for exampl e, SxSy = l- S z, SySz = i Sx, SZSX = '-Sy (1.131) Express S± in terms of Sx and Sy, (see (1.127)), apply the anticommutation relation (1.129) to show that S+S-+S-S+ = l. (1.132) By combining (1.132) and (1.128) we find that, for spin-1/2, S+S_ = ± + Sz, S-S+ = ^ - S z. (1.133) Now, if |±) are the eigenstates of Sz corresponding to the eigenvalues ±1/2 then verify that s+ = l+)(- l> = l~)(+li = 2 0 ”^ ”^ _ l_ )(_ l)i (1.134a) 5+ | - ) = |+), 5_|+) = | - ), 5+|+) = 0, 5 _ | - ) = 0, 52 =0. S+Sz = ~ S +, SZS+ = ^S+ (1.134b) Next, let Se = e.S be the operator corresponding to the spin component in direction e. Let p(±, e) be the probability that the outcome of measurement of the component in the direction e of a spin-1/2 in state \φ) is ±1/2. On applying (1.60) it is straightforward to show that P(±> e) = l ( ± | ^ ) | 2 = ( ψ - ± S e 'tpj . (1.135) Consider next a system consisting of two spin-1/2 subsystems. Let the eigen vectors | ± α, 1) of the component of spin 1 in direction a be the basis for the Hilbert space of that spin and let the eigenvectors | ± b, 2) of the compo nent S[2^ of the second spin in direction b constitute the basis vectors for the states of the second spin. A state of a system consisting of these two spins is then a linear combination of the states | ± a, ±6) = | ± a, 1) <g> | ± b, 2). This 1.6 Systems of One and Two Spin-l/2s 25 linear combination can be decomposed to express any state of two spin-1/2s in the form 7Γ \ψ(α)) = cos(a)|a1; —a 2) — sin(a)| — a 1,a 2), 0 < a < -. (1.136) This is known as the Schmidt decomposition [12] . This state reduces to a product of the states for the two spins for a = 0 but is an entangled state for d/0. The extent of entanglement may be measured by the absolute value of the correlation function C(ai,a2) = ( S g S g ) - {Sg>)(Sg). (1.137) For the state (1.136), it can be verified that |<7(αι, α2)| = sin2(2a)/4. (1.138) This shows that the maximal entanglement is achieved for a = π/4. In what follows, we consider the maximally entangled singlet state corresponding to a i = 0,2 = z (say) and a = π/4 in (1.136): 1 Γ, 1°) - V 2 1+’ ~ I- ’ w h e r e | ±, ± ) i s t h e e i g e n s t a t e o f S'!1 · * a n d Si z>. The expectation value of S: in this state is straightforwardly zero. By expressing Sx}y in terms of S± using (1.127) and on applying (1.134b), we find that (5W) = (5W) = (5W) = 0, 5 W5 W) = ( S ^ S ^ ) = 0 ( 2 ) (1.139) ?(*) (1.140) Express the vectors in terms of their Cartesian components and use these results to show that, in the state (1.139), n ?( 2) 0) = 0, 0) = — a ■ b (1.141) We now find t h e probability pa,b { ^ ’, ) that the outcomes of measurements on the component of spin 1 in the direction a and that of 2 in the direction b are, respectively, the eigenvalues ei^/2 and e[2^/2 (ei1^, = ±1). Note that those two measurements are compatible as the corresponding operators, being the operators acting on two different spins, commute. On using (1.60), that probability may be shown to be given by P... (4'>; 42)) = (°| (1 + (| Apply (1.141) to get 42)) = \ [1 - 4 υ42)« · b] ■ , ,(2) 0(2) - + eb ΰ b )l°) ( 1.142) ( 1.143) 26 1. Basic Quantum Mechanics From this we infer that the probability of finding two spins in the same direction is zero, i.e. if a ■ b = 1 then pa,b( +, +) = Pa,b(~, ~) = 0. From the point of view of the discussion to follow, we consider the prob abilities for the pairs of directions from a set of three directions a, b and c. Use (1.143), to shown that Pa,h( + , + ) + Pb,c{+, + ) — Pa,c{+, + ) 2 I uab \ . .2 I @bc \ - 2 sin I ) + sin | — | — sin (1.144) where 9ab,9bc,eac are the angles between the directions identified by the respective subscripts. With this we conclude the discussion of the methods of quantum me chanics relevant to us. Next we turn our attention to the important issue of identifying the characteristic non-classical features of the quantum theory. 1.7 Wave—Particle Duality Classical mechanics deals with two types of dynamical systems, particles and fields. A particle is an entity localized in space and time. A field, on the other hand, is described by a function E(x,t ), called the field amplitude defined at a continuum of points in space. An important property of the fields is that the resultant amplitude due to two fields at a space-time point is the sum of their amplitudes: E(x, t) = Ei (x, t) + E2(x,t). (1.145) This is the principle of superposition of the field amplitudes. The field am plitude may be expressed as a Fourier series in time. Consider a field whose amplitude has the Fourier expansion E(x,t) = —j= A(x) exp(—iu)t) + A*(x) exp(iwt) , (1.146) V 2 L J i nvol vi ng onl y one f r equency. Thi s de s c r i bes a wave of f r equenc y ω. A quantity of experimental interest is the intensity of the field. The intensity of the wave of a given frequency is the average of the modulus square of its amplitude over a period. The intensity of the wave described by (1.146) is thus I(x, t) = |^4(ar)|2. Consider the superposition of two waves of the same frequency. On expressing each of the amplitudes E\( x,t ) and E2(x,t ) as in (1.146), the intensity of the superposed waves is seen to be given by I = h + 12 + cos(φ(χ)), (1.147) where I t = |^4j(a;)|2, A^ x ) = |^4i(x)| exp(i0l (x)) is the intensity and ampli tude of the individual waves, and φ(χ) = φι (x) — φ2(χ) is the phase difference between them. This shows that the intensity I due to superposition of two waves is not a simple sum of the intensities I\ and I 2 of the superposed waves; it is modified by the addition of an interference term which is the last term in 1.7 Wave-Particle Duality 27 (1.147). There is no interference of this kind in the classical formalism if the entities involved are particles. The phenomenon of interference in classical mechanics is a characteristic of waves. Now, recall that the quantum theory describes the state of motion of a particle in terms of a state vector and, like waves, the quantities of experi mental interest in quantum mechanics, the expectation values, are quadratic in the state vector. Also, like the wave amplitudes, the state resulting from combination of states is a linear superposition of the corresponding state vec tors. Hence, like waves, the expectation values of the observables in a state which is a linear superposition of two states may exhibit the phenomenon of interference. In other words, in quantum theory, a system can exhibit dual character: the particle-like character of space-time localization and the wave like character of interference. The wave-particle duality, however, turns out to be complementary. It means that in an experiment a system would exhibit either the particle-like or the wave-like character but not both. We elaborate on these issues in the following. For the sake of simplicity, consider a system described by the state vector \ψ) which is a superposition of the states \rpi) (i = 1, 2) i.e. \Φ) = ΪΨι) + 1^2)· (1.148) The expectation value of an observable A in this state is given by ( i ) = (ψ\Α\ψ) = (ψ^Αΐ ψι ) + {%p2\A\ip2) + 2\('ψ1\Α\'ψ2)\οο8(φ), (1.149) where φ is the phase of the complex number (ψι\Α\ψ2). This shows that the expectation value of an observable A in a state \ψ), which is formed by a linear superposition of the states \ψι) and \ip2 ) is not a simple sum of its expectation values in the states l^i) and |Ψ2 ) alone but is modified by the addition of an interference term (which is the last term in (1.149)). Clearly, the interference effects may be exhibited if the observable has non-zero matrix element between the two states. The observable may, for example, be |α)(α| where |a) is an eigenstate of A with eigenvalue a. The expectation values in (1.149) are then the probabilities of observing the system in state |a). The probability amplitudes in quantum mechanics thus exhibit the inter ference phenomenon of waves. In order to understand the origin of quantum interference, note that the fact that the state \φ) is a superposition of two states means that the system can exist in one or the other state. We will see that interference is due to the lack of information about the state in which the system existed at the time of observation. In order to see that, let the system, before it is observed, pass through a detector whose state is changed differently on interaction with the two superposed states. Let \do) be the state of the detector before interaction and let \di) (i = 1,2) be its state after interaction with the state labeled i. The combined state of the observed system (hereafter called system S) and the detector after the interaction may be expressed as m = hMMi) + \Ψϊ )\Φ2.), (1.150) 28 1. Basic Quantum Mechanics We are, however, interested in the results of measurements on the system S alone. As explained in Sect. 1.5, those results are determined by the density operator ps = Tr(j [ | i') ( i'| | where Trd denotes trace over the detector states. Verify that ps = \Ψι)(Ψι\ + 1-02> (-021 + h A i X M ^ M i ) + \i>2){i>i\{d\\d2), (1.151) The expect at i on value of a system observable A alone is then given by (A) = {φλ\Α\φγ) + {ip2\A\i})2) + 2Κβ^(ψι\Α\ψ2)(άι\ά2γ). (1.152) The interference term in this case has an additional factor {d^d?,). This fac tor determines the overlap between the states in which the detector is left as a result of its interaction with the two superposed states of the system. Since \(di\d2 )\ < 1, the process of detection reduces the contribution of the interference term. The two superposed states are evidently distinguishable unambiguously if the states in which they leave the detector on interaction with it are orthogonal, i.e. if (di\d2) = 0. The interference term in (1.152) then vanishes. This shows that there is no interference if the superposed state in which the system existed before the time of observation is known. In other words, the interference is lost if we know by which path (welcher weg in Ger man) the system arrived at the detector. Now, in the classical mechanical description, a particle follows a definite trajectory or path. Hence, knowing unambiguously the path of an object is a particle-like property. From this it may, therefore, be inferred that the interference, which is a wave-like phe nomenon, is lost if the path, which is a particle-like characteristic, is known. Hence, though what is perceived as a particle in classical mechanics, may exhibit wave-like interference phenomenon, the wave-particle duality is com plementary. We will see in Chap. 5 that even what are perceived as waves in classical mechanics may exhibit particle-like dual character. The loss of interference due to the process of detection may at times be attributed to a change in the observed part of the superposed states brought about by their interaction with the detector. The details of the changes so brought about depend upon the particular situation at hand. However, it is at times possible to conceive detectors which reveal which-path information by interacting with that operator of the system which commutes with the one under observation. Such detectors, therefore, do not alter the observed part of the state of the system. The loss of interference in that case can not be attributed to any detector influenced state alteration mechanism. An example of such a detector in the context of quantum optics may be found in [13]. In the following, we refer to the states | d») of the detector resulting from its interaction with the superposed states of the system under observation as the which-path states. We have s ee n above how t h e pr oc e s s of d e t e c t i o n may r e duce t h e ef f ect of i nt e r f e r e nc e e x h i b i t e d by obs e r va t i ons ma d e on a s ys t e m i n a s u p e r p o s e d 1.8 Measurement Postulate and Paradoxes of Quantum Theory 29 state. Let us now examine the possibility of observing interference in a joint measurement of a system observable A and a detector observable D. In the combined state of the system and the detector, given by (1.150), the expec tation value in question is evidently 2 (Ψ\ί)Α\Ψ) = ^ \ (ipi\A\ipi }\2\(di\D\di }\2 i = 1 + 2 R e | ^ i | J4|'i/>2)(di|.D|<i2) · (1.153) This implies t h a t t he j oint observation of t he system observable A and the detector observable D would exhibit interference effects between the super posed states l^i) and if {ψ\\Α\ψ2 ) Φ 0 i-e· if A exhibits interference in the absence of detection, and if (di\D\d2 ) φ 0 i.e. if D has a non-vanishing matrix element between which-path detector states. Recall from (1.152) that if the which-path detector states are orthogonal to each other then the system observable A alone would not exhibit interference effects. However, its joint observation with a detector observable D may exhibit such an effect even if (oil|<^2 ) = 0 but if (di\D\d2 ) Φ 0. Since the condition {d\\d2 ) = 0 provides which-path information, the condition (di\D\d2) φ 0 means that the detec tor observable D is an eraser, called the quantum eraser, of the which-path information [14]. It should be emphasized that the interference in this case is not in A but in joint observation of A with the detector observable D. For examples of the possibility of realization of quantum eraser in the context of quantum optics, see [14, 15]. 1.8 Measurement Postulate and Paradoxes of Quantum Theory The validity of a postulate of a physical theory is judged, of course, by its ability to describe the observed physical phenomena. However, in spite of its success on this count, the postulate (postulate 4) which relates the theoretical predictions with experimental observations, has been at the centre of contro versy right since its conception. The controversy revolves mainly around the questions involving (a) that postulate’s consistency with other postulates, and (b) its denial of the objective reality. Those issues are brought out forcefully in the thought experiment (Gedankenexperiment in German) of Schrodinger and that of Einstein, Podolsky and Rosen (EPR). In the following we outline briefly first the issue of consistency of the process of measurement envisaged by the postulate in question with other postulates of the quantum theory and follow it up with a discussion of the two paradoxes: the Schrodinger's Cat Paradox and the EPR Paradox. 30 1. Basic Quantum Mechanics 1.8.1 The Measurement Problem A measurement is an outcome of interaction between the system under ob servation (hereafter called the object) and an apparatus designed to measure an observable. Since there is no restriction in the postulates of the quantum theory on the kind of physical systems that it seeks to describe, it is expected to describe an apparatus as well as it describes the object. If that be so, the process of measurement envisaged by the quantum theory should be consis tent with its other postulates applied to the interacting system of the object and the apparatus. In order to examine whether or not it is so, consider the act of measurement of an observable B. Let {| )} be the set of orthonormal eigenvectors of B corresponding to the eigenvalues {&*} {i = 1,... ,n) where, for the sake of simplicity, we assume that those eigenvectors are denumerable and the eigenvalues non degenerate (see Chap. 10). Any state of the object is then expressible as Similarly, let the state space of the apparatus be spanned by the orthonor mal set {|ai)} and let the apparatus be initially in a pure state. Recall that, according to the measurement postulate, the state of the object on interac tion with the apparatus is reduced to one of the eigenstates, say \bk), of the observable being measured with probability | ('01^'fe) |2 = la fc|2 and that the outcome of the measurement is the corresponding eigenvalue bk ■ This implies that the state of the combined system of the object and the apparatus after the interaction be described by the density operator For details see the papers in [11]. The density operator (1.155) characterizes a mixture of states {1^,6^)} with probability {\a i\2 }. However, recall from Sect. 1.5 that a pure state remains pure under a unitary transformation. Since the initial state of the combined system is a pure state whose evolution is governed by unitary Schrodinger evolution operator, the state of the com bined system at any time will be pure. Note also that the state reduction is an irreversible process whereas a unitary evolution is reversible. The mea surement postulate thus seems to be out of tune with other postulates. The non-unitary, irreversible, evolution may, however, arise in the limit of non- denumerably infinite number of degrees of freedom of the environment with which a system interacts (see Chap. 8). Quantum theory thus seems to have different laws depending upon whether a particular process is a measurement or not. What, then, distin'- guishes an apparatus from any other quantum mechanical system? Although the quantum theory does not provide an answer to it, it is generally ac cepted that an apparatus is a macroscopic system. Each macroscopically dis- n (1.154) i=1 (1.155) 1.8 Measurement Postulate and Paradoxes of Quantum Theory 31 tinguishable state is a statistical mixture of several microscopic states. Hence a macroscopic state should be described, not by a state vector, but by a den sity operator. However, even if it is assumed that the apparatus is initially in a mixed state, the Schrodinger evolution still does not lead to the form (1.155) expected of a process of measurement [11]. The cause of the measurement paradox thus seems to be rooted in linear ity (resulting from superposition principle) and the unitarity of the quantum evolution. It would thus appear that by postulating a different kind of evo lution when a quantum system interacts with an apparatus may resolve the paradox. Indeed, a non-linear time-evolution for object-apparatus has been advocated by some authors [11], However, a theory whose laws depend upon whether or not a system is an apparatus can not be regarded as satisfactory. Finally, it may be argued that the state reduction is not a physical pro cess, that the quantum theory predicts only the statistical average of many observations and that state reduction postulate is needed to make statistical predictions about the outcome of individual experiments. The state vector provides just a means of calculating the statistical outcomes of the process of measurement made on identically prepared systems. It describes an en semble of identically prepared systems and not any individual system. The paradoxes discussed below indeed bring out emphatically the inadequacy of the state vector in describing the outcome of individual measurements. 1.8.2 Schrodinger’s Cat Paradox Following Schrodinger [16], assume that a cat is pinned up in a steel chamber. Let there be a mechanism (details of which are unimportant for the present) which releases a poisonous gas in the chamber when hit by a spin pointing in the +z-direction but not if the spin points in the -z-direction. The release of the poisonous gas is assumed to kill the cat. Let the said mechanism be hit by a spin-1/2 whose states in the ±z-directions are denoted by |±). Let | d) denote the state of the dead cat and 1Z) that of the living cat. If a spin in state |+) hits the triggering mechanism then quantum mechanics predicts that the state of the combined system of the cat and the spin is |+, d). In any subsequent experiment, the cat will be found dead with certainty. Similarly, if that mechanism is hit by a spin in state | —) then the quantum mechanical state of the combined system of the cat and the spin would be | —, Z), and in any subsequent experiment the cat will be found alive with certainty. Those predictions agree with what is the common perception of reality or the objective reality. The situation is, however, different if the triggering mechanism is hit with a spin in a superposition state, say, in the state γ^1/2(|+) — |—)). According to quantum mechanics, the state of the combined system of the cat and the spin would be (1.156) 32 1. Basic Quantum Mechanics This state is a superposition of the states in one of which the cat is dead and in the other in which the cat is alive with equal probability. According to the measurement postulate, the cat in a process of measurement will be found dead or alive with equal probability and that no statement can be made about the state of the cat until a measurement is performed, i.e. the state of the cat is decided by the act of measurement. The act of measurement could be the act of looking at the cat after the event is over. Till then, the cat remains in a state of suspended animation! This is contrary to the objective reality according to which the state of the cat is decided according to how the spin hits the triggering mechanism and not by any subsequent act of measurement. The subsequent act of measurement only ascertains that state. It would, therefore, seem that a macroscopic object, like a cat, may not exist in a superposition state. A state that is a superposition of macroscopic states is often referred to as a cat state. Which quantum states should we accept as macroscopic though may be a matter of debate. However, the coherent states (introduced in Chap. 3) of a harmonic oscillator and those of spins are widely recognized to be macroscopic. It is because those states are most classical; in fact the only pure states of respective systems possessing classical properties (in the sense explained in Chap. 4). A number of models for generating coherent superposition of field coherent states [17] and those of spin states [18] in the context of quantum optics have been proposed. 1.8.3 EPR Paradox Here we present experimentally important Bohm’s [19] version of the EPR paradox [20]. It considers a maximally entangled state (1.139) of two spin- l/2s. Now, let the two spins fly apart and let the z-component of spin 1 be determined experimentally after it ceases to interact with the spin 2. Accord ing to the measurement postulate, spin 1 then collapses to an eigenstate of its z-component. Since the combined state of the spins is given by (1.139), it follows that if the state of spin 1 is found to be |+) (|—)) then that of spin 2 is certainly | —) (|+))· According to the measurement postulate, the state of spin 2 in this case is reduced to an eigenstate of its z-component. However, we can, instead, measure some other component, say, the x- component of 1. The eigenstates |±,α;) of the z-component of a spin-1/2 are related with those of its z-component by the relation Substitute the resulting expression in (1.139) to show that the state of the spins assumes the form The reader may verify by expressing Sx in terms of S± that these are indeed the eigenstates of Sx. Invert this relation to express |±) in terms of |±, x). (1.158) Hence, if the outcome of a measurement of the x-component of spin 1 is 1/2 (—1/2) then it follows that the spin 2 is certainly in the state |—,x) (|+,x)). According to the measurement postulate, the state of spin 2 in this case is reduced to an eigenstate of its x-component. Observe that the state reduction of spin 2 in the example above is caused, not by a measurement performed on it, but by the one performed on another spin not interacting with it at the time of the measurement. This shows that the process of state reduction of a spin may be controlled by the process of measurement performed on another spin with which it interacted in the past but with which it has no interaction at the time of measurement. Since none of the spins knows while it interacts with the other, which of its component is going to be measured by a subsequent act of measurement, the said con trolling influence by the process of measurement envisaged by the quantum theory is non-local. Moreover, an experiment can be performed to determine simultaneously one component of spin 1 and the same or the other component of the spin 2. This is permissible because all the operators corresponding to one spin commute with all the operators corresponding to the other and, according to the quantum theory, the value of the observables whose corresponding operators commute can be determined simultaneously with certainty. Let then the z-component of spin 1 and the z-component of spin 2 be measured simultaneously. As discussed above, the knowledge of the z-component of spin 1 provides precise information about the state of that component of spin 2 as well. However, the z-component of spin 2 is also known precisely by virtue of the measurement performed on it. We thus know simultaneously the precise values of two non-commuting observables. That is in contradiction with the quantum theoretical dictum that two non-commuting observables can not possess precise values simultaneously. This thought experiment exemplifies a situations wherein, it is possible, in principle, to determine with certainty the values of two non-commuting operators simultaneously. As mentioned above, this is incompatible with the description of a system in terms of a state vector. Therefore, EPR [20] argue that, the description in terms of a state vector is incomplete. According to them, a necessary requirement for the description given by a theory to be complete is that every element of the physical reality must have a counterpart in the physical theory. Their criterion for determining the elements of physi cal reality is that if, without in any way disturbing a system, we can predict with certainty (i.e. with probability unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this phys ical quantity. Since, in the example above, it is possible to determine with certainty the values of two non-commuting components of a spin without dis turbing it, there is an element of physical reality corresponding to those spin components. However, according to the description of the spin in terms of a state vector, the values of non-commuting operators can not be determined 1.8 Measurement Postulate and Paradoxes of Quantum Theory 33 34 1. Basic Quantum Mechanics simultaneously with certainty. Hence, the said elements of the physical reality have no counterpart in the description in terms of a state vector. Such a de scription is, therefore, incomplete. They left open the question of whether or not a complete description exists but concluded by saying that they believe that it does. An attempt in the direction of providing a complete description of a phys ical system is in terms of the hidden variables theory discussed next. 1.9 Local Hidden Variables Theory The paradoxes discussed above may be resolved by assuming that in each re alization the system carries definitive value of all its observables distributed among different realizations with the probability assigned by the measure ment postulate. This is the basic premise of the hidden variables theory. It attributes the quantum indeterminism to some ‘hidden’ influence analogous to that due to microscopic constituents in a macroscopic classical statistical system. It assumes that, with a complete knowledge of those hidden variables, the outcome of the observations can be predicted with certainty and that the probabilistic nature of the quantum predictions is a result of our ignorance of such variables. If it is also assumed that the measuring devices at two different places are neither correlated with each other nor with the object under observation then the hidden variables theory is called a local hidden variables theory (LHVT). We do not discuss the hidden variables theory in all its details (for a survey, see [21]) and restrict ourselves to highlighting the implications of the LHVT only. The incompatibility of the LHVT with the quantum theory can be seen at the outset. For, it implies that in each realization of an experiment the system carries definitive information about every observable whether commuting or not. According to quantum mechanics, a definitive information about non commuting observables is precluded. Thus, in the hidden variables theory, we can define joint probability distributions involving even non-commuting observables. In the quantum theory, there is no concept of a joint probabil ity distribution involving non-commuting observables. Let us, nevertheless, investigate some observable consequences of the LHVT and compare them with the predictions of the quantum theory. The simplest and the most widely studied system, as regards the verifiable consequences of the LHVT are concerned, is a system of two spin-1/2 particles in a maximally entangled state such as (1.139). For the two spins in that state, we have in (1.144) the quantum theoretic relationship between pa,b(+, +), Pb,c(+, +) and pa,c(+, +) where pa,b(ea, £b) is the probability that the outcome of measurement on the component of spin 1 in the direction a and that of spin 2 in the direction b are, respectively, the eigenvalues ea/2 and Cb/2 (ea,e6 = ±1). Here we derive that relationship following the local hidden variables theory. 1.9 Local Hidden Variables Theory 35 To that end, let P (e, . „ . ) be the joint probability that the spin 1 has the components measuring ei^/2, ej^/2, e£^/2 along the directions a, b, c and that ei2^/2, c ^/2, ^ j 2, are the components of the spin 2 along the same directions where each of the e assumes values ±1. Now, the probability of finding two spins in the same direction, say the direction a, may be expressed in terms of the joint probability introduced above as the state (1.139) then, by virtue of (1.143), p a,a ( e a',ea ) = 0. Since classical probabilities are non-negative, it follows that p a a ( e a',ea ) = 0 can hold only if each of the joint probability under the summation in (1.159) is zero. On using the same argument for the probabilities in the other two directions, it follows that the P's with the same sign at the same place on the two sides of the semicolon is zero i.e. Again, positivity of the probabilities on the right hand side of (1.162) implies the inequality known as Bel l ’s inequality [22]. Its derivation given here is due to Wigner [23]. Recall that the corresponding quantum theoretic result is the equality (1.144). That equality need not respect the inequality (1.163). A violation of (1.163) for any choice of directions would constitute a rebuttal of LHVT. It is not difficult to see that if, for example, the three vectors are coplanar and 9ab = Obc = 7r/3, 9ac = 27r/3, then the value of (1.144) is —1/4 which violates (1.163). (1.159) where the summation is over all the e ^ ’s and the e ^ ’s. If the spins are in by summing over all the e's other than ei1·* and e ^. Similar argument for Pb,c (4^1 e c2 ^ ) and p a ^c along with the use of (1.160) yields Pa,b(+', + ) = -P(+> —, +; —; +> —) + -P( +, —j —> — ί +> +)> Pb,c(+', +) = - P( +i +5— j +) + 35(—,+,—;+,—,+), Pa,c(+', +) = - P( +,+5— ί - ,+) + - P( +,—!+>+)· (1.161) These lead to the relation Po,h(+; +) + Pb,c(+', +) - Po,c(+; +) — 35( +,) + P( — ( 1.1 6 2 ) P a,b ( +\ +) + Pb,c(+! +) — Pa,c{+', + ) > 0 (1.163) 36 1. Basic Quantum Mechanics It has been shown that any pure entangled state of two spin-l/2s violates corresponding Bell’s inequality [12, 24, 25], Similar inequalities involving four directions have also been derived [26]. A more powerful criterion for compar ing the predictions of the two theories in the form of an equality is obtained for a system of three spin-l/2s [27, 28]. Violations of Bell-type inequalities have been observed experimentally (see the papers reprinted in [11]). They thus establish that the LHVT can not account for quantum correlations. In other words, the nature of quantum indeterminism is not the same as that of a classical statistical mechanical system. Note that the assumptions involved in deriving the results of hidden vari ables theory above are locality and the positivity of the probabilities. By lifting any one of those restrictions we can make the hidden variables theory agree with the quantum theoretic predictions. However, relaxing the require ment of locality would mean that the states of the measuring devices are correlated with the states of the object and with each other. It has the ques tionable feature of denying a description of isolated systems. The negative probabilities [29] are, of course, unphysical. However, as discussed in Chap. 4, they can be useful as calculational tools and in identifying signatures of quan tum effects. 2. Algebra of the Exponential Operator The exponential operator, i.e. the exponential function of an operator, defined in (1.19) by way of a series expansion, is of paramount interest in mathemat ical physics. In view of its importance, we discuss in this chapter some useful algebraic operations involving an exponential operator. The operations dis cussed here are (i) the parametric differentiation of an exponential operator, (ii) its reduction to a polynomial when the exponent is finite-dimensional, (ii) similarity transformation, defined below, by an exponential operator, (iii) the operation of disentangling an exponential, and (iv) evaluation of time-ordered exponential integral. 2.1 Parametric Differentiation of the Exponential Consider the exponential function exp(A(<)) of an operator which is a func tion of a scalar t. In this section we evaluate its derivative with respect to t. To that end, use the series expansion (1.19) of the exponential and the formula (1.40) for the derivative of Am(t) with respect to t to get (2,) n=0k—0 v ' Now, (i) interchange the n and k summations in (2.1) using OO OO Σ Σ ^ = Σ Σ ^ = Σ Σ ^ + ^ ’ (2·2) n~0k=0 k—0n=k k=0n=0 (ii) multiply and divide the resulting summand by nlk\, (iii) rewrite n\kl/(n+ k + 1)! in terms of an integral using (A.18), and finally (iv) carry the sum mations over n and k to arrive at the Sneddon’s formula ^ e x p [ i ( i ) ] = J e x p[uA]^ exp[(1 - u)A]d u. (2.3) If A(t) = At then this yields the familiar c-number result ^ exp(Jii) = Jiexp(Jii) = exp(At)A. (2.4) 38 2. Exponential Operator 2.2 Exponential of a Finite-Dimensional Operator In this section we show that the exponential of an n-dimensional operator X may be expressed as a polynomial of degree at most n — 1 in X. To that end, we recall eq.(10.24) which states that an n-dimensional operator satisfies the equation n Π (a- - Ai) = 0, (2.5) i = 1 Ai being the eigenvalues of X. This is a polynomial of degree n in X. Recall also that the equation (2.5) is not necessarily the minimum polynomial equa tion of X. Let N < n be the degree of the minimum polynomial equation of X. We may rewrite such an equation in the form N - l * N = Σ (2·6) m=0 Here am are known in terms of the A’s. Using (2.6), we can express the nth and higher powers of X as linear combinations of its powers up to N — 1. Consequently, we may write N - l βχρ(ΘΧ) = Σ Cm(e)Xm. (2.7) 771=0 Note that Ck(0) = SkO. (2.8) To detrmine the unknown functions Cm{9), differentiate (2.7) with respect to Θ to obtain N N - l Σ Cm_i(0)Xm = Σ Cm{e)Xm. (2.9) 771=1 m=0 The dot over a quantity denotes derivative with respect to Θ. Now, use (2.6) for X N and compare equal powers of X in (2.9) to arrive at the equation Ck(e) = akCN- 1(e) + Ck- 1(0), C-!(0)= 0, (2.10) The set of equations generated by (2.10) for k = 0,..., N — 1 determine Ck. Their solution, with (2.8) as the initial condition, on substitution in (2.7) gives the desired expression of the exponential as a polynomial. As an example, consider a spin-1/2. Any of its component Sa along a direction a is a two-dimensional operator (n = 2) which can assume the values ±1/2. The equation (2.5) in this case reads (2.11) It is also the minimum polynomial equation for Sa■ On comparing this with (2.6) we note that ao = 1/4 and a x = 0. Since N = n = 2, the equations (2.10) reduce to C0 (θ) = \θ 1(θ), C i ( 0 ) = C o(0)· (2-12) The solution of these, with the condition (2.8), on substitution in (2.7) yields exp(0Sa) = cosh + 2sinh Sa. (2-13) We leave it as an excersise for the reader to show that if Ja is a component of a spin-1 then exp(#Ja) = 1 + sinh(#) Ja + (cosh(#) — 1) J 2. (2-14) This may be established by noting that the eigenvalues of Ja are 0, ±1 and that the equation (2.5) in this case assumes the form Ja(J 2 — 1) = 0. 2.3 Lie Algebraic Similarity Transformations 39 2.3 Lie Algebraic Similarity Transformations Let S be a non-singular operator. The transformation defined by S~lA S = B (2.15) is called the similarity transformation of A by S. By invoking the definition of the power of an operator we see that, for a positive integer m, s ^ A s = S ^ A mS. (2.16) As a consequence of this we note that if F(A) is a function expandable in a power series of A then S-'F ( A) S = F . (2.17) As an application of the similarity transform, note that if S'^1 exists then we can shuffle S to the right (left) in the product S'A (A5) by means of the relations SA = ( ^ i ^ - 1)^, AS = S i S -'A S ) (2.18) involving a similarity transformation. Of particular interest to us is the similarity transformation (2.15) when S = exp(-P). As an important example, recall from (1.77) that if a sys tem is described by a time-independent hamiltonian H then the time- evolution of an operator A is determined by the similarity transformation A(t) = exp( i Ht/h) Aexp( —i Ht/h). The similarity transformation by an ex ponential operator may be expressed as 40 2. Exponential Operator A (Θ) = exp i#p) A exp (θΡ^ = Α - θ [ Ρ, A] + ^ [P, [P,A (- β γ L PA = exp n = 0 -0LP j o „ „ j „ LPA = A, LPA = P, A ,LPA = Lnp- 1[P,A}. ( 2.19) (2.2 0) To pr ove ( 2.19), show t h a t A(ff) obeys the differential equation (ά/άθ)Α(θ) = —ΣρΑ(θ) whose solution is evidently (2.19). The evaluation of (2.19) is generally a formidable task unless the series terminates at a low order. A systematic approach for its evaluation may be developed when A and P are the elements of a Lie algebra: A linear vector space of operators is said to constitute a Lie algebra if it is closed under the operation of commutation, i.e. if the commutator of any two operators in the space also lies in that space. Consider a vector space spanned by a complete set of operators X\, ■ ■ ■ ,X n. This space would constitute a Lie algebra if X ,, X -, ^ ^ CijkXki k= 1 (2.21) Cijk being c-numbers called the structure constants of the algebra. An algebra is characterized by its structure constants. The operators Χ ι,· · ·,Χ η are called the generators of the algebra. Now, consider the similarity transformation Xi (Θ) = exp Oz'j Xi exp . Di f f e r e nt i a t i on of t h i s e q u a t i o n wi t h r e s p e c t t o Θ yields d -^Xi (Θ) = exp (-6>Z) Xi, Z exp H · Let Z be an element of the given algebra expressible as Ζ = γ/αιΧι, (2.22) (2.23) (2.24) i=1 (Xi being c-numbers. Substitution of (2.24) in (2.23) and use of the commu tation relation (2.21) leads to a set of first order coupled ordinary linear differential equations for |Xj(0) j: άθ Xi (θ) = Σ ^ ] cijkaj Xk (Θ). ( 2.25) fc=l 3=1 2.3 Lie Algebraic Similarity Transformations 41 Solution of (2.25) gives Xi (Θ) in terms of a linear combination of Xl(0), · · ·, i ( f l ) = X i,-,t In the next section we solve (2.25) explicitly for some algebras which en compass a number of hamiltonians of frequent occurrence in quantum optics. 2.3.1 Harmonic Oscillator Algebra The set of operators ( α^,ά,Ν = α^α,Ι^ obeying the commutation relations [a, = 1, N, a = —ά, N, a 1 = a) (2.26) generates the harmonic oscillator algebra. It is so called because the hamil tonian of a harmonic oscillator (h.o.) coupled linearly to a driving force is expressible as a linear combination of the elements of this algebra. The last two commutation relations in (2.26) are consequences of the first relation called the canonical or bosonic commutation relation. The operators a, at are also called boson operators. An element Zh in this algebra is expressible as Zho = α\ά + α2αϊ + α3 ά^ά, (2.27) the a's being c-numbers. Consider the similarity transformation ά{θ) = exp aexp (ΘΖh0) . (2.28) It is straightforward to show that ^α(θ) = αΆα(θ)+α2. (2.29) This is solved by α(θ) = βχρ(α!3 0 )ά H exp (α^θ) — 1 . (2.30) «3 L Next we derive results for some frequently encountered special cases of (2.30). 1. Set « 3 = 0, Θ = 1 to reduce (2.30) to exp {— ( a i a + a 2a} ) } aexp { ( αχά + a 2a^)} = a + a2. (2-31) Rearrange this to arrive at the commutation relation [a, exp (αιά + ο;2,ά^)] = a2 exp (αιά + 0:2 *^) . (2.32) Set a\ = 0 and differentiate this m times with respect to a2 at a2 = 0 to get [a, atm] = ma*™-1. (2.33) 2. On setting αχ = a 2 = 0, 0:3 = 1, (2.30) yields exp (—flat a) a exp (θα* a) = exp(θ)ά. (2.3 4 ) The system of interest in quantum optics described by the harmonic oscillator algebra is the electromagnetic field (see Chap. 6 ). 2.3.2 The S U( 2) Algebra The Lie algebra generated by the spin operators Sx,Sy, Sz obeying the com mutation relations (1.126) is called the SU(2) algebra. SXJ stands for special unitary. This nomenclature has its origin in the group theory. However, the group theoretic aspects of the algebras are of no relevance to us here. Follow ing the notation of Sect. 1.6., we denote by Sa = a ■ S = axSx + aySy + azSz the spin operator along the direction a. Use this decomposition and (1.126) to show that the commutation relation between the components in two arbitrary directions a and b is Sa, Sb = i ( a x b) ■ S. (2.35) Now, consider the similarity transformation Sa(0) = exp iθη · S') a · S'exp (ι θη ■ S') , to · to = 1. (2.36) Differentiate this with respect to Θ and use (2.35) to show that ^ S a(9) = ( n x a ) - S ( 9 ). (2.37) The equation satisfied by (to x a) ■ S(0), can similarly be found to read (to x a) ■ S(0) = exp i 0n-S^ [to x (to x a)] · S'exp ^ t o - S') . (2.38) Now, use the relation a x (6 x c) = (a · c)b — (a · b)c to rewrite (2.38) as (to x a) ■ S{0) = - S a{0) + (a ■ n)(n · S). (2.39) ασ In writing this we have made use of the fact that exp ( —ϊ θη · S') to · S' exp (ϊ θη - S') = to · S' (2.40) is independent of Θ. Solution of the coupled set (2.37) and (2.39) yields exp ^ — ϊ θη ■ s ) a · S exp (ίθη ■ s ) = cos (Θ) a ■ S + (to x a) · Ssin (Θ) + [1 — cos(0)] ^ro · s)^to · a ). (2.41) If to · a = 0 then, also on invoking (2.35), the expression above reduces to exp ^ — ϊ θη ■ s ) a · S exp (ίθη ■ s ) = cos (Θ) a ■ S — i[n · S, a ■ S] sin (Θ). (2.42) Some special cases of this are enumerated below. 1. If n ■ S = Sz, a ■ S = Sx then exp ( — itf.S'z) Sx exp (iOSx'j = cos (Θ) Sz + Sy sin (Θ) . (2.43) 42 2. Exponential Operator 5±(0) = exp S± exp (i05z) = exp(Ti0)5±. (2.44) This may be derived by noting that d5-t(0)/d0 = =Fi5-t(0). 3. 5 2(0) = exp ( —i0S+) Sz exp (i05+) = Sz + i05+. (2.45) This may be derived by noting that d52(0)/d0 = i5+. Multiply (2.45) on the left by exp(i05+), differentiate it m times with respect to 0 at 0 = 0 and show that [52,5™] =mS™ . 4. 5 _ (0) = e xp ( - 1 0 5 + ) 5 _ exp ( i 0 5 + ) = 5 _ - 2i 0 5 2 + 025 +. ( 2.46) Th i s ma y b e der i ve d by showi ng t h a t d 5 _ - ( 0 )/d 0 = —2 i 5 2(0) wher e 5 2(0) i s as i n ( 2.45). Mu l t i pl y (2.46) on t h e l ef t by e x p ( i 0 5 + ), di f f e r e nt i a t e i t to t i me s wi t h r e s p e c t t o 0 a t 0 = 0 a n d show t h a t [ 5 _, 5 ™ ] = — 2 t o 5 + - 1 5 2 - m ( m - l ) ^ ” 1 . Th e s y s t e m of i n t e r e s t i n q u a n t u m o p t i c s t h a t i s a n e l eme nt of SU(2) is that of a collection of two-level atoms (see Chap. 7). 2.3 Lie Algebraic Similarity Transformations 43 2. 2.3.3 The 5C/(1,1) Algebra The 5 t/( l,l ) algebra is generated by hermitian operators K x, K y, and K z obeying the commutation relations = - i K x K,„ K, = \Kr K x, K r i K„ (2.47) Not e t h a t t h e di f f er ence bet we en t h e c o mmu t a t i o n r e l a t i o n s for SU(1,1) and the relations (1.126) for SU(2) is in the sign of the commutator of the x and y components. Like the case of SU(2) operators, the 5 [/( l,l ) operators can also be thought of as components of a vector in a three dimensional space. The space for SU( 1,1) operators, however, is not the Euclidean space of SU(2) operators. It is the so called (2+l)-dimensional Minkowski space in which the dot and the cross products between vectors a and b are defined by ο, ’ b — axbx + ciyby Qizbzi X b ) i — ^ ^ ^ i j k Q'j b k i ϊ Φ z ‘-> j,k =x,y,z (θ, X b)2 = ^ ' ^zjkO'jbk· ( 2.48a) ( 2.48b) j,k =x,y,z Here exyz is +1 (—1) for even (odd) permutation of x,y,z\ eijk = 0 if any two of the i,j,k are same. Note that the scalar product so defined does not have the properties of the Euclidean scalar product as the "norm" a ■ a of a vector a according to (2.48a) need not be positive. Keeping in mind the definitions (2.48a) and (2.48b) of the dot and the cross products, show that 44 2. Exponential Operator a K, b K i (a x b) ■ K. (2.49) By following the steps outlined above for the case of SU( 2), establish also that exp ( —ϊ θη ■ K ) a ■ K exp (ίθη ■ K^j = cosh(0)a · K + (n x a) sinh(0) — [cosh(0) — l](n · a) n ■ K. (2.50) It would turn out to be useful to introduce the operators k ± = k x ± i k y, k + = k I. (2.51) Apply (2.47) to see that k z, k ±j = ±k±, \k+, £ _ ] = - 2 k z. (2.52) Some similarity transforms involving SU( 1,1) of frequent occurrence are enu merated below. 1. 2. Κ±(Θ) = exp ( ~ i 0 k z^J if±exp (ί θΚζ) = exp(=FiΘ)Κ±. (2.53) This may be derived by noting that dk±( 0)/d0 = =pi.k±(9). Κ ζ(θ) = exp ( ~ί θΚ+^ K z exp (ί θΚ+^ = K Z + iΘΚ+. (2.54) This may be derived by checking that d k z (6)/d6 = ik +. 3. Κ_( θ ) =βχ ρ ( - ί θ Κ+^ K - βχρ(ί6·ΛΓ+) =K_ + 2iΘΚΖ - Θ2Κ +. (2.55) This may be derived by verifying that dK_ (θ)/άθ = 2ιΚζ(θ) where Κ ζ(θ) is as in (2.54). The hamiltonian of interest in quantum optics that is an element of 517(1,1) is the one that describes the process of two-photon down-conversion (see Chap. 7). 2.3 Lie Algebraic Similarity Transformations 45 2.3.4 The S U( m) Algebra The algebra S U(2) introduced above is a special case of the SU(τη) algebra. It is generated by a set of m2 operators {At] }, i,j = 1, · · · ,m obeying the commutation relations A-ij, A — 5jkAu SuAkj, z, j — 1,2,· · ·, τη. (2.56) Verify that the operator τη ά = Σ Α» (2·57) commutes with all Aij. Hence the number of independent generators of SU(m) is m 2 — 1. A similarity transformation of a generator by an arbi trary element of the algebra would involve solving a set of up to πι2 — 1 coupled equations. However, it will be seen in the next subsection that, by using the so called bosonic representation, the number of coupled equations required to be solved for an arbitrary similarity transformation in SU (m) is τη. As a simple example, we consider the similarity transformation generated by a linear combination of the diagonal operators An: (0) = exp ^ - 0 ^ aiAi-^j A^ exp ^ 0 ^ ai Ai ^j . (2.58) It is easy to verify that άΑ^(θ)/άθ = (ctj — ai)Aij(6) so that A^ (0) = exp {0 (ctj — cti)} A^. (2.59) The hamiltonian of an m-level atom interacting with classical e.m. field and a class of optical parametric processes constitute realizations of SU(m) [30], 2.3.5 The S U ( m,n ) Algebra The SU( 2), SU( 1,1) and SU(m) algebras introduced above are special cases of the SU( m,n) algebra with SU(m) = SU(m,0). The group theoretic as pects of SU( m,n) are of no relevance to us here. For us, it is adequate to introduce SU( m, n) in terms of what is known as its bosonic representation. The SU( m, n) operators in this representation are bilinear combinations of the boson operators {at, ά£; bp, Sj,} (k = 1,..., m;p = 1,..., n) obeying dk·, &i Ski, bp, % 'pq, bp = 0, I ak, 0. The s e t o f (m + n) 2 bilinear combinations of the form - a]c- Xj k Lpq bp>q, Zj p ■ CL j bp ( 2.60) ( 2.61) al ong wi t h t h e i r h e r mi t i a n conj uga t e s g e n e r a t e SU(m,n). These operators commute with the effective number operator m n JV = 2 > J a i - £ & t v (2.62) 2 — 1 1 Hence t he number of independent generators of SU( m,n) is (m + η)2 — 1. It should be emphasized that the representation of SU( m,n) in terms of bilinear combinations of bosonic operators does not encompass all its representations. Consider the similarity transformation Α(θ) = exp ( - 9ZmnJ Aexp (eZmnj . (2.63) Let Zmn be an element of SU(m, n) expressed as τη η τη n Zmn — fipqbpbq + Σ Σ Ϊ Μ ρ i,j = 1 p,q—l i =\ p= 1 τη n - Σ Σ ^ Μ · <2-S4) 2 = 1 p—1 Verify invoking the commutation relations that ί τη n —θί(θ) = Y^ ai j aj i e) - Σ ^ & ρ Ο 9)’ i = i p = i , n m = - Σ Λ - Σ Λ 9)· (2·65) q —1 i = l These constitute a set of τη+rz linear coupled equations. Since any element of SU(m,n) is a bilinear combination of {Si}, {bp} and their hermitian conju gates, any SU( m,n) similarity transformation is determined by the solution ofm + n coupled linear equations (2.65) and the property (2.17). For a dis cussion of solution of (2.65) see [30], In the following we consider its special cases: 1. Verify that S+ = a\a2, 5_ = Sz — ^{α\άι - a^a2) (2.66) obey the commutation relations (1.128) of S U(2). Consider the similarity transformation άί(θ) = exp (iΘΖ-^j άί exp [—ι θΖ· ^ , (2.67) Z2 = ωια^άι + ω2ά2ά2 + ξ + ά ^ + ξ-ά^όι (2.68) On differentiating (2.67) with respect to Θ we find that άι(θ) and α2(θ) obey the coupled equations 46 2. Exponential Operator 2.3 Lie Algebraic Similarity Transformations 47 ΐ ^ ά ι ( 0 ) = ωιάι(θ) + ξ+&2{θ), i ^ a 2(6l) = ω2ά2(θ) + ξ-άχ(θ). (2.69) These are easily solvable. Their solution may be used in conjunction with (2.66) and (2.17) to find similarity transformation of any SU(2) operator. In particular, for ωχ — ω2 = 0, (2.69) are solved by άχ(θ) = cos ( ι/ξ +ξ_0) αχ - i y ^ s i n a2, ά2φ) = cos «2 - (VC+C^9) αι· (2·70) 2. Verify that K+ =a)b\ K-=ba, Kz ='^{asa + tfb+ 1) (2.71) obey the commutation relations (2.52) of SU( 1,1). Consider the similar ity transformation Α(θ) = exp (ϊθΖχχ^ A exp ιθΖχχ^ , (2.72) Ζχχ = ωαα*α + LOb&b + ξ+a^V1 + ξ-ba. (2.73) It is straightforward to show that •^0(6») = ωαα(θ) + ξ+#(θ), ΐ ^ ( θ ) = - ω $ ( θ ) - ξ - ά ( θ ). (2.74) These are easily solvable. Their solution may be used in conjunction with (2.71) and (2.17) to find similarity transformation of any S?7(l, 1) operator. In particular, for ωα = = 0, (2.74) are solved by ά(θ) = cosh « - i y ^ s i n h (>/£+£_6») b\ b\6) = cosh (VC+C^61) ^ + i ^ s i n h a. 3. Ver i f y t h a t t h e SU( 1,1) commutation relations (2.52) are obeyed also by the bilinear combinations K+ = ^ - = y, K z = ^ a + 1/2) (2.76) of a boson operator with itself. Consider the similarity transformation (2.75) 48 2. Exponential Operator Α(θ) = exp A exp iOZ^j , (2.77) (2.78) It then follows that ί^ά(0) = ωαα(θ) + ξ+ά1(θ), = ~ ω^ ( θ) ~ ξ-ά{θ). (2.79) The s e a r e eas i l y sol vabl e. The i r s o l u t i o n f or ωα = 0 reads ά(θ) = cosh (2.80) The hamiltonians of a variety of important optical parametric processes are realizations of the elements of SU( m,n) [30]. 2.4 Disentangling an Exponential By disentangling an exponential we mean expressing the exponential of a sum of operators in terms of a product of the exponentials of operators. If A and B are given operators then the problem of disentanglement consists in finding operators C\, C2, ■ ■ · such that details). The expansion (2.81) in general involves an infinite number of C'ns. Finding analytical expression for the C'ns is generally a formidable task. How ever, as elaborated below, the number of C^s is finite if A and B are elements of a finite-dimensional Lie algebra. exp + B — exp exp exp (Ci J exp · ■ ■ (2-81) The Cn are combinations of repeated commutator of A and B (see [31] for Consider a Lie algebra generated by X\, ■ ■ ■, X n. Let exp Θ = exp [/i(0)Xi ■ · ■ exp |/n(0)X„] . (2.82) = 1 Note that MO) = o. (2.83) Now, differentiate (2.82) with respect to Θ and multiply the resulting ex pression on the right with the inverse of (2.82) (constructed using (1.26)) to get 2.4 Disentangling an Exponential 49 i=1 - f ^ X! + /2(0)exp [ΜΘ)Χ. +/n(6,){exp X'2 exp + ·· x exp f m x i ι{θ)Χη-ι ■ exp exp - Xn (2.84) On carrying the similarity transformations, the right hand side of this equa tion reduces to a linear combination of the X^s. A comparison of the coeffi cients of the X's in the resulting equation then leads to differential equations for the {fi{0)}. The equations so obtained are, in general, non-linear. Their solution determines {fi{0)}. We fol l ow t h e ge ne r a l pr oc e dur e o u t l i n e d h e r e t o t h e a l gebr as i n t r o d u c e d i n pr e vi ous s ec t i ons. 2.4.1 T h e H a r mo n i c Os c i l l a t o r A l g e b r a Let a, a) be the boson operators and let exp [θ {αχό, + a2ά^ά + α^ά^}] = exp [fi{9)a1] exp [/2(θ)ά*ά] exp [/3(0)ά] exp [/4]. (2.85) The procedure outlined above alongwith the results of Sect. 2.3.1 yield Λ — /1/2 = a 3, h = «2, h = ai exp {f 2) h ~ /1/3 exp ( -/2) = 0. (2.86) The solution of these equations reads fi = — [exp(a20) - 1], f 2 = α2θ, — [ e x p ( a 20) a2 h = — [exp (α2θ) - 1], a2 fi = —^ [exp (α2θ) - α2θ - 1]. (2.87) In particular, for a 2 that 0, 0 = 1, it follows by substituting (2.87) in (2.85) exp \a.ia + ά^] = exp [α^α^] exp [αιά] exp = exp [αιά] exp [α^ά^] exp --αχαΆ (2.88) The last line above is obtained by invoking (2.18), (2.17) and (2.31). In many applications, we need to expand a function F( a\ a) in powers of a and a* such that all the a operators lie on the right of all the at operators. Such an expansion is called normally ordered expansion. The expansion of F(a^, a) is said to be antinormally ordered if all the at operators appear on the right of all the a operators. Note that the first line of (2.88) is in normal ordered form whereas the second line there is its antinormally ordered form. Next we derive those forms for the exponential of a)a. To that end, let exp Ιθά^α] = V (2.89) L J ml m —0 where χ(θ) is an unknown function. To find that function, differentiate (2.89) with respect to Θ to get a*Sexp [ea'al = ^ V (2.90) L J d Θ ^ m\ K m = 0 Now, (i) rewrite t he exponential on t he left hand side using t he series expan sion (2.89), (ii) use (2.33) t o write aa)m = a‘[ma-\-rna[m~l, and (iii) compare the coefficients of a)m+1am+1 to arrive at « =*<«) + !. (2.9D Its solution for a;(0) = 0 reads χ(θ) = exp(0) - 1. (2.92) On substituting this in (2.89) we get exp [θα^α] = V (6XP(g) ~ at mam. (2.93) L J 777,! m—0 The ant i normal form of t he exponential of o'a may be derived by starting with the expansion TTL i /I \ exp \θαα)} = V (2.94) L ' ml m —0 Follow t he steps outlined above for determining t he normal ordered form of t he exponential of a) a and show that exp [0ά^ά] = exp(—Θ) ^ ---——ama^m. (2.95) m=0 2.4.2 The S U (2) Algebra Let the exponential of an element of SU(2) be disentangled in the form exp | $ {«+5+ + a zSz + a _ 5 _ | j = exp (/+(*)£+) exp (/2(0)&) exp (/_ (0 ) S_ ) = exp ( φ- ( θ) §- ^ exp ( - φ ζ(θ)ε>ζ^ exp (<£+ (0)S+) . (2.96) 50 2. Exponential Operator We outline below a derivation of the first equation above. The second equation follows by noting that the disentangling relation is a consequence of only the commutation relations and that if S± —> S+ then the commutation relations of (*?+,— Sz) are the same as those of (S ±,SZ). Hence </>±,2(α+, a_, az) = fzf,z(a-,a+, - a z). By following t he procedure outlined in t he beginning of this section we obtain non-linear differential equations /+ ~ f +f z~ /+/- exP[-fz] = ia+, (2.97a) fz + 2/+/- exp[—/z] = iaz, (2.97b) /_ e x p [ -/2] = i a_. (2.97c) Substitution of (2.97c) in (2.97a) and (2.97b) gives /+ - f + f z ~ i f l a - = ia+, (2.98a) f z + 2i f +a - = i az. (2.98b) Elimination of f z between these equations leads to the Riccati equation i/+ + a z/+ - « -/+ + « + = 0. (2.99) Follow the standard method for solving a Riccati equation or verify by direct substitution that (2.99), with /+(0) = 0, is solved by /+ = Ψ r r m ’ (2·100) 1 1 cos(l ιθ) — i az sm( l ι θ)/211 Γ2=α+α_ + °ί. (2.101) The function f z can now be determined by combining (2.100) and (2.98b). The solution of the resulting equation yields 2.4 Disentangling an Exponential 51 f z = - 2 In coδ(Γιθ) -\— sin(r16>) 111 ( 2.102) The expression for obtained by combining (2.102) and (2.97c), reads - i a - sin(r i2.103) Γ χ 0 0 8( 7^1 0) — i a z s i n ( r i f f )/2 Γ ± 2.4.3 S U ( 1,1 ) A l g e b r a L e t t h e e x p o n e n t i a l o f a n e l e m e n t o f SU (1,1) be disentangled in the form exp l^a+K+ + azKz + = exp[0+ ( 0) ii'+] exp[φζ(θ)Κζ\ exp[0_(6*)ii_]. (2.104) The procedure outlined in the beginning of this section leads to following non-linear differential equations for unknown functions φ±(θ),φζ(θ): φ+ - φ+φζ + φ2+φ - exp[-0 2] = α+, φζ - 2Φ+Φ- exp[~φζ\ = 0ίζ, φ- exp[—φζ] = α-. (2.105) Following the method outlined in the last subsection we find that φ+ — φ2+α _ — φ+αζ — a + = 0. (2.106) The solution of this equation and its use in solving the equations for φζ and φ- yields a + sinh(/2 0 ) 52 2. Exponential Operator Φ+ = Γ2 cosh(r29) - az ύτύι(Γ2θ)/2Γ2 ’ = ~ 21n a- az cosh(Γ2Θ) - — sinh(Γ2Θ) 2 sinh(T2 0 ) Γ2 cosh(Γ2Θ) - az sinh(J2 0 )/2 r 2 ’ r 2 a l (2.107) (2.108) 2.5 T i me - Or d e r e d E x p o n e n t i a l I n t e g r a l We have s een i n t h e l a s t c h a p t e r t h a t t h e s t u d y o f t h e mot i on of a q u a n t u m s ys t e m gover ned by a t i me - de pe nde nt h a mi l t o n i a n r equi r e s e va l ua t i on of a t i me - o r d e r e d e xpo n e n t i a l i nt egr a l. I f Χχ, ■ ■ ■, X n generate a Lie algebra then we may write Ψ = exp I . dr ai{T )Xj • exp (f n(t )Xn) ( 2.109) On di f f e r e n t i a t i n g t h i s wi t h r e s pe c t t o t and on multiplying the resulting expression on the right by the inverse of (2.109) (constructed using (1.26)) we arrive essentially at the equation (2.84) (with Θ there identified for the present as t) except that whereas the the a's appearing there are independent of the differentiation parameter Θ, they are now dependent on the differentiation parameter t. Explicit results for some algebras of interest are derived below. 2.5.1 Harmonic Oscillator Algebra In this case, let 2.5 Time-Ordered Exponential Integral 53 Ψ exp / dr {αι (τ )ά + α2(τ)ά^ά + α3 (τ)ά^} .Jo = exP [fi (t )af ] exp [f2(t)a]a\ exp [/3 (ί)ά] exp[/4 (i)]. (2.110) As argued above, the s are determined by solving (2.86) but the a j ( i ) ’s appearing there are now time-dependent. The solution of those equations is, therefore, no longer given by (2.87). It may be shown that the solution of (2.8 6 ) for time-dependent s is h(t) = [ d r' Jo exp a 2(r ) dr α 3 (τ), f 2 (t) = / a 2(r)dr, r1 J 0 h{ t ) = [ a\ (T) exp[/2 (T)]dr, f 4(t) = [ /i ( r ) a 1 (r)dr. (2.1 1 1 ) Jo Jo These may be evaluated for the a(t)'s for the problem at hand. The expres sions (2.111), of course, reduce to (2.87) if the a's are time-independent. 2.5.2 S U( 2) Algebra Let the time-ordered integral involving an element of SU(2) be expressed as exp i J d r | α + (τ)S+ + az (r)Sz + a _ ( r ) 5 -1 | exp ^/+(ί)5+ exp \j z {t)Sz exp _ (2.112) The equations obeyed by f ±(t ) and f z (t) are the same as (2.97a)-(2.97c) but with time-dependent ai(t)'s. However, note that the steps leading to (2.99) are not influenced by the time-dependence of the a's. Hence /+ even in the present circumstance obeys (2.99) but now with time-dependent coefficients. That equation can not be solved in general. It can be handled analytically or numerically depending on the a(t)'s for the problem at hand. 2.5.3 The SU( 1,1) Algebra Let the time-ordered integral of the exponential of an element of SU( l, 1) be expressed as rt exp = exp i J dτ ^ α +( τ ) Κ+ + αζ( τ ) Κζ + ^ + ( i ) i i + e xp ψζ(ί)Κζ e x p j φ-(ί)Κ. (2.113) Th e φ±( ί ),φζ(ί) obey (2.105) but the α»(ί);s are now time-dependent. The φ+, however, still obeys (2.106) which can not be solved analytically for general time-dependence of the ai(t)'s. It can be handled analytically or numerically depending on the a(t)'s for the problem at hand. 3. Representations of Some Lie Algebras Comparison of the quantum theory with experiments is made by evaluating expectation values of observables. Evaluation of expectation values is carried in a c-number representation of the vector space of the states of the system by choosing a suitable basis. A basis that immediately suggests itself for in vestigating time-evolution of a system is the one spanned by the eigenvectors of the hamiltonian governing its motion. As has been pointed out in the last chapter, hamiltonians encountered frequently in quantum optics can be clas sified as elements of the Lie algebras or of their direct products. Hence, the problem of representation of a quantum optical system in terms of the eigen vectors of its hamiltonian reduces to that of finding eigenvectors of hermitian elements of the Lie algebras introduced in the last chapter. However, it will be seen that there are other bases which prove to be of not only mathematical interest but also of importance in understanding various physics aspects. In this chapter we address the question of constructing representations of some Lie algebras of interest in quantum optics. 3.1 Representation by Eigenvectors and Group Parameters The bases of a Lie algebra may be classified as (i) the bases constituted by the eigenvectors of the elements of the algebra, (ii) the bases labeled by the parameters characterizing the associated group to be defined in the sequel. 3.1.1 Bases Constituted by Eigenvectors The most convenient bases are, of course, the ones formed by complete sets of orthonormal vectors. Since the eigenvectors of a hermitian operator constitute a complete orthonormal set of vectors, a way of constructing an orthonormal basis of an algebra is in terms of the eigenvectors of a hermitian generator of the algebra. Consider the eigenvalue equation Χ|λ) = λ|λ), X = X i (3.1) for a hermitian element X in the algebra. The set of its eigenvectors (|λ)} constitutes an orthonormal basis. We can construct different eigenbases by 56 3. Representations of Some Lie Algebras solving (3.1) for different X in the algebra. However, not all the sets so obtained are inequivalent in the sense discussed below. On operating (3.1) with a unitary operator U it follows that ( ϋ χ ϋ ^ Ε/|λ) = λί7|λ). (3.2) From this we infer that C/|A) is an eigenvector of the hermitian operator UXUt with the same eigenvalue λ. Now, let U be given by £ ( { « } ) = exp j i ^ «;*;j, = *}, (3-3) wh e r e { α } ξ ( a i,..., a n) is a set of real constants and {Xj } are the genera tors of the given algebra. It is not difficult to show that the set of all U ({a}), parameterized by the set of real numbers {a}, constitute a group. It is called the Lie group associated with the given Lie algebra. In particular, we note the group property 17 ({a}) 17 ({a'}) = l/( { a >,a') } ) (3.4) where a” (a, a') is a function of a and a'. The relation (3.4) states that the product of two group elements is also an element of the group. Let us now reexamine (3.1) and (3.2). The operator U {{a}) X U 1* ({a}) in (3.2), being a similarity transform of X belongs, as discussed in Chap. 2, to the algebra of X. The operators X and U ({a})XC/t ({a}), related by a unitary transformation in the associated group, are said to be unitarily equivalent. Their corresponding eigenstates {|λ)} and {ί/|λ)} constitute uni tarily equivalent representations. Unitarily inequivalent bases are constructed by solving the eigenvalue problem for unitarily inequivalent generators of the algebra. Another instructive way of representing an algebra, discussed next, is in terms of the bases labeled by the group parameters {a}. 3.1.2 Bases Labeled by Group Parameters Let |Vo) be a vector in the space in which the elements of the Lie algebra generated by {Xi } act. We show that the states IVO ({«})) = exp E a'-V' ι \ψ0) = ύ ( { α } )\ψ0), X j = x ), (3.5) j =i constitute a complete set as a function of {a}. To that end, consider the operator A = = J d μ({α}) I t o ({a})) (Φο ({«}) | (3.6) 3.1 Representation by Eigenvectors and Group Parameters 57 Here Λμ( {(>:}) is an invariant measure on the parameter space of the group [32]. Now, (i) rewrite |·0ο ({α })) in (3-6) as in (3.5), (ii) carry the similarity trans formation of the resulting expression by U({a'}), and (iii) use (3.4) to arrive at Next, transform the variables {a} of integration in this to new variables {a"}. On using the fact that the measure is an invariant of the group, i.e. d/x({a}) = άμ({α"}), the integral in (3.7) reduces to the one in (3.6), so that determined. This constant is fixed by taking the expectation value of (3.6) On redefining the measure by suitably scaling it by the constant c, emerges the completeness relation for the states I'i/'ol'^}) parameterized by {a}. However, not all the a's need be essential for characterizing the state \Ma } ) · F°r > 'f the fiducial state |ϊ/?ο) is an eigenstate of some of the genera tors, say, X m, X m+i,... then those generators must commute and thus form a subalgebra, say, h. The group generated by the operators in h is called the stability group or stationary group of |i/o)· The transformation U ({a}) can then be written as a product of the stability transformation and the transformation generated by the elements not belonging to h. The stability transformation, acting on \ψο), gives rise to only a phase factor. Hence the variables am, am+\.. associated with those transformations do not play any essential role. The states (3.5), without those operators in i7({a}) of which |·0θ) is an eigenstate, are called the Perelomov coherent states or the gen eralized coherent states [33]. Note also that, by virtue of the group property (3.4), the action of a group element on the coherent state results in another coherent state. Using the resolution of identity, (3.10), any state \Ψ) can be expanded in terms of the generalized coherent states as U ({a'})ii7t({a'}) = J dM{a})[i7(V ({«},{«'})) X |V'(0))(^(0)|i7t ( {a"( {a},{a'}) j . (3.7) i 7 ( { a } ) i t/t ( { a } ) = i, (3.8) i.e. ( { a } ), A = 0. Since i7({a}) is an arbitrary element of an irreducible representation of the group, it follows from the Schur’s Lemma that A must be a constant multiple of unity, i.e. A = cl where c is a constant to be (with A = cl) in a state \Φ) so that (3.9) (3.10) 58 3. Representations of Some Lie Algebras W) = J άμ({α})\ψ0 ({a}))(V>0 ({<*}) |#}· (3.11) The function ({α}) \Ψ) provides a representation of the state l^) in terms of the generalized coherent states. The set of generalized coherent states is, in fact, overcomplete in the sense that it contains subsets which are complete (see [34] for details). Perelomov’s definition generalizes the concept of coherent states. The con cept that it seeks to generalize is, in fact, of the Glauber coherent states (see Sect. 3.2) introduced by Glauber [35] as those states of the electromagnetic field which give maximum contrast in a two-slit interference pattern (see Chap. 6 ). These states and their generalizations have since played a signifi cant role in diverse fields [34, 36]. The guiding characteristics for identifying similar states of other systems are that (i) the Glauber coherent state |a) is labeled by a continuously varying parameter a, (ii) the scalar product (β\α) is continuous as a function of the labels of the states, (iii) the set of states |a) is complete, (iv) ja) is an eigenstate of the harmonic oscillator (h.o.) an nihilation operator (see (3.24)), (v) it is generated by the h.o. unitary group transformation on the vacuum state (see (3.27)) and, (vi) it is an uncorre lated equal variance minimum uncertainty state (UEVMUS)(as defined circa (1.69)) of (q, p). T h e f e a t u r e s ( i ) - ( i i i ) c ha r a c t e r i z e a c o h e r e n t s t a t e of a ny s ys t em. Accor d i ngl y, mi ni mum r e qui r e me nt s for a s t a t e | {z}) o f a n y s ys t e m t o be cal l ed a c o h e r e n t s t a t e a r e t h a t {z} be a set of continuously varying parameter; ({z'jllz}) be continuous as a function of the labels of the states and; the set of states |{z}) as a function of {z} should be complete. While there may be more than one set of states with these features, the tag ’coherent’ can be pinned only to a set possessing some specified properties. For example, as we will see in Sect. 3.2, there are more than one complete sets of continuously labeled h.o. states but the name coherent is given to the set of states in which the two-slit interference pattern exhibits maximum contrast. However, in the absence of a knowledge of any similar property for other systems, the prop erties defining the coherent states has been a debatable issue. In the case of systems describable by the hamiltonians belonging to one or the other Lie algebra, the definition of coherent states is extended by generalizing to other systems one or the other methods (iv)-(vi) of construction of the Glauber coherent states. Those methods, though equivalent for constructing h.o. co herent states, are not so for other systems. For an extension of the concept of coherent states to a general one-dimensional potential, see [37]. In the following we restrict our attention to the coherent states of the Lie groups. The generalization based on (iv) consists in defining a coherent state as an eigenstate of an annihilation operator of the algebra. This approach, however, is restrictive as not all the algebras contain an operator whose eigenvalue problem is solved by continuously labeled states. As an example, note that the only solution of the eigenvalue equation S_ \'ψ) = 0 of the SU(2 ) algebra is 3.1 Representation by Eigenvectors and Group Parameters 59 \φ) = 15, —S) (see Sect. 3.3 for the notation). On the other hand, as we will see in Sect. 3.4, the S U(1,1 ) annihilation operator K_ does admit continuously labeled eigenstates. However, as discussed in that section, those solutions do not possess analog of the group property (v). Perelomov’s definition of a coherent state, as given above, is clearly a generalization of the property (v). It holds for any algebra but ignores (vi). The property (vi) contains in it the essential physics of the coherent states. For, the notion of minimum uncertainty states is associated with the states in which a system behaves closest to its classical counterpart. Search for such states dates back to the early days of quantum mechanics. Schrodinger [38] labeled the states whose wave-packets do not spread in time under the h.o. potential as the most classical states of the h.o. Those are nothing but the Glauber coherent states. The coherent states of the systems described by the Lie algebras and possessing the property analogous to (vi) may be constructed by restricting the choice of the fiducial state in Perelomov’s definition to minimum uncertainty states of the operators in the algebra in a way described below (see also [39]-[41]). This approach unifies the properties (iv)-(vi) of the h.o. coherent states. In view of preceding delibrationns, let |-0o) in (3-5) be a minimum uncer tainty state of a pair of non-commuting hermitian generators, say, Χχ, X 2 in the given algebra. Recall from (1.66) that a minimum uncertainty state l^o) of Xi, X 2 solves Χ1+ϊλΧ2]\φ0)=ζ\φ0). (3.12) It is straightforward to verify by operating (3.12) by U ({a}) that the gen eralized coherent state (3.5) is a minimum uncertainty state of the pair ϋ ({a}) Xi f f l ({a}), U ({a}) X 2U^ ({a})· The coherent states (3.5) obtained with \ψ0) as a solution of (3.12) may, therefore, be called minimum uncer tainty coherent states (MUCS). Recall from discussion circa (1.69) that if λ = ± 1 then Xi, X 2 are uncorrelated and their variances A X\ and A X 2 are equal. The MUCS corresponding to λ = ± 1 may, therefore, be named uncor- related equal variance minimum uncertainty coherent states (UEVMUCS). We can thus construct different sets of MUCS by choosing the minimum uncertainty states corresponding to different pairs of non-commuting gen erators as the fiducial states. However, not all such sets are unitarily in equivalent. For, as remarked above, if \ψ0) is a minimum uncertainty state of the pair Χ 2^ then ϋ\ψο) is a minimum uncertainty state of the pair ( ύ Χ ι ί π, I I X2U^. Thus the set of MUCS obtained by choosing the minimum uncertainty states of a pair of generators (Xi, X 2^j as the fiducial state is unitarily equivalent to that constructed by choosing the minimum un certainty states of the pair of generators ( ϋ Χ χ Φ, IJX2I J ^ as the fiducial state. Hence, a pair of generators ( X i, X2J is said to be unitarily equivalent 60 3. Representations of Some Lie Algebras to another pair ( x,, X j j if there exists a unitary transformation U in the associated group such that (Xi, X 2^ = ( ϋ Χ ί ϋ ^, t j X j i j ^, i.e. if Xi and Xj are unitarily equivalent, respectively, to Χχ and X 2 by the same unitary transformation in the associated group. A complete set of MUCS of a given algebra consists of all unitarily inequivalent MUCS. In the following we construct inequivalent representations of some algebras of interest in quantum optics. 3.2 Representations of Harmonic Oscillator Algebra Recall from Chap. 2 that the harmonic oscillator algebra is generated by the operators {a,a) ,a) a,I } obeying the commutation relations (2.26). Let q= ^=(a + a)) , p = - a) , [q,p} = i. (3.13) The set (q,p, a^a, I) constitutes hermitian generators of the h.o. algebra. Now, by inspecting (2.30) it may be verified that there does not exist any h.o. group transformation that can transform any linear combination of (q,p) to a)a. The set (q,p) and the operator a)a are therefore unitarily inequivalent. Bearing this in mind we construct below the orthonormal bases and the MUCS of the harmonic oscillator algebra. 3.2.1 Orthonormal Bases As a consequence of the observation above, this algebra admits two sets of unitarily inequivalent bases: one formed by the eigenstates of at a and the other by the eigenstates of q or of p. E i g e n s t a t e s o f af a. Recall that the eigenstates |n) of a)a, called the number or Fock states, are such that a) a\n) = n\n), n = 0,1,2,... (3-14) The completeness and orthonormality relations for these states are OO Σ\η)(η\ = Ι, {m\n)=6mn. (3.15) 71 = 0 Using the commutation relations, it is straightforward to show that a\n) = y/n\n — 1 ), a)\n) = Vn + 1| n + 1). (3.16) These, on repeated application, yield = ^™|n) = y^±^|n + m>. (3.17) 3.2 Representations of Harmonic Oscillator Algebra 61 The operator a (a^) thus lowers (raises) the number state and is, therefore, called lowering or annihilation (raising or creation) operator. Note that ά|0 ) = 0. (3.18) The state |0) is called the vacuum state. In applications, it is at times useful to know the expression of the product \m)(n\ in terms of the h.o. operators. We derive the desired expression first for |0)(0|. To that end, invoke (1.35) to note that Now, express the exponential operator above in the normal ordered form using (2.93) so that This expresses |0)(0| in terms of normally-ordered h.o. operators. Now, (3.17) implies that |m)(n| = _ L = a t m|0 )(0 |fi". (3.21) vm!n! Combination of this with (3.20) expresses \m)(n\ in terms of normally ordered product of the h.o. operators. Eigenstates of q. The other set of unitarily inequivalent orthonormal basis is formed, as stated above, by the eigenstates of q or of p. The eigenstates of q follow as a special case of the eigenvalue equation (3.42) solved in the next subsection. 3.2.2 Minimum Uncertainty Coherent States The inequivalent pairs of generators of inequivalent sets of minimum un certainty states of harmonic oscillator are evidently (q,p), and (q, ά^ά) (or (p, «t«)). The corresponding minimum uncertainty states are constructed as follows. MUCS for the Pair (q,p ). The MUCS for the pair (q,p) are given by substituting for \ψ) in OC m—0 |0) ( 0|. ( 3.19) OO m ^ ( - ) ( 3.20) | {/3}} = e xp [~ϊ {βι α)α + β1ά + β 2α)}] \φ), (βχ real) the solution |λ, a) of the equation for the MUS of q,p: ( 3.2 2 ) 62 3. Representations of Some Lie Algebras [q + ϊλρ] |λ, a) = α\/2|λ, a) (3.23) and by dropping from the exponent in (3.22) the operators of which |A, a) is an eigenstate. We construct first the UEVMUS corresponding to A = 1. On invoking the definition (3.13), (3.23) for A = 1 reduces to the eigenvalue equation (|α) ξ |1,α)) α\α) = a\a) (3.24) for the h.o. annihilation operator. Using (2.31), this may be rewritten as D(a)aD\a)\a) = 0. (3.25) where the unitary displacement operator D(a) is defined by D(a) = exp(aa^ — a* a). (3.26) On comparing (3.25) with (3.18), it follows that D^(a)\a) = |0 ) i.e. |α)=£>(α)|0>. (3.27) Note that |a) has the structure of a Perelomov coherent state, for, it is ex pressible as a result of transformation of a state, in this case the vacuum state |0), by the operator D(a) which is a unitary group element, not containing the generator a*a of which |0) is an eigenstate. The substitution of (3.27) in (3.22) leads to, by virtue of the group property, the state of the same form. The UEVMUCS (3.27) is the so called Glauber or h.o. coherent state. On disentangling the exponential in the normal ordered form by employing (2.88), and on using (3.17), (3.27) leads to the expansion OO γγ^ I a) = exp ( ~ M 7 2) Σ (3·28) z' Vm! 771 — 0 of the coherent state in terms of the number states. The expression (3.10) for the resolution of unity in this case assumes the form ^ J d2a\a) ( a\= I. (3.29) Here the integration extends over the whole complex plane. This result may be verified (i) by using in its left hand side the expansion (3.28), (ii) on carrying the integration using (A.25), and (iii) on invoking the completeness of the number states. A useful expression for the trace of an operator A is obtained by operating (3.29) with A and carrying the operation of trace. On applying (1.30), we find that T r ( i ) = J d2a(a\A\a). (3.30) Note that, by virtue of the definition (3.24), (a\a)man\a) = a*man. (3.31) Hence, if it is given that (a\A\a) = J 2 c mna*man (3.32) m,n then the operator form of A would read A = Y j cmna'man. (3.33) m,n Now, as a consequence of (3.29), any state \φ) in the space of the h.o. may be expressed as |ip) = — J d2 a\a) (α\φ). (3.34) The complex number (a\ip) represents |φ) in the basis of the coherent states. On using (3.28) for \a), we see that °° (a |φ) = exp ( - | a |2/2) ^ = exP (“ M 2/2) V’i»*)· (3·35) The factor exp (—| a |2/2) is only a normalization factor. The state |φ) may thus be represented by the function φ(α*): 3.2 Re pr e s e nt at i ons o f Har moni c Os c i l l at or Al gebra 63 OO a φ(α*) = (α\φ), \a) = ^ ~/=,\m)· (3·36) This implies that (τη\φ) = - ^ = ^ —φ{ζ) (3.37) Vm! dzm z=o Now, on taking the matrix element of (3.29) in the states |φ) and |φ), the scalar product between the states may be expressed as (φ\φ) = — J ά2α(φ\α)(α\φ) = — J d2a exp (— \a\2) φ(α)φ(α*). (3.38) This shows that |φ) is normalizable if — J d2a exp (—|α|2) \φ(α)\2 < oo. (3.39) By using the definition (3.36) of |a), show that ( a | a f = a*(a\, ( a | a = ^ ( a |. (3.40) Accordingly, if Φ (ά^,ά) is a function of the h.o. operators then (α|Φ (af, a) |φ) = φ ( α *, φ(α*). (3.41) Note that this requires the existence of all derivatives of φ(α*). That is en sured by the requirement that φ(α*) be an entire function. Thus, in the basis 64 3. Representations of Some Lie Algebras of the coherent states, a state vector is represented by an entire function, normalizable in the sense of (3.39). The action of an operator on a vector is represented as a differential operator on the corresponding function. This is known as the function-space or Bargmann representation [42], We have thus constructed the UEVMUCS by using the solution of (3.23) for A = 1. We will see that the UEVMUCS corresponding to λ = —1 do not exist. Next, we construct the minimum uncertainty coherent states of a har monic oscillator corresponding to the solution of (3.23) for λ φ 1. That equation, rewritten in terms of a, (by suitably redefining β) reads [(1 + A)« + (1 - A)af] |λ, β) = Τ2/3|λ, β). (3.42) On applying (3.41), (3.42) assumes the form (1 + λ)^ + (1 - A)z (ζ·\λ,β) = ν2β(ζηλ,β). (3.43) This is evidently solved by (ζ*|λ, β) = De xp 1 - λ 2 , ν ϊ β -z -z 2(1 + A) 1 + A (3.44) with D as the normalization constant. Now, apply (A.29) to show that (ζ*\λ,β) would satisfy the normalizability condition (3.39) if Re (A) > 0. (3.45) This implies in particular that (3.42) does not admit normalizable solution if A = —1, i.e. at does not admit normalizable right eigenvectors. If (3.45) holds then an application of (A.29) shows that 1 \D\ ~ J d2al(z*|A,/3)|2 x exp [(-β 2(1 - A*)/{2(1 + A)(A + A*)}) + c.c.] . (3.46) This determines the constant D. Now, by using (A.33), (3.44) may be ex pressed in terms of the Hermite polynomials as 1 — A \m/2rr ( β <*·λ « = ° Σ ^ ( ^ ) <3 · 47> On c ombi ni ng t h i s wi t h ( 3.37) i t fol l ows t h a t t h e MUCS s at i s f yi ng ( 3.42) i s gi ven i n t h e n u mb e r s t a t e r e p r e s e n t a t i o n by i *.« - ( ^ Γ |m)' <3-48) 3.2 Representations of Harmonic Oscillator Algebra 65 This is normalizable, of course, only if (3.45) holds. Let us now examine the case Re(A) = 0 which separates the admissible and inadmissible regions. For the sake of illustration, we let A = i and evaluate (λβ'\Χ,β) using (3.38). Write the integration variable a as x + \y and carry out the integration over x to obtain (λ, β'\\, β) ~ Γ dye xp\V2i yt f - β'*)}. (3.49) J —OO Now, if the /3’s are complex then (3.49) diverges whereas it reduces to a delta function if the /3’s are real. Hence, the solution of (3.42) is (delta-function) normalizable for Re(A) = 0 if β is real. That is, of coursc, as it should be because for Re(A) = 0, the eigenvalue equation (3.42) becomes an equation for a hermitian operator whose eigenvalues ought to be real. An instructive form of the solution of (3.42) is obtained by rewriting it as α(ξ)\ξ,α) =α\ξ,α), ξ = exp(i0)|£|, (3.50) α(ξ) = cosh(|£|)a + exp(i^) sinh(|£|)at. (3.51) On comparing the forms (3.50) and (3.42) we note that U ifh /( ΐ + λ)(ΐ + λ*) cosh(iei) = W 2(A + A.} ■ exp(i^) sinh(|£|) = ( 1 — A)<' ^ ) 2(1 + A)(A + A*): S(i) = exp a = β\ , (3.52) MY ( 1 + A)(A + A*) v ; By employing (2.80), (3.50) may be rewritten as α&(ξ)\ξ,α)=α&(ξ)\ξ,α) (3.53) where 5(ξ), called the squeezing operator, is defined by i ( r « 2 - e « t2) . (3.54) Comparison of (3.53) and (3.24) implies that \ξ,α) =Ξ(ξ)\α) =S(OD(a)\0). (3.55) Note that 5(£) is an element, not of the h.o., but of the S U(1,1) group. The meaning of the term squeezing is clarified below. For a detailed discussion of the properties of these states and the quantum optical processes for their generation, see [43, 44], It is instructive to examine the probability pm that the state |£, a) has excitation number m: 66 3. Representations of Some Lie Algebras Pm = |(m|£,a}|2. (3.56) This may be evaluated by using (3.48) and the relations (3.52) between (Λ, β) and (ξ,α). In Fig. 3.1 we have plotted prn as a function of m for a = 6, sinh(|£|) = 3.5. Notice the oscillatory behaviour of pm. We compare it with Fig. 3.1. Plot of Pm as a function of m for the oscillator in the squeezed state (3.55) with sinh(|£|) = 3.5, a = 6 (solid curve). Long dashed curve is for the coherent state with a — 6. Short dashed curve is for the thermal state with n = 36. pm = |(m |a) | 2 in the coherent state and with pm = (m\pt\l\'m) in the state of the oscillator in equilibrium with a bath at temperature T described by Pth = βχρ(-/3ά+ά)/ΤΓ[βχρ(-/3«+ά)], β = Τιω0/k BT, (3.57) ke being the Boltzmann constant and ωο the frequency of the oscillator. We also define Ti exp(— β) = (3.58) n + 1 Long dashes in t he Fig. 3.1 are for t he coherent s t at e | a) with a = 6 whereas small dashes are for the thermal state corresponding to ή = 36. The pm for the coherent state is a Poissonian centered at | a| 2 = 36. The pm is a mono- tonically decreasing function for the thermal state. The oscillatory behaviour of pm for the squeezed state is a signature of its non-classicality [45] in the sense explained in Chap. 4. The operator averages in state |ξ, a) may be found by noting that ( £,a | F ( a,a t ) | £,a ) = ( a\^ ) F ( a,a ^ )\a ) = < a | F ( 0 ( 0,a t ( 0 ) l « ). (3-59) 3.2 Representations of Harmonic Oscillator Algebra 67 the ά(ξ) being given by (3.51) with ξ —> —ξ. The matrix element in the last A coherent state corresponds to ξ = 0. Hence, in a coherent state, A X 2 (Θ) = 1/2. A state in which Δ Χ 2(Θ) < 1/2 for some Θ is called a squeezed state. We will see in Chap. 4 that a squeezed state is non-classical in the sense explained there. On examining (3.62) we note that it is posible to have Δ Χ 2(Θ) < 1/2 in the MUS |£, a). For example, if 2θ + φ = 0 then Δ Χ 2(Θ) = exp(—2|£|)/2 < The concept of squeezing introduced above is based on an application of the uncertainty relation. This concept, however, arises in the context of search for minimizing error in a process of measurement imposed by the quantum theory. For, the quantum theory assigns, through the uncertainty relation, an inherent error to the measured value of an observable. This error is independent of the one contributed by external factors like those due to the limitations of the apparatus and the environment. It is, therefore, im perative to know how to minimize the intrinsic quantum imprecision in a process of measurement. By analyzing some simple measurement processes it has been shown that if a process of measurement by means of a h.o. is carried by finding average of an observable Χ(θ) defined in (3.61) then the fundamental quantum-theoretic error in the measurement of that average is related directly with Δ Χ 2(Θ) [43, 46]. Hence, smaller the Δ Χ 2(Θ), better is the sensitivity of measurement. Since Δ Χ 2(Θ) in a squeezed state is smaller than that in a coherent state, the squeezed state provides better precision in measurement than that in a coherent state. For detailed examples and comparative estimates of errors in various states, see [46]. MUS for t h e P a i r (p, ά^α). Next we construct the minimum uncertainty states for the pair (ρ,αϊα) which is equivalent to the pair (q,a*a). The equa tion (3.12) then reads line above may be evaluated by expressing F in normal-ordered form in a, ol*. For example, the average occupation number in the state |£, a) is (ά^ά) = (a| {cosh(|£|)at — exp(—\φ) sinh(|£|)a} x {cosh(|£|)« — exp(i</>) sinh(|£|)«t} |a) = \a\2 {cosh2(|£|) + sinh2(|£|)} +sinh2(|£|) - cosh(|£|) sinh(|£|) {exp(—i ^ ) a2 + c.c.} (3.60) Squeezed S ta t e s of Harmonic Oscillator. Let (3.61) Then [Χ( θ + π/2), X(0)] = i. Using (3.59), verify that 2cosh(|£|)sinh(|£|)cos(0 + 2Θ) . (3.62) 1/2. 68 3. Representations of Some Lie Algebras [α^α + iAp] |ip) = a\ip). (3.63) By using (2.31) this may be rewritten as A2' exp (Aq) a t a + y βχ ρ ( - Aq) \ip) = a\ip), (3.64) which is solved by 1 1pm) = exp (A q) \ to). (3.65) We leave it to the reader to check that substitution of this for \ip) in (3.22) results in [{/3}) in the form Σ>(β,λ )[m) where Ι)(β 7 Χ) is an exponent of a linear combination of a, . The operator exp (Aq) in (3.65), and consequently Ό(β,λ), are unitary if A = ±i. The state | ipm) is then of the form Σ)(β)\τη). These states were investigated in [47] as a generalization of the Glauber coherent states. We have thus constructed all the unitarily inequivalent MUCS of the harmonic oscillator algebra. Next, we construct such states for the SU(2) algebra. 3.3 Representations of S U (2) Recall from chapter 2 that the SU{2) algebra is generated by the operators (sx, Sy, sjj ξ S which obey the commutation relations (1.126). It may be verified by inspecting (2.41) that these operators can be transformed to each other by an SU(2) unitary group element. Hence all the generators of this algebra are unitarily equivalent. In this case, therefore, the orthonormal representations corresponding to the eigenstates of any of the three operators are equivalent and so are the continuous bases generated by any pair of generators. 3.3.1 Orthonormal Representation An S U(n) algebra admits η — 1 operators, called the Casimir operators, which commute with all the operators of that algebra. For the SU(2) there is one Casimir operator. It is the total spin operator S 2 = S 2 + S 2 + S 2Z. (3.66) The vector space of the algebra is, therefore, reducible to a sum of the sub spaces each characterized by an eigenvalue of S 2. The eigenvalues of S 2 are known to be given by S( S + 1) where S — 1/2,1,3/2,... and the eigenvalues of any hermitian S U(2) operator, say Sz, are given by m = —S, - 5 + 1,..., 5. Hence, simultaneous eigenstates |5,m) of 52 and Sz defined by S2\S,m) = S(S + l)\S,m), 5 = 1/2,1,... Sz\S,m) = m\S,m), m = — 5, — S + 1,..., 5 (3.67) constitute a basis for the 25 + 1 dimensional space of the spin states. The set of states |5, m) is complete and orthonormal: s y, \S,m)(S,m\ = I, (S,m\S,n) = Smn. (3.68) m — — S Consider now the operators S± defined in (1.127). By expressing Sx, Sy in terms of S± and, by using (3.67) to write S 2 = 5 ( 5 + 1), (3.66) reads 5 ( 5 + 1) = ^ 5+5_ + 5_5+ + S 2 = 5+5_ - Sz + S 2Z. (3.69) Next, using the commutation relations (1.128) of S± with Sz, verify that 3.3 Representations of SU( 2) 69 5 + | S', m) = \/( S — m) ( S + m + 1) 15, m + 1), 5_ 15, m) = -\/(5 + m)(5 — to + 1)|5, m — 1). (3.70) The operators 5+ thus act as raising and lowering operators of the eigenstates of Sz. By repeated application of (3.70), we find that c f c i o \ /( 5 — m )!( 5 + to + fc!) 5 + | S ·"*> = V ( S — m — fc)!(S + m )! IS'm + fc>· g | = + (3.71) ; y (5 + m - f c )!( 5 - m )!' The action of 5X, Sy on 15, to) is determined by expressing them in terms of 5±. Note, in particular, that 5+|5, 5) = 0, 5_|5, - 5 ) = 0, S ^ +1 = 0. (3.72) As stated before, there is no other unitarily inequivalent set of orthonormal states in this algebra. The eigenstates of any hermitian operator in the algebra can be found by constructing the transformation coupling it with Sz. We have a l r e a d y e nc ount e r e d i n Cha p. 1 t h e ca s e of a s pi n- 1/2, i.e. of a s pi n of t o t a l s pi n q u a n t u m numbe r 5 = 1/2. We wi l l have occas i ons t o de a l al so wi t h a c ol l ec t i on of N spin-l/2s. Such a collection is described by the collective spin operators N 0μ = Σ Ψ (3-73) i= 1 where Sμ 1 is the μί,Ί component of the spin-1/2 labeled i. It is straight forward to verify that (3.73) obey the commutation relations (1.126). The state space of such a system is evidently spanned by \mi, τη2, ■ ■ ■, tojv), mi, m 2,... ,mjv = ±1/2. The set of these states is reducible to a sum of sets of states such that the states in each set transform amongst each other. Each such set is characterized by a total spin quantum number 5 capable of assuming the values 5 = N/2, N/2 — 1,..., 0 or 1/2 depending on whether N is even or odd. The states in a set, characterized by 5, may thus be labeled by IS1, m) (m = —5, —S + 1,..., 5). Next we construct the minimum uncertainty states of SU(2). 70 3. Re pr e s e nt a t i o ns o f Some Li e Al ge br as 3.3.2 Mi n i m u m U n c e r t a i n t y C o h e r e n t S t a t e s As me nt i one d above, t h e r e is onl y one pa i r - c l a s s i n t h i s case whi ch i s ge ne r a t e d by any t wo, s ay (Sx,Sy), of the three generators. Its minimum uncer tainty coherent states are constructed by substituting for |ψ) in I M ) exp i -(- fJjySy “I- μζ Sz ^ the solution |A, z) of the equation Sx +iXSy \\,z) = z\\,z) (3.74) (3.75) and by dropping from the exponent in (3.74) the operators of which | A, z) is an eigenstate. The equation (3.75) determines the minimum uncertainty states of the pair Sx,Sy. Cons i de r f i r s t t h e c as e of UEVMUS c or r e s pondi ng t o λ = ±1. In this case, (3.75) assumes the form 5±|±, z) = z\±, z). (3.76) By virtue of (3.70) and (3.72), it is solvable only if z = 0 with |±, 0) = \S, ±5). By substituting this solution in (3.74) and by dropping from the exponent the operator Sz of which IS1, ±5) is an eigenstate gives the UEVMUCS of SU(2) called the SU(2) or spin coherent state. For the sake of definiteness, we let IS1, S) be the fiducial state and rewrite (3.74) as \θ,φ) = ϋ(θ,φ)\3,3), U(θ, φ) = exp exp ^ (βχρ(ϊ0)5_ - ex p ( -i 0)5+) i0 (sin(^)S'x — cos( 0 ) 5 ^ , (3.77) (3.78) 0 < θ < π, 0 < φ < 2π. Similar results follow if the fiducial state is chosen instead to be the state |S, - S ). On disentangling the exponential according to (2.96) and on applying (3.71), (3.77) leads to 2 s \Ω) = \μ) = Σ ( l + H 2) 5 ^ V (2 S - m )!m! (25)! -15,5 - m ), (3.79) where μ = exp(iφ) tan . (3.80) The scalar product between the spin coherent states is given by w = (1 + M y ( 1 + | „ | Τ + (3 81> In t he following we enumerate some propert ies of t he spin coherent states. 1. The spin coherent st at e is an eigenstate of t he spin component in t he direction (θ, φ). To see this, operarte the eigenvalue equation (3.67) of Sz corresponding to m = 5 by U and rewtrite it as ϋ ( M ) & £ f ( M ) ] U(0,<l>)\S,S) = St j ( 0^ )\S,S). (3.82) Evaluate the similarity transformation above by using (2.42) and invoke the definition (3.77) to get 5 ( M ) | M > = S|M>, (3.83) βφ,φ) = ϋ (β,φ&ϋ'φ,φ) = sin(0) cos (φ)£ΐχ + sin(0) sin(<^)5y + cos(0)5z. (3.84) This is, of course, the component of the spin in the direction (θ, φ). The equation (3.83) shows that the spin coherent state is an eigenstate cor responding to the eigenvalue S of the spin component in the direction (θ,φ). 2. E x p e c t a t i o n val ues of o pe r a t or s i n a s pi n c o h e r e n t s t a t e may be e va l ua t e d by n o t i n g t h a t (i9, <1>\F ({5μ}) \θ, φ) = (S, S\&( 0, φ)Ρ ({5μ)}) ϋ(θ, φ)\S, S) = { S,S\F ( { S ^ ( - e ^ ) )\S,S ), (3.85) Ξμ(θ,φ) = ϋ(θ,φ)3μϋ\θ,φ). (3.86) The similarity transformation above may be carried using the results of Sect. 2.3.2. 3. Consider a spin component S 1- = exSx+eySy, (e^.+ey = 1), in a direction orthogonal to the z direction. Note that (θ,φ\^φ,φ)\θ,φ) = (5,515^15,5) = 0. (3.87) By virtue of the fact that S x (6, φ) is obtained as a result of a unitary transformation of the spin component perpandicular to Sz, it follows that § ± (θ,φ) is perpandicular to Ξ(θ, φ), i.e. in a direction orthogonal to the direction of the spin coherent state. The expression (3.87) shows that the average of a spin component in a direction orthogonal to the direction 3.3 Representations of SU( 2) 71 of a spin coherent state is zero in that state. We evaluate fluctuations in the said orthogonal components by starting with the equation (S,S\(exSx + eySy)2\S,S) = {S,S\S+S- +S-S+\S,S)/4: = S/2. (3.88) In writing this we have made use of (3.69). This implies that (S,S\&(e,<f>)S±2(e,<l>)0(e,<f>)\S,S) = (θ,ψ\3±2(θ,ψ)\θ,φ) = f · (3- 89) Thi s, t o g e t h e r wi t h ( 3.87) shows t h a t t h e va r i a nc e Δ S ±2 in any compo nent orthogonal to the direction of the spin coherent state has the same value S/2. 4. Th e expr e s s i on for t h e r e s ol ut i on of i d e n t i t y i n t h e pr e s e nt case r e a ds ^ / ϊ τ τ ξ ρ ψ M W = ^ /d" | β>β' = 7· (3-90> άΩ = sin(0)d0dφ. Using this relation, a state \φ) is represented by (μ\φ) or by t h e f u n c t i o n φ{μ*) = [μ\φ) of μ* where \μ) is unnormalized spin coherent state: 72 3. Representations of Some Lie Algebras 2 S / φ(μ*) = (μ\φ), S,S- m). (3.91) m—0 V ^ ' On i nvoki ng (3.67) and (3.70), i t ma y al so be ver i f i ed t h a t ( μ | 5 + = ± ( μ\, ( μ | 5 _ = [2 5 μ * ( μ | & = ( ^ - μ * ^ ) ( μ |. ( 3.92) Th u s, i n t h e s pi n cohe r ent s t a t e s r e p r e s e n t a t i o n, vec t or s ar e r e pr e s e nt e d by a compl ex val ue d f unc t i on a nd o p e r a t o r s on a ve c t or by t h e di f f e r ent i al o p e r a t o r s. Si nce S'+'s + 1 = 0, the expressions (3.92) imply that φ{μ*) is a polynomial of degree 2S in μ*. We have t h u s c o n s t r u c t e d t h e UEVMUCS c or r e s pondi ng t o t h e s ol ut i on of ( 3.75) f or A = ± 1. I t s s ol ut i on f or λ φ ± 1 my be derived by writing it as exp(—0) S- + exp(0)5'+j |θ,ζ) = ζ\θ,ζ). (3.93) Use (2.44) and (2.42) to reexpress this as Szf\e,z) = Z-T\e,z), (3.94) where T = exp exp ( - 0 S Z) . (3.95) 3.3 Representations of SU( 2) 73 The equation (3.94) shows that Τ\θ,ζ) is an eigenstate of Sz. Hence z = 2n (—5 < n < 5) and, with |θ,ζ) —> \θ,η), \θ, η) = T ~ 1\S, η) — exp (6SZ) exp i — Syj |5, ή). (3.96) This is a squeezed spin state. The meaning of squeezing of a spin state is discussed below. This state will turn out to be non-classical in the sense explained in Chap. 4. The properties of these states are studied in details in [48]. Next, we derive the expression for \θ,η) in terms of the eigenststes of Sz. To that end, let S 2 S |θ,η ) = Σ Cnm\S,rn) = J 2 CnS-P\S,S ~ p ) (3.97) m— S p —0 The Cnm may be determined by substituting this in (3.93) to construct the recursion relation for Cnm. It is straightforward to verify that (iV = 25) exp( - θ) ^/ρ( Ν - p + l )CnS-p+i + exp(θ)\/(ρ + 1 ){N - p)CnS- p- i = 2 nCnS- P. (3.98) Define fnp = CnS-p/y/pl{N-p)\. (3.99) On substituting this in (3.98) we obtain exp(-0)(iV - p + l ) f np- i + exp{θ)(ρ + l ) f np+i = 2nf np. (3.100) This is the same as the recursion relation (10.66) solved in Chap. 10. The exact solution (10.69) of (10.66) in the present case assumes the form Up = (-)pAnexp(-p0) R fc Σ (p - k)\k\(n + N/2 - k)\{N/2 - n - p + k)l (3.101) Th e c o n s t a n t An is to be found by the normalization condition. We leave it to the reader to derive this result directly by using (3.96). Note that pm = |CVim| 2 gives the probability of finding the spin in the state |τη). We substitute (3.101) in (3.99) and evaluate pm. The numerical results for pm as a function of m for a system of N = 20 spins in the state |0,O) with exp(20) = 5 are presented by a solid line in Fig. 3.2. We notice that, much like pm as a function of m in Fig. 3.1 for a harmonic oscillator in its squeezed state, pm in Fig. 3.2 for the squeezed spin state exhibits an oscillatory behaviour. We will see in Chap. 12 that \θ, 0) is the steady state of a system of even number of two-level atoms in contact with a squeezed reservoir. We compare this with the equilibrium state of N spins in contact with a thermal reservoir. Such a spin state is characterized by 0.4 74 3. Representations of Some Lie Algebras Fig. 3.2. Plot of pm as a function of m for the spins in the squeezed state (3.96) with exp(20) = 5, n = 0, N = 20 (solid curve). The dashed curve is for the thermal state with n = 5. pth = exp(—/352 )/Tr[exp(—/3iS2)], (3.102) the β being as in (3.57). The corresponding probability of occupation of the state jS1, m) is Pm = (m\pth\m) = exp(—/377i)/Tr[exp(—/JS^)], (3.103) N Tr(exp(-/3S*) = exp(-/3S') ^ exP (βρ) p=0 . exp{(iV + 1 )β} = exp(-/3 S)- 1 (3.104) exp(/3) - 1 Th e b e h a v i o u r of pm as a function of m given by (3.103) is displayed by a dashed line in Fig. 3.2 for n = 5 (n defined as in (3.58)) and N = 20. The oscillatory behaviour of pm is a signature of non-classicality of the state in question. Spin Squeezing. Recall from the discussion of Sect. 3.2 that a squeezed state of the h.o. is defined as the state in which the variance in a linear combination of q and p is less than its value in the coherent state. Extension of this definition to a spin, however, requires more careful considerations because, whereas the variance of a linear combination of q and p in a h.o. coherent |a) state is independent of a, the variance in the generators of spin in a spin coherent state \θ, φ) depends on ( θ , φ). I n o r d e r t o e x t e n d t h e c onc ept o f s quee z i ng t o a s ys t e m of s pi ns, we n o t e f r om ( 3.89) t h a t t h e v a r i a nc e i n a s pi n c o mp o n e n t i n a ny di r e c t i o n or t h o g o n a l 3.3 Representations of SU( 2) 7 5 t o t h e a v e r a g e d i r e c t i o n (θ,φ) of a spin in its coherent state |θ,φ) is S/2. A squeezed spin state may then be defined as a state in which the variance A S ^ 2 in a spin component in some direction orthogonal to the average direction ( S) in that state is less than |(£)|/2 [49]: (AS^)2 < M i (SQ(I)). (3.105) The concept of spin squeezing may also be based on the problem of sensitivity of measurement. Some simple processes of measurement using a spin have been analyzed [39, 50, 51]. For an experiment on measuring the quantum noise in a spin system, see [52]. In those processes, the measurement of a quantity a is carried by coupling it with linear spin operators. The value of a is related with the averages of an observable Sa associated with the component of spin in direction a. It is then shown that the error in the measurement of a is given by Zk* = zASa/| ( ^ ) | (3.106) where S x is the observable corresponding to the spin component in a di rection orthogonal to a. The actual direction depends on the details of the interaction. It can be verified that the minimum value of ( Δα) 2 in the spin coherent state is 1/2S'. Hence the parameter £ = 2 S(zASa ) 2/| < ^ ) | 2 (3.107) is defined as a measure of sensitivity of measurements involving a spin. As remarked before, its minimum value in a spin coherent state is £min,coh = Ι Α squeezed spin state may be defined as the one in which ξ is less than unity, i.e. a state in which ξ = 2 S(ZiSQ) 2/| ( ^ ) | 2 < 1 (SQ(II)). (3.108) This criterion of squeezing is not the same as the criterion SQ(I) of (3.105). However, since the average of any spin component is less than S, it follows that SQ(II) implies SQ(I). The criterion SQ(II) is also referred to as spec troscopic squeezing. The squeezed spin state (3.96) satisfies SQ(I) as well as SQ(II) [53], Note that U in the definition (3.77) of the spin coherent state is expressible as a product of N spin operators. The state \S,S) therein is also a product of N spin-1/2 states. Hence the spins in the state |θ,φ) are uncorrelated. However, although the operator T defined in (3.95) is a product of individual spins but not the state |s,n) if η φ ±s. Hence, spins in MUS (3.96) are correlated if η φ ±s. In general, the spins in states which satisfy SQ(I) or SQ(II) are correlated. For details of evaluation of ξ and the spin-spin correlation function for various states, see [5 3 ]. 76 3. Representations of Some Lie Algebras 3.4 Representations of SC7(1,1) Recall from Chap. 2 that the S'i7(l,l) algebra is generated by K = ( Kx, K y,K z) which obey the commutation relations (2.47). Note from Sect. 2.3.3 that the norm of vectors in a direction which is a linear combination of x and y directions is positive but the one in the direction z is negative. Hence, linear combinations of directions x and y are equivalent to each other but not to 2. Accordingly, the compnents K x,K y are equivalent to each other but not with K z. A third inequivalent class is formed by combination of K z with K x or Ky which gives zero norm. There are thus three classes of orthogonal bases generated by the hermitian operators of the algebra: one equivalent to the eigenbasis of K z, second equivalent to that of K x and third equivalent to K x + K z. The Casimir invariant for SU( 1,1) is Q = kl + k l~ k 2z = k+k_-k 2 z + k z. (3.109) The vector space of £{7(1,1) is, therefore, reducible to a sum of invariant subspaces each labeled by an eigenvalue of Q. Here we are concerned only with one-mode and two-mode bosonic realizations of S'i7(l,l) defined in (2.71) and (2.76). Verify that the Casimir operator (3.109) in the two-mode bosonic representation assumes the form Q = k ( l ~ K ), k = ^ ( t f a - t f b + 1), (3.110) whereas for one-mode realization it reads Q = ^ · (3.111) The operator k in (3.110) is related with the difference in the occupation number of the two modes which, as a consequence of the fact that k is an invariant, remains unchanged in an SU( 1,1) process. 3.4.1 Orthonormal Bases As stated before, in this case there are three inequivalent representations: one equivalent to the eigenbasis of K z, second to that of k x, and third to kx + kz. Consider the eigenvectors of K z. The expressions (2.71) and (2.76) for k z in the bosonic representations and the corresponding expressions (3.110) and (3.111) for Q suggest that the eigenstates \χ[πι,Κ)) of k z are such that Q\X(m,K)) = K(1 - K)\X(m,K)), Kz\x{m,K)) = (m +K)\x(m,K)), m = 0,1,2,... (3.112) where, for the two-mode realization, \x (m,K)) = \m + 2K-\,m), (3.113) 3.4 Representations of SU( 1,1) 77 the |m, n) being a simultaneous eigenstate of the operators a)a and b% with to and n as respective eigenvalues. For one-mode realization, |x(to, K ) ) = |2m), |2to + 1). , (3.114) Also, by comparing (3.112) with (3.111), check that K = 1/4, 3/4 for one mode realization. It may also be verified by invoking the definitions (2.71) and (2.76) that K-\x(m,K)) = y/m{m + 2K - l)jχ(τη - 1 ,K)), K+\X{m, K)) = y/(m + 1 )(m + 2Κ)\χ(τη + 1, K )). (3.115) Note that Κ_\χ(0,Κ))=0. (3.116) Repeated application of (3.115) yields + <3,Π> Note that, in the single-mode case, K± couple \m) with \m ± 2). Hence the space of states in this case reduces to a sum of odd and even number states. Also, from the relationship (2.76) between Kz and a^a, we see that -ftfz|2m) = (m + l/4)|2m), K z\2m + 1) = (m + 3/4)|2m + 1 ). On comparing these results with (3.112) we infer that K = 1/4 (K = 3/4) for even (odd) number states leading to the correspondence \χ(τη, 1/4)) —► |2to), |x(m,3/4)) —>· |2to + 1). (3.118) The eigenstates of K x in terms of the eigenstates of Kz introduced above are obtained as the special case A = 0 of (3.119) below. 3.4.2 Minimum Uncertainty Coherent States The classes of minimum uncertainty states of S U(1,1) are: One corresponding to the pair ( Kx,K y) and another to the pair class (KX,K Z). MU S o f t h e P a i r ( K x,K y ). The MUS in this case are the solutions of |λ, z) = z|A, z). (3.119) Consider first the case of the UEVMUS corresponding to A = 1. The equation (3.119) in this case reduces to the eigenvalue equation (with 11, z) ξ \z)) k -\z ) = z\z) (3.120) for the lowering operator. Let KX - iXKy 78 3. Representations of Some Lie Algebras |z) = 5 3 Cm\x(m, K)). (3.121) m=0 Subst i t ut e t hi s in (3.120) and use (3.115) t o arrive at t he recursion relation \f (τη 4- l )(w + 2K)Cm+i — zCm. (3.122) This is easily solvable. Its solution, on substitution in (3.121) yields 7 r ===== \χ(τη,K) ), (3.123) m=o V r (m + 1)r (m + 2 K) A be i ng t h e no r ma l i z a t i o n c o ns t a nt. Th e s t a t e \ξ,η, z) = exp ξ Κ+ - ξ*Κ^ + ϊ ηΚζ |^r), 77 real (3.124) generated by the SU( 1,1 ) group transformation is, by definition, the SU( 1,1 ) UEVMUCS. A special case of (3.124) of particular interest is the state |£, 77,0) corresponding to \z = 0) = |χ(0, K)). The |χ(0, K)) is an eigenstate of K z. The relevant part of the coherent state (3.124) then is 10 = exp [ ^ + - CK- ] |x(0, K)) (3.125) Note that for single-mode realization of SU (1,1) this is the same as the MUS |£,0) of the harmonic oscillator defined in (3.55). Now, disentangle the exponential above using (2.104) and expand the resulting exponentials. Next, use (3.116), (3.112) and (3.117) to obtain (with μ = exp(i0) tanh(|£|)) m = ( 1 - w y e < λ/ K ) ) ■ ( 3 J 2 6 ) The expr e s s i on ( 3.10) f or t h e r e s ol ut i on of u n i t y i n t h i s cas e as s umes t h e f or m Γ ( 2 ^ - 1 )/( 1 - | μ | 2 ) 2 |μ)(μ| = /· (3· 127) Here the integration is over the unit disc |μ| 2 < 1. This relation may be verified by expanding the states as in (3.126) and on carrying the integration. The resolution of unity is also admitted by the states |z) of (3.123) [54]. The states \z) thus fulfill minimum requirements listed in Sect. 3.1 for labeling a state as a coherent state. Note that this is an eigenstate of the annihilation operator of SU( 1,1) like the Glauber coherent state is an eigenstate of the annihilation operator of the h.o. The states |z) of (3.123) have, therefore, been recognized as the SU( 1,1) coherent states parameterized by z [54]. The properties of their two-mode realization and quantum optical processes for their generation have been identified by Agarwal [55] who named them pair coherent states. Note, however, that the states |z) are not coherent in the sense of Perelomov because the variable z is not a group parameter. That is in contrast with the fact that the parameter a defining the h.o. coherent state 3.4 Representations of SU( l, 1) 79 |a) is the h.o. group parameter. Recall also that the action of a h.o. group transformation on a h.o. coherent state |a) results in another h.o. coherent under the action of an SC/(1,1) transformation. The state |z) is, therefore, not a group parameter related coherent state. We reserve the name SU( 1,1) coherent states for the UEVMUCS (3.125). The completeness relation (3.127) shows that any SU(l, 1) state |φ) may be represented by (μ|φ) or by the function φ(μ*) = (μ|φ) of μ* where |μ) is the unnormalized £{7(1,1) coherent state: On using (3.127), the scalar product of two states in this representation turns out to be given by These relations are useful in converting the £(7(1,1) operator equations in to differential equations by means of the correspondence This, along with (3.130), demands φ(μ) to be analytic in the disc |μ| 2 < 1 if it is to be an admissible function. The representation (3.132) along with the analyticity requirement pro vides a simple means for solving £f7( 1,1) equations. This method has been used in [40, 41] to solve (3.119) for Λ φ 1 and to find the MUS of the other pair class, namely, the pair (KZ,K X). We conclude by mentioning that (3.119) admits normalizable solutions only if Re(A) > 0 and that its solution is delta- function normalizable if Re(A) = 0. Hence (3.119) does not admit acceptable solution for A = —1. Consequently, the states (3.125) corresponding to A = 1 are the only UEVMUCS of SU( 1,1) for the pair (Kx, K y). We refer to [40, 41] for details. state. However, the state |z) of (3.123) does not transform to another |z') φ(μ*) = (μ\φ), \μ ) = Σ μ η m= 0 oc (φ\φ) = Γ (2^ _ i) J ^ { ΐ -\μ?ΐ Κ 2 φ( μ ) Φ ^ ) (3.129) Th i s shows t h a t |φ) is normalizable if r(2K — l) / dV ( i -!μ|2) 2Κ 2 I2 < oo· (3.130) On using (3.115) and (3.128), it is easy to show that (μ|^ - = έ (μ1, (μ|7Μ μ*2έ +2* μ*](μΙ ( ^ = \μ * ~ + κ](μ\. (3.131) (μ|Φ ( Κ +,Κ -,Κ Ζ) |φ) (3.132) 4. Quasiprobabilities and Non-classical States Recall that a classical dynamical system may be described by a phase space probability distribution function f {{q},{p}), ({<?} = qi, q2, · — , Qn] {p} = pi,p2, ■ ■ ■ ,P n ) which is such that {p})dNqdNp gives the probability that the system is in a volume element dNqdNp centered around ({</}, {p})· In the quantum mechanical description of a dynamical system, however, the phase space coordinates qi and pi can not be ascribed definite values simulta neously . Hence the concept of phase space distribution function does not ex ist for a quantum system. It is, however, possible to construct for a quantum system functions, called quasiprobability distributions (QPDs), resembling the classical phase space distribution functions. A QPD provides insight into quantum-classical correspondence as well as useful means of calculations. 4.1 Phase Space Distribution Functions For the sake of simplicity, consider a one dimensional dynamical system de scribed classically by a phase space distribution function f (q,p,t ). It deter mines the average of any function A(q,p) by means of the relation (A(q,p)} ci = j dqdpA(q,p)f (q,p,t ). (4.1) The quantum mechanical description of a system, on the other hand, is con tained in its density operator p which determines the average of any_ function A(q,p) by the relation (4(<?,p))qm = Tr pA(q,p) . (4.2) Now, assume that A is a function of q alone and carry the operation of trace in the equation above in the basis of the eigenstates | q) of q to get {A(q))qm = J dqA(q)(q\p\q). (4.3) This is of the form (4.1) of the classical phase space average. Similar result follows by working in the basis of the eigenstates |p) of p in case A is a function only of p. However, as argued in Sect. 1.1, there is no state which 82 4. Quasiprobabilities and Non-classical States is simultaneously an eigenstate of q and p. Hence the foregoing procedure can not yield an analog of the classical phase space distribution function in quantum mechanics. However, through (4.2), the density operator determines the average of any function of the operators q, p. Note also that the classical distribution function may be expressed in terms of the averages of a complete set of func tions of q and p. This suggests that we may be able to construct a quantum analog of the classical distribution function by expressing the latter in terms of the average of a suitably chosen complete set of functions and by identify ing those classical averages as quantum mechanical ones. In order to explore this possibility we rewrite a classical distribution as This expresses the distribution function f (q,p,t ) in terms of the average of a complete set of functions of q and p. Now, to construct the quantum analog of f{o.iPi t), (i) express the exponential under the average in (4.4) as a sum of products of the form qmpn, (ii) replace the c-numbers q,p by the operators q,p, and (iii) replace the classical average by the quantum average defined as in (4.2). The basic difficulty in administering this prescription lies, however, in the fact that, due to non-commutivity of q,p, there are several different operator forms of a c-number product qmpn if to, η φ 0. Those different forms correspond to different ways of ordering q and p. For example, q2p may be represented by any of the forms: q2p, qpq, pq2 or by their linear combination X\q2p + x 2qpq + %3pq2 where Xi are arbitrary subject to the condition χ λ +X2 +X3 = 1. This condition ensures that the linear combination in question reduces to q2p when the operators are replaced by c-numbers. In general, we formally represent a c-number product as an operator as where Q(qmpn) defines a linear combination of to q's and n p’s. Use the cor respondence (4.5) to replace the exponential of c-numbers under the average in (4.4) and follow the step (iii) above to get Different choices of the correspondence Ω lead to different f ^n\q,p,t ), each f(<l,P,t)= J dq'dp'3{q-q')S(p - p')f(q' ,p' ,t) = ± f dq'dp'dkdlexp[i{k(q~ q') + l{p - p')}]f{q',p',t) -ξ—^ J dkdlexp(ikq)exp(Up){exp(—i kq)exp(—Up))ci. (4.4) qmpn -► Ω (qmpn) (4.5) J / qm Th i s i s a q u a n t u m a na l og of t h e cl as s i ca l p h a s e s pac e d i s t r i b u t i o n f unc t i on. 4.1 Phase Space Distribution Functions 83 called a quasiprobability distribution (QPD). It is designated a quasiprobabil ity to emphasize the fact that it is a mathematical construct and not a true phase space distribution function as no such function exists for a quantum system. Let us now examine the results of different ordering prescriptions. In order to investigate various operator orderings, it is convenient to ex press q,p in terms of the creation and annihilation operators a, a) and to transform suitably to complex variables so as to rewrite (4.6) as f {Q)(a,a*) = ^ [ d2£ exp [ί(α£ + α*ξ* e x p ( —ί άξ) e x p ( —i a ^ * ) | p j. (4.7) x T r Now, l et Ω {exp(i£a) βχρ(ί£*ά^)} = Π^=1[βχρ(ία^ξά) βχρ(ί/3^ξ*ά^)], (4.8) where aj, β j are complex numbers such that αχ+· ■ -+ctN = βι~\ \-βΝ = 1 · This condition ensures that (4.8) reduces to an identity when a —> a, a c-number. By applying (2.88) repeatedly, we may combine the product of the exponentials in (4.8) in to a single exponential. We note that each such combination would contribute a c-number exponential whose exponent is proportional to |£|2. As a result, we may rewrite (4.8) as 7 7 ^ ι [ β χ ρ ( ϊ α ^ α ) β χ ρ ( ϊ ^ Γ ^ ) ] = exp ~ | | £ | 2 exp {i(£a + ξ *^ ) } . (4.9) Here s is a complex number related with products of the a's and the β ’s. Although the exact expression of s in terms of the a's and the β's may be derived, it is inessential. The ordering for s = 0 is called the Weyl ordering. For the reason mentioned after (4.12), it is also called the symmetric order ing. In applications, it is often useful to know the form of operators in the normal or antinormal ordering introduced in Sect. 2.4. By applying (2.88), the exponential operator in (4.9) may be put in the antinormal or the normal ordering as exp{i(£a + f a 1)} = exp Q l £ | 2) exp(i£a)exp(i£*af ) = exp exp(i£*af) exp(i£a). (4.10) The ordering corresponding to different choices of the a's and the β's in (4.8) thus reduces to the one defined in terms of just a complex number s. It is referred to as the s-ordering [56]. The operator ordering corresponding to a c-number form ama*n in the s-ordering may be derived by noting that ama*n -► Ω3 (ama'n) m r exp H l£|2) “ p <1(£a+ £‘“’» L w <4·π > 84 4. Quasiprobabilities and Non-classical States As an example, the s-ordering of the powers of the operators in terms of the normal ordering is obtained by expressing the exponential operator in (4.11) in the normal-ordered form using (4.10) and then carrying the operation of differentiation. We leave it to the reader to show that k Σ k= 0 s + 1 mini (to — k)l(n — k)lkl a [n~ ka m~ k (4.12) I t may be s hown t h a t i f s = 0 then the operator on the right hand side above may be expressed as a symmetric combination (see [57] for a proof). In the symmetric ordering, a product X1 X2 ■ ■ ■ x m of variables is replaced by the sum of all possible permutations of the product divided by the total number of such permutations. The QPD in the s-ordering, obtained by substituting (4.9), read with (4.8), in (4.7) reads /(s) {a, a*) = j d2£ exp {i(a£ + α*ξ*)} x exp (-fl£l2) Tr {-*(** + atn } p] (4.13) This relation may be inverted by using (A.10) and (A.11) to express p in terms of the QPD as Tr [βχρ{-ϊ(άξ + ά^*)}ρ] = G(£,£*)exp ( f l£|2) > (4.14) G(£,n = / d2a ^( α,α*) βχρ{- ΐ ( αξ + α*ξ*)}. ( 4.1 5 ) Now, d i s e n t a n g l e t h e e x p o n e n t i a l u n d e r t h e t r a c e i n ( 4.1 4 ) i n t o t h e a n t i n o r m a l f o r m u s i n g ( 4.1 0 ). Us e t h e r e l a t i o n T r [ e x p ( —ιξά) exp(—ίξ*α^)ρ| = Tr [exp(—ϊξά^)ρβχρ(—i^*a)] (4-16) to carry the operation of trace in the resulting expression by applying (3.30). It will be found that 1 d2/3exp {—i(/3£ + β*ξ*)} {β\ρ\β} G ( £,T ) e x p { - ^'4 ( 4.17) Thi s, on a p p l y i n g ( A.10) a nd ( A.11), l ea ds t o t h e expr es s i on (β\ρ\β) = ~ f d2ξ G ( ξ,ξ * ) e x v { m + β*ξ*)} 7Γ _ x exp ( 4.18) f or t h e d e n s i t y ma t r i x i n t h e c ohe r ent s t a t e r e p r e s e n t a t i o n. On i nvoki ng ( 3.32) a n d ( 3.33), t h e o p e r a t o r f or m of (4.18) i s f ound t o r e a d 4.1 Phase Space Distribution Functions 85 s 1 “ '' exp(iaT£*) βχρ(ίάξ) p = ^ J ά2ξ G (£,£*) exp = ~ ί ά2ζ G (£,£*) exp ( f l ^ l 2) βχρ[ϊ(άξ + afD ] d2 C G ( C,r ) e x p s + 1 l£l5 exp(ia£) exp( i a ^*). (4.19) This determines t he relationship between t he density oper at or and its various phase space representations t hrough G (£,£*) defined in (4.15). The relationship between different phase space representatives /and may be derived by substituting (4.14) and (4.15), with s replaced by t, in (4.13) and on carrying the ^-integration we find that, provided Re(s) > Re(i), 2\α-β\2' f(°\a,a*) = n(s — t) d β exp (4.20) This is the desired relation between two phase space distributions. In practical applications, we need to convert equations involving products of p with ama^n in to c-number equations (see Chaps.13,14). The c-number equivalent of amp is obtained by replacing p in (4.13) by amp. On applying (4.10) and using the cyclic property of the trace, we note that Tr [exp {—i(a£ + afD } amp] = e xp( —| £ |2/2)Tr [exp(—ia£)am/5exp(—iaf£*)] = exp(H£|2/2 )- gn '8(-ίξ) Substitute this in (4.13) to obtain Tr [exp(—ia£)pexp(—ία^ξ*)] . (4.21) 1 am p —> — / d ^ a + i .s + 1 Γ exp {i(a£ + a*£*} x exp j - | | £ | 2} Tr [exp {-ί(ά£ + ^ ξ * ) } p] π τ έ ]" /“ ’ < « · « * > · <4 · 2 2 > t h e f(s\a, a*) being the QPD of p. The phase space equivalent of a)mp may be derived in a similar manner but by starting with the anti-normal form of the exponential operator under the trace in (4.13). Verify that s — I d aJmp a + 2 da ( 4.23) Th e r e l a t i o n s ( 4.22), ( 4.23) a nd t h e i r h e r mi t i a n c onj uga t e s may be us ed t o fi nd t h e p ha s e s pac e e qui val e nt o f a ny c o mb i n a t i o n fi(a, ai)pf 2 (a , a1") in terms of a differential operator on the phase space representative of p. 86 4. Quasiprobabilities and Non-classical States Operator Averages. The foregoing considerations for a density operator p may be extended to construct the phase space representation A^s\a,a * ) of any operator A by way of the correspondence p —»· A and —> Α^^α,α*) in (4.13) so that AW ( α, a ) = d2£ exp {i(a£ + a*£*)} xexp { “ |£|2} Tr βχρ{-ΐ(αξ + αϊξ*)}4 (4.24) The expression for A in terms of its phase space representative, obtained similarly using (4.19), reads A = - 7Γ d2£ Ga (ξ,ξ*) exp ( | |£|2) exp [i(a£ + af£*)] with Ga (£,C)= / d2a 4 (s)(a,a;*)exp{-i(a£ + a*£*)}. (4.25) (4.26) The average of A may now be expressed in terms of a phase space integral similar in appearance to the classical expression (4.1). To that end, multiply (4.19) by A, substitute for G (£,£*) from (4.15) and take its trace to obtain Tr I Ap = ~ f d2£d2ct fs (a, a*) exp [-i(a£ + a*£*] exp ^ fl 2) xTr A exp {i(a£ + <r£*}] π / d2a (a,a*) A( s·* (a, a*). ( 4.27) I n wr i t i n g t h e l a s t l i ne above, we have us e d t h e de f i ni t i on (4.24). Th e e x p r e s si on ( 4.27) i s e vi de nt l y of t h e f or m (4.1) of a cl as s i cal aver age. Th e f unc t i on Λ( - «) (a, a*) is said to be conjugate to (a, a*). We thus see that the expectation value of an operator is the phase space integral of the product of any of its phase space function with its conjugate representative of the density operator. Next, we list some properties of the phase space representation for some particular values of s of special interest. 1. Let ά2β Ρ(β,β*)\β)(β\. (4.28) This is known as the P-function representation of a density operator. Substitute this in (4.13) and express the exponential operator under the trace there in the normal-ordering. Let s = —1. Use the cyclic property of the trace to obtain 4.1 Phase Space Distribution Functions 87 /(_1) («, «*) P (β, β*) exp {i(a£ + a*£*)} = Z2 ί ά2ξ f ά2β x T r 1 β χ ρ ( - ϊ ά ξ )\β) (β\ exp( -iafr ) = ^ 2 J d^ J ά2βΡ( β,β*) βχ ρ[ ί { ( α~β) ξ +( α*- β*) ξ *} } = P{ a,a*). (4.29) The phase space representative for s = —1 is thus the P-function. The P-function representation for an antinormally ordered product can be found easily by using the completeness relation (3.29). For, on operating that relation with an on the left and a^m on the right we find that = - f d2a a na*m\a)(a\. (4.30) π On comparing this with (4.28) we see that the P-function for ana)m is a na*m/n. 2. Ne xt, s e t s = 1 in (4.17) and compare it with (4.15) to note that -(β\ρ\β) = ί ι1)(β,β’) = <2(β,β*)· (4-31) 7Γ Thus !{β\β,β*) for s = 1 is simply the matrix element of the operator in the coherent states representation. It is also known as the Q-function or the Husimi function. The Q-function of a^man is clearly a*ma n/n. - As a consequence of (4.29), (4.31) and (4.27), it follows that the trace of a product of two operators is the phase space integral of the P-function of one with the Q-function of the other. Thus, if the density operator is represented by its P-function then J d2βΡ (,β, β*) β*™βη = {a)man). (4.32) This shows that the phase space average of β*ηιβη with the P-function of the density operator gives the average of normally ordered operators. On the other hand, if the density operator is represented by its Q-function then J d2/3Q (β, β*) /r m/T = (a"atm). (4.33) This shows that the phase space average of /3*m/3ra with the Q-function of the density operator gives the average of antinormally ordered operators. 3. The phase space distribution function /( s)(/3,/3*) corresponding to s = 0 is called the Wigner function. It is usually denoted by W(j.3,β*) = f(°\/.3,β*). We infer from (4.27) that the trace of a product of two oper ators is the phase space integral of the product of their Wigner functions. Hence, if the density operator is represented in terms of its Wigner func tion then J ά2β\νφ,β*)β*™βη = i - < ( a"a t - ) s), (4.34) the suffix s denotes symmeterized operator product, i.e. the sum of all products formed by permutation of rri a t ’s and 11 a ’s and Ns is the number of such permutations. The relation (4.9) for an operator representation of a c-number function may be generalized by replacing a by a linear combination b = ηιά + Ω {exp(i^a) exp(i£*at)} = Π^L1[exp{iajξb)exp{iβjξ*b^)}. (4.35) The phase space distribution functions for such a generalized rule can be constructed by following the procedure outlined above for the special case ηι = 1,772 = 0. The theory of quasi-distributions of the canonical operators is developed in its generality in [58]. 88 4. Quasiprobabilities and Non-classical States 4.2 Phase Space Representation of Spins Consider a system of spins. The spin observables obey non canonical com mutation relations. As shown in Chap. 2, we can represent spin operators as bilinear combinations of two canonical operators and thus extend the con siderations of the Sect. 4.1 to construct the QPD for a spin [59]. However, a direct approach is to exploit the fact that a spin-S traverses the surface of a sphere. Hence, a spin may be described classically by a distribution function /{θ,φ) of the polar and the azimuthal angles. In order to construct its quan tum analog, /( θ,φ) is expressed in terms of the averages of a complete set of functions. A convenient set in this case is the set of spherical harmonics Y l m { 9, φ), (L=0,1,...), (M=-L,-L I 1,... ,L) so that ί{θ,φ) = J Ηΐη{θ,)άθ,άφ,δ{φ - φ')δ{οοδ(θ) - cos(0'))/(0', <£') / OC L s i n ( 0') d 0'd ^' ^ Σ Υί Μ{θ,φ)ΥΙΜ{θ',φ')!{θ',φ') L = 0 M = - L oo L = Σ Σ Υί Μ{θ,φ){Υ£Μ{θ,φ)). (4.36) L=0 M — — L I n wr i t i n g t h e s econd l i ne above we have i nvoked t h e compl et ene s s r e l a t i o n ( A.37) of t h e s phe r i c a l har moni c s. Th e QP D f or a s ys t e m of s pi ns i s o b t a i n e d by r e pl a c i ng t h e cl as s i cal aver age of t h e s phe r i c a l ha r moni c s i n t hi s e q u a t i o n by t h e q u a n t u m me c ha ni cal e x p e c t a t i o n val ue of a p p r o p r i a t e o p e r a t o r s. T h e o p e r a t o r s a p p r o p r i a t e f or t h i s pur pos e e v i d e n t l y a r e t h e ones whi ch t r a n s f o r m 4.2 Phase Space Representation of Spins 89 under rotation in the same way as do the spherical harmonics. Now, recall that the operators corresponding to an integral spin may be represented by differential operators in θ, φ on the functions of θ, φ. We know the commuta tion relations between integral spin operators in the (θ, φ ) representation and the Y l m (&7 Φ) [60]. The defining property of the operators we are looking for corresponding to Y l m ( θ, Φ) in (4.36) is that their commutation relations with the spin operators be the same as the those between the spherical harmonics and the spin operators in the (θ, φ ) representation. Hence, the desired opera tors should have the form T k q with K = 0,1,... and Q = — K, —K + l,..., K and they should be such that Sz, TKQ = Q T k q, S±, T K q J = \/{ K =F Q){K ± Q + l ) f K±i Q· (4-37) These are the commutation relations of Y k q ( 0 ^ ) with the spin operators in the (θ, φ ) representation. We recall that, for a system of total spin quantum number S [60], s Tk q = £ ( ~ ) s - m( - ) ^"C ^ f lQ|m,£ }(n,£ t, (4.38) m,n=— S K = 0,1,, 2S; Q = —K, —K + 1,..., K. Substitution of (4.38) in (4.37) shows tha t C^ s_KnQ obey the recursion relation of Wigner or Clebsch-Gordan coeffieients[60\. The operators T k q, called state multipole operators, consti tute a complete set. Their orthonormality relation is |60] Tr [Tl m T k q = 5k l 5Mq. (4.39) We note also the property T ]KQ = ( - ) QT k - q · (4.40) Hence, any spin operator, for example, a density operator, may be expressed as 2 S K Ρ = Έ Έ ( f KQ) f KQ· (4·41) k =oq =- k Now, the QPD of spins is obtained by identifying the classical average of Y l m { 0,$ ) over { θ,φ ) as an average of T l m 'n the state described by p by means of the relation ( Y l m{ 0, φ)) = J?lmTt ξ Q l m ( T l m ^ J , (4.42) &lm being a free constant. Substitution of this in (4.36) results in different quasiprobability distribution functions 2 s L 90 4. Quas i pr obabi l i t i e s and Non- c l as s i c al St a t e s ί {Ω\θ,Φ) = Σ Σ Y l m ( 0, 0 ) i?L M T r L = 0 M = - L Ti l m P ( 4.4 3 ) for di f f e r ent choi ces of t h e val ue of Ω^μ · By using the orthogonality relation (A.36) of the spherical harmonics, (4.43) may be inverted to obtain Tr '■l m P ΩLM / sin(0)dθ ά φ ^ ( θ,φ ) Υ £ Μ(θ,φ). (4.44) S u b s t i t u t i o n of t h i s i n (4.41) expr es s es p in terms of the $(Ω\θ,φ). T h e p h a s e s pac e d i s t r i b u t i o n (4.43) i s n o t nor ma l i ze d. We nor mal i ze i t by no t i n g t h a t s i n (0 )d θ ά φ ^ Ω\θ,φ) = ^ 2 ^ ϊ β 0 0' (4.45) In arriving at this result, we have inserted (4.43) for }(Ω\θ,φ) and carried the integration using (A.36) by setting K = Q = 0 in it, and used (A.35) for yoo($> Φ) along with the relation ~ 1 (4.46) (oo — V2 S + 1' Now, in analogy with (4.43), we may define the phase space representation Α(ωϊ (θ, φ) of any spin operator A as Α^Ω\θ,φ) = Σ Σ Υεμ{Θ,Φ)Ως μ Tr |V2 ma | . L=0 M — — L Th e f u n c t i o n Α ^ ^ θ,φ ) determines A through the relations 2S A = Σ Σ IV f\Mk TLM, Tr p t ■ L MJ L = 0 M — — L Ω^μ / sin(0)dfcty Α^Ω\θ,φ ) Υ Ι Μ{θ,φ). ( 4.47) ( 4.48a) (4.48b) Ne xt, we expr e s s t h e t r a c e of a p r o d u c t of t wo o p e r a t o r s i n t e r ms of t h e i r phas e s pa c e r e p r e s e n t a t i o n. To t h a t end, mu l t i p l y ( 4.48a) by B and take the trace to get Tr AB Tr Tlm B L,M ] T ( - ) MIY [t I m A\ Tr [Tl _MB L,M ( 4.49) We have a pp l i e d ( 4.40) i n wr i t i ng t h e s ec ond l i ne above. Use (4.48b) a n d t h e p r o p e r t y ( A.34) t o r e duce ( 4.49) t o t h e f or m 4.2 Phase Space Representation of Spins 91 Tr AB [ sm(9)d9d<j> [ Άη(θ')άθ'άφ' V —------- J J ULMi l L _M YLM(e, Φ)ΥΙμ (Θ\ Φ')ΑΜ(Θ', φ') Β ^'\θ, φ)]. (4.50) Tr AB If = 1 then an application of the completeness relation (A.37) to (4.50) leads to = J 5ίη(θ)άθάφΑ^Ω\θ,φ ) Β ^ { θ,φ ), (4.51) where is the phase space functions corresponding to fiLM = 1/QL _M. The functions and are said to be conjugate to each other. The equation (4.51) shows that the trace of the product of two operators is the phase space integral of the phase space representation of one with the con jugate representation of the other. In particular, if one of the operators, say, A is the density operator then (4.51) determines the quantum expectation value of B in terms of the phase space integral. Now, in analogy with the considerations of the last section, let p = J 5ίη(θ)άθάφΡ(θ,φ)\θ,φ){θ,φ\, (4-52) be the P-function representation. It implies that P l m ) = J sin(0)d0dφΡ(θ, φ)(θ, φ\Τΐ Μ\θ, φ). (4.53) It is known that [61] (θ, φ\ΤΐΜ\θ, φ) = f LMYZM(0, Φ) (4.54) where I l m = ( —) £'~ Μ λ/4 π ( 2,S') · (4.55) ^ ' \J(2S — L)\(2S + L + 1)! Substitute (4.54) in (4.53) and insert the resulting expression in (4.43). Set Q l m = w i t l 1 Ωlm = I l m - (4.56) The summation over L,M then is simply the completeness relation (A.37). It then follows that = Ρ(θ,φ). (4.57) Thus, the QPD corresponding to the Ol m given by (4.56) is the P-function. Verify that the normalization factor (4.45) in this case is unity. Next, take the matrix element of (4.41) in the spin coherent state to obtain 2 s κ (i9,φ\ρ\θ,φ) = Σ J ] (τ*κ<3) ( θ,φ\τ κ(3\θ,, K = 0 Q ——K 2 S Κ . = Σ Σ [nKQfKQ / sm(e')dW K = 0 Q = - K J x/(i 2)(0', φ')Υ*Κ0{ θ φ') Υ Κ(ϊ(θ, φ) ]. (4.58) In writing the second line above we have invoked (4.44) and (4.54). Now, if we let βχ,Μ = with 4 m = /i m (4.59) then the summation in (4.58) reduces to the completeness relation (A.37) leading to the identification /^ ( θ,φ ) = (θ,φ\ρ\θ,φ). (4.60) Thus, the QPD for the Ω^ μ corresponding to (4.59) is the diagonal matrix element of the density operator. The corresponding normalization factor may be evaluated by using (4.45). The normalized form of (4.60), called the Q- function is then defined by 2S -μ i <3(θ,φ)= 4π {θ, φ\ρ\θ, φ). (4.61) Next, the Wigner function for spins is defined, following the deliberations of the last section, as the function which is its own conjugate i.e. the one corresponding to Ω^μ = 1: ^ lm = 1· (4-62) Let /^ ( θ,φ ) be the corresponding QPD. Its normalization factor may be found using (4.45). The normalized form of the QPD corresponding to (4.62) reading 92 4. Quasiprobabilities and Non-classical States / 2 S' -I- 1 νΤ(θ,Φ) = )/—£ Γ/{1)(θ,φ) (4.63) is called the Wigner function for spins. As an example, we give explicit expressions for the Q, P and the Wigner function W for a spin-1/2. Using the expressions for the Clebsch-Gordan coefficients [60], (4.38) yields 3oo = -y=, Tiq = V2SZ, i n = - S+, i i _ i = S-. (4.64) Use of these and relevant expressions for Yl m(6, φ) and Ql m give [61] 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components 93 Ρ(θ,φ) = — [1-6(3).η(-θ,φ)} w(e,4,) = ±- 1 + 2'/3(S).n(e, φ) (4.65) t h e η(θ, φ) = (sin(0 ) cos (φ), sin(0 ) sin (φ), cos(0 )) being the unit vector in the direction (θ,φ). For a s pi n, i t i s pos s i bl e t o def i ne, bes i des t h e pha s e s pace d i s t r i b u t i o n s i n t r o d u c e d above, a n o t h e r cl ass of q u a s i p r o b a b i l i t i e s. I t is t h e d i s t r i b u t i o n of t h e ei genval ues of i t s non- c ommut i ng c o mpone nt s di s cus s ed next. 4.3 Qu a s i p r o b a b i l i t i y D i s t r i b u t i o n s f or Ei g e n v a l u e s o f S p i n C o mp o n e n t s Any c o mpone nt of a s p i n - 1/2 can a s s ume t wo val ues, namel y, ± 1/2. Hence, we ca n c o n s t r u c t a cl assi cal ana l og of a s p i n -1/2 by treating a spin com ponent Sa = S.a in any direction a as a two-state random variable ca pable of assuming the values ±1/2. The classical statistical description of such a system is provided by the probability pm(ea, eb,... ,em) that the spin components in the directions a,b,... ,m assume values ea/2, e j/2,...,em/2 where ea, eb,..., em = ±1. The probability distribution of the spin compo nents Sa, Sb,..., Sm is then given by f m(Sa, ...,S m) = E * (!- S e)...* ( ^ - S m) M t o ). (4·66) w where {e} ξ ea,..., em. In the spirit of the approach developed in the last section, the quantum quasiprobabilities may be constructed from the classical distributions expressed in terms of the averages of the spin variables. To derive the expression for the classical distributions in terms of the averages of the spin variables, multiply (4.66) by (Saa a + l/2 ) · · · (Sma m + l/2), where aa, ct/,, ■.., am = ±1, and integrate over Sa, Sb,..., Sm to obtain 2 ± ^ ^b^b^ * * * = 2 ^ E ^ W X 1 + faa«)( 1 + eba b)....( 1 + emam). (4.67) to The angular brackets denote the average with respect to f m(Sa, Sb,· ■ ■, Srn). Now, on noting that the possible values of ε,α:, are ±1, it follows that the right hand side of (4.67) is non-zero only when = 1 for all i so that Pm({f}) = ( ( 2 '^°e°^ ^ 2 ^b € > ^2 (4.68) 94 4. Quasiprobabilities and Non-classical States This is the desired expression for the probability distribution in terms of the averages. The quantum analog of (4.68), constructed by replacing the c-number variables Sa, Sb, ■ ■ ■, Sm by the operators Sa, Sb, ■ ■ ·, Sm, may be written for mally as KG — + Sa€a j ( - + Sb^b 1 <{(i — + Sb^b )}< Smem ] }c q ), (4.69) where the suffix CO stands for ’chosen ordering’ of the product of the oper ators and p is the quantum density operator of the spin. The p^({e}) is the joint quasiprobability distribution for the eigenvalues of the components of spin-1/2 in the directions a, b..., m. Th e o p e r a t o r or de r i ng pr obl em, of cour s e, does n o t a r i s e i f t h e d i s t r i b u t i o n is s ought onl y f or one component, say, t h e c ompone nt al ong a di r e c t i on a. The equation (4.69) then reads Pi(ea) = Tr ^ + Saf a I P ( 4.70) Now, t a k e t h e t r a c e i n t h i s e q u a t i o n i n t h e e i g e n s t a t e s ja, ± 1/2 ) of Sa t o show t h a t P?( + ) = 1 2 ( 4.71) Si nce | a, is a s pi n cohe r ent s t a t e (see Sect. 3.3), ( 4.71) is p r o p o r t i o n a l t o t h e Q function. The quasiprobability for more than one component would depend upon the chosen ordering. Here we discuss only the completely symmetric ordering defined after (4.12). It turns out to be useful in formulating a criterion for a spin system to be labeled as classical or non classical. On using the anti commutation relation (1.130), the correspondence between some c-number and the operator products in symmetric ordering reads SaSb -»· i ( SaSb + SaS b) = ^ sasbsc ~ (Sa(SbSc + scsb) + ( sbsc + ScSb)Sa +(a ->■ b, b ->■ c, c —> a) + (a ->■ c, c ->■ b, b ->■ a)j a · bSc + c ■ a§b + b ■ c5aJ. (4.72) Of particular interest is the quasiprobability for three orthogonal components. On using (4.72), the QPD (4.69) for three orthogonal components in the symmetric ordering reads 4.4 Classical and Non-classical States 95 1 Tr [(I + Saea + Sb£b + Sc ( 4.73) 2 2 ‘ L1 2 The preceding considerations can be generalized t o a system of N spin- ^ 7)}) that the components l/2s. The quasiprobability pqm{{e%, ),..., emU) of t h e s pi ns, l a be l e d 1,2...,TV, assume values { e ^ j, e(j). mX·?) } along the directions {α^,ίΡ, of (4.69), by . ,m 3} is given, on a straightforward generalization ( u U) ■ j = i N n{ J= 1 O') Λ iCi)J J a(j) a(J' ϋ) O') c0 ) .U) Saih^ 0 ) 0 ) ) L D (4.74) The QPD for three orthogonal components in the symmetric ordering, ob tained by generalizing (4.73), reads O') c0) ,0) \\ i) > fcc(i) -O 22N Tr N n[ J = 1 ^ I C ,0 ) „ I “ «(j) eaU) Q . ,0 ) ^bO>ebu) nr (j) 5cW)eJ(J) (4.75) By following the steps leading from (4.70) to (4.71) and by invoking the fact that a spin coherent state for a system of N spin-1/2s is the product of the spin coherent state for each of the spins, it can be established that the Q-function for a system of N spin-l/2s is the same as i>^r({+}iv)· We will see below that the QPDs introduced in this section play an im portant role in the scheme for identifying non classical states. 4.4 Classical and Non-classical States In Chap. 1 we discussed the merits of the suggestion that the quantum inde terminism may be regarded as a classical statistical one arising from a lack of any knowledge about the dynamics of some hidden variables. States ex hibiting properties not attributable to any classical statistical description are termed non-classical. Here we outline an approach for classifying the states as classical or non-classical based on the concept of quasiprobability distribution functions introduced earlier in this chapter. 4.4.1 Non-classical States of Electromagnetic Field We showed in Sect. 4.1 that each QPD in phase space determines operator averages in certain order unique to it. We found that the QPD ({q,p}) 96 4. Quasiprobabilities and Non-classical States in s-ordering acts as a phase space distribution function for determining the averages of operators in the ordering — s conjugate to s. If ({qsp}) for a state is a classical distribution function, i.e. if it is normalizable and non negative everywhere then the corresponding state is classical with respect to the measurement of operator averages in — s ordering. This means that the moments of operators in — s ordering in the said state will exhibit all the properties of the moments of a classical distribution. If ({g,p}) assumes negative values or is non-normalizable then it is non-classical The moments of operators in — s ordering in that case may exhibit purely quantum char acteristics, i.e. the properties not expected from the moments of a classical distribution. Such non classical characteristic are exhibited in the form of violation of some inequalities between the moments expected of a classical distribution. Note that the Q-function, being the diagonal element of the density ma trix, is always positive and normalizable. That function corresponds to s = 1 which characterizes the normal ordering. Hence every state is classical with respect to measurements of anti-norinally ordered products corresponding to s = —1. The functions ({?,£>}) for other orderings may become non- classical. What value of s should one choose to classify the states? The answer to this question is provided by the mechanism by which experimental mea surements are made. Here we confine our attention to the issue of characteri zation of the states of the electromagnetic (e.m.) field. We will see in Chap. 6 that the e.m. field is described by the harmonic oscillator operators and that the experimental measurements on it consist in measuring the expectation values of those operators in normal ordering. Since the normal order corre sponds to s = 1, the QPD appropriate for classifying the states the e.m. field as classical or non-classical is the one corresponding to s = — 1. As shown in (4.29), the QPD corresponding to s = — 1 is the P-function. A state of the e.m. field is accordingly called classical if its P-function is classical. Else it is labeled non-classical. In terms of this criterion, it can be proved that the coherent state is the only pure state of the e.m. field that is classical [62]. To that end, let p be the density operator of a pure state of the field and let P (a, a*) be the corresponding P-function. Since p describes a pure state, p = p2. The normalizability demands that If P(z,z*) is a classical distribution function then it must be non-negative and normalizable as in (4.76). However, if that be so then the right hand side in (4.7 7 ) is always less than one unless (4.76) Now, the expression (4.28) of p in terms of P(a,a*) yields Tr [p2] = J d 2a d2p exp [~\a - β\2} Ρ(α,α*) Ρ( β, β*). (4.77) 4.4 Classical and Non-classical States 97 P(a,a*) = δ(a-a!)δ(a·-a.*,). (4-78) Since, by virtue of (4.76), the left hand side of (4.77) is required to be always unity, it follows that (4.76) and (4.77) are consistent only if (4.78) holds which is the expression for the P-function of the coherent state la'). The MUS may be shown to be the only pure states of the e.m. field whose Wigner function is non-negative [63]. The experimentally measurable quantities are, of course, the moments. Experimental observation of non-classicality of the P-function should, there fore, be translated in to the properties of the moments. Some such properties are discussed in Chap. 6. 4.4.2 Non-classical S tates of S p i n - l/2 s In the absence of any guideline based on the method of measurement, other considerations come in to play for classifying the states of a system of spins as classical or non-classical. A consideration is suggested by the deliberations in Chap. 1 regarding non-classical characteristics of a pair of spin-1 /2s. We found there that the non-classicality of a pair of spin-1/2 s is linked with correlations between them. An uncorrelated state of spin-l/2s, and hence any state of a single spin-1/2, is classical. Recall from Chap. 3 that an uncorrelated pure state of a system of spins is its coherent state. We may, therefore, base the classification scheme on the premise that it should classify any uncorrelated state as classical. However, this scheme is not helped by the phase space distribution of a spin, introduced in the Sect. 4.2. For, (4.65) shows that the P and W functions are negative even for a spin in the eigenstate | ± 1/2) which is a spin coherent state. However, the QPD of the eigenvalues of spin components, introduced in Sect. 4.3, do lead to a scheme of classification of states according to the following criterion [25]: A quantum state of a system of N spin-l/2s is classical if the joint quasiprob ability for the eigenvalues of the components of each spin in three mutually orthogonal directions, one of which is the average direction of that spin, is classical in the symmetric ordering of the operators. It is non-classical if any of those. m-spin (m < N ) joint quasiprobabilities is negative in the said or dering. Th i s c r i t e r i o n i dent i f i es (i ) t h e cohe r ent s t a t e s of N spin-l/2s as classical, (ii) any state of single spin-1/2 as classical; (iii) any pure entangled state of two spin-l/2 s as non-classical: this is in conformity with the finding in Chap. 1 that any such state violates Bell’s inequality which is a signature of non-classicality; (iv) squeezed spin states as non-classical. For details, see [25]. 5. Theory of Stochastic Processes In a number of situations, the forces acting on a system are non-deterministic. The dynamical variables of the system then are random functions of time. The behaviour of such variables can be described only statistically. The problem of studying the statistical behaviour of random functions of time is the subject of the theory of stochastic processes. In this chapter we summarize some concepts and the operational techniques of the theory of stochastic processes. For details, we refer to [64]- [67]. 5.1 Probability Distributions Consider a process described by a real valued function ξ(ί) of time t. If ξ(ί) traces unpredictable paths as a function of t in different realizations of an experiment then it is called a random function and the process described by it as a stochastic process. Let &(i) be the functional form of the path traced by ξ(ί) in the ith realization of an experiment. The set of all possible realizations £i(t), £2( ^) 1 · · · °f constitutes an ensemble of ζ(ί). The theory of stochastic processes deals with the problem of determining the ensemble- averaged values {f (£( t i ),---,£(t n))). Cons i de r f i r s t t h e pr obl e m of d e t e r mi n i n g {f ( £( t ) ) ) · Let p i ( x,t ) d x be the probability that the value of £(t) at time t lies in the interval [x,x+dx]. The single time probability densi t y (henceforth also called single time probability) Pi (x, t) is defined by Pi(x,t) = [5{x - i(t))) (5.1) where the average is over all realizations of £(t). The average of a function of ξ(ί) may be determined in terms of pi ( x,t ) by noting that (/(£(*))) = ( J f ( x ) S ( x ~ £ ( t ) ) ) d x = J pi ( x,t ) f ( x) dx. (5.2) The integration is over the range of realizable values of ξ(ί). In a simi lar way, the average of a function of £(ti), ■ ■ ■ ,ξ(ί η) is determined by the n-time joint probability density (henceforth also called n-time probability) Pn(xn,t n;...-,xi,t i ) = p„ ({xi,ti}) defined by Pn ({Xi,U}) = (5(a;i - ξ( ί ι ) ) ■■■δ{χη - ξ(ί „))>. (5.3) The pn ({Xi, ti}) da;i · · ■ dxn is the probability that ξ(ί) assumes values ξ(ίι), ..., ξ(ίη) at times ti, ■ ■ ■ ,t n lying in the intervals [x\,x\ + da^],..., [xn,xn + d^n] respectively. The n-time averages are determined by the rela tion (/(£(ίι)>· ··>£(*"))) = J dxi ■■■ J d x n Pn d x j,^ } ) f (xu ... ,x n). (5.4) A stochastic process is completely described by the infinite hierarchy of the probability densities pn({xi, h}). (n = 1,2,...). However, note that Pk (Xn+k, tn+k i ’ ' ' , Xn j t n) J dxn — i dXipn-\-f» ({Xj ,t j }). (5.5) This is the compatibility condition relating fc-time probability with n+fc-time probabilities. A stochastic process is called stationary if its probability densities are invariant under time translation, i.e. if Pn({Xi,ti}) = Pn({xi,k + T}), (5.6) where T is arbitrary. On setting T = —t i we see that Pi(x,t) =Pi(x,0), P 2 ( x i,t i;x2,h ) = P2(xi,0;x2,t 2 - h). (5.7) This shows that single time probability of a stationary process is time- independent and that its two-time probability depends only on the time difference. Now, by invoking the representation (A.l) of the delta function, (5.3) may be rewritten as Pn {{Xi,U}) ^ \ n poo poo f 71 \ — j J dU i -"J dunCn ({iii,ti})exp ί - i (5.8) where the Fourier coefficients Cn({ui,ti}) of pn({xi, ti}), 100 5. The o r y o f St o c ha s t i c Pr oc e s s e s Cn({ui,ti } = ^exp ^ i ^Wf e ^( i f e ) ^ = V — j--------:( i « i r i ---(i un)m’* ( r i ( i i )"- r ’*(in)) (5-9) Λ 7711 . * * · ΎΥΧγϊ . is called the characteristic function. An average (ξ1"1 (ii) · · · £m” (ira)) is called a moment of ξ (t ) and the number m x + · · · + mn its order. The concept of characteristic function enables us to determine the probability densities in the problems specified in terms of the moments of a random function. Note from (5.9 ) that the characteristic function generates the moments by means of the relation 5.1 Probability Distributions 101 Π ^'( ω ) = Π Yj=i i =i d m, ( 5.1 0 ) Uj = 0 Now, if ξ(ί) assumes the same value at any time t in all its realizations with out any correlation between its values at different times then any moment (£mi (ii) · · · ξ1ΎΙη(ίη)} factorizes in to the product (£(ti))mi · · · (ξ(ίη))ηΐη· We may classify the stochastic processes according to the deviation of its mo ments from their factorized value in terms of lower order moments. A measure of this deviation is provided by the cumul ant s ((£mi (ii) ■ ■ ■ζπΐη(ί η))) defined by . (5.11) ,=o The number mi + · · · + mn is the order of the cumulant in (5.11). This defi nition of cumulants is equivalent with the following expansion of the charac teristic function in terms of the cumulants Cn ({wj , ti }) OO = exp[ ^ { m i -0 } mi! · · ■ m„! mi + ■ · · + m„ φ 0. (5-12) By expressing an integral as a sum followed by the use of (5.12), it may be shown that [64] rt exp = exp υ,(τ)ξ(τ)άτ t ^ p rt rt T - t dri ■·· / άτηη{τι) ■ ■ ■ u{rn) χ({ξ{τι)···ξ{τη))) ■ ( 5.13) I t s houl d b e e mpha s i z e d t h a t t h e q u a n t i t y i ns i de a doubl e a ngul a r br a c ke t d e n o t i n g a c u mu l a n t i s no t a n al ge br a i c o p e r a n d. I t i s onl y a symbol i ndi c a t i ng t h e h i ghe s t mome nt i n t h e e xpr es s i on of a c umul a nt i n t e r ms of t h e mome nt s. Two l owes t - or der cumul a nt s i n t e r ms of t h e mome nt s a r e m u ) ) ) = ( m ), a m m ) ) = m x m - <£&)> <£&)> = σ ( u,^ ). (5.1 4 ) The first equation above shows that the first order cumulant is the mean whereas the second order cumulant is the variance if tj = ti and a covari ance if tj φ ti. The covariance is a measure of correlation between the val ues of the random function at two different times. The elements a(t i,t j ) 102 5. Theory of Stochastic Processes (i,j = 1,2,· · · ) constitute the correlation matrix σ. It is a symmetric ma trix. By virtue of the positivity of the probability densities, we can identify the average (ζ(ί ί)ξ(^)) as a scalar product which fulfills the axioms of a scalar product listed in Sect. 1.1. Consequently, the generalized Schwarz’s in equality (1.7) in this case reads det(<r) > 0 where the matrix σ has a(U,tj ) (■i,j = 1, 2,..., N) ( N=l,2,...) as its elements. As an example, for N = 2 it states that a(t i,t i )a(t j,t j ) > a2(ti,tj). Now, i f al l t h e mome nt s of a pr oces s f a c t or i z e i n t o t h e p r o d u c t s o f {£(£*)) t h e n Cn({u}) = exp(hii(£(£i))) · · · exp(m„ {£(£„))). Insert this in (5.11) and verify that {{ξ"11 (ti) ■ ■ ■ ξπΐη (tn))) = 0 if m i + ·■· + m n > 2. This property characterizes a deterministic process. The cumulant expansion enables us to define the process next to a completely factorized one as the one for which ((ξτηι(ti) ■ ■ ■ £m"(tn)}} = 0 if toi + · · · + m n > 3. It is called a Gaussian process. On expressing {(£(ti)£(tj))) in terms of a(t i,t j ) as in (5.14), the characteristic function (5.12) for a Gaussian process assumes the form Cn({ui,U}) = exp I ^ ^ 2a( t i,t j ) ui uj j . (5.15) \ i ϋ / We c a n f i nd t h e mome nt s by s u b s t i t u t i n g t h i s i n ( 5.10). As a n exampl e, l et us d e t e r mi n e s i ngl e t i me mome nt s as s umi ng {£(£*)) = 0. To t h a t end, e xpa nd C\(ui) in (5.15) in powers of u and insert it in (5.10). Carry the required differentiation to show that single time moments of a Gaussian process are given by <£2m) = ( f ) m ^, <£2m+1) = 0. (5.16) Next, we derive the n-time probability distribution by inserting (5.15) in (5.8). We assume σ to be positive and carry the integration over {ui} by applying (A.22) to obtain Pn({Xi,ti}) 1 1 ( X * - (?>) ( X - « ) y/(2π)η det(<r) exp (5.17) Here X and ξ are columns formed by n elements x x,..., xn and £(t i ),..., ζ(ί η) respectively, and A T denotes transpose of A. Ne xt we o u t l i n e t h e s cheme f or cl as s i f yi ng s t o c h a s t i c pr oces s es a c c or di ng t o t h e i r c or r e l a t i ons i n t i me. I t l eads t o t h e c onc e pt of Mar kov pr oces s es whi ch enc ompa s s a l a r ge va r i e t y of pr oc es s es of p r a c t i c a l i nt e r e s t. 5.2 Ma r k o v P r o c e s s e s We n o t e d i n Sect. 5.1 t h a t a s t o c h a s t i c pr oc e s s i s c h a r a c t e r i z e d by t h e i nf i ni t e h i e r a r c hy of t h e p r o b a b i l i t y de ns i t i es ρη({χ%, ij})· Of considerable importance 5.2 Markov Processes 103 is also the concept of conditional probabilities. It enables us to classify the processes according to their correlations at different times The probability density of finding ξ(ί) in the intervals \xr, xr + dxr], ■ ■ ■ \xn, xn+dxn] respectively at times t r,..., t n under the condition that £(i) has known values Xi,..., x r~i at times 11,..., 1 (ti < t 2 < ■ ■ ■ < t n) is called the conditional probability density p( x n,t n;... ;xr,t r\xr- i,t r- i;... ;xi,t i ). We note that, for tn > ■ ■ · > ti, Pn({xi,ti}) = p(xn,tn\xn-i,tn~i;...; Xi, 11 ) p„_i ({Xi, t i }). (5.18) Consider the conditional probability p ( x n, t n\xn- i, i „ _i;...; Xi, ti) for find ing ξ(ί) in the interval [xn, x n + d x n] at time t n when its value at the earlier times i „ - i,... ,t i is known respectively to be xn- i,..., x\. If this probability at any time t n is independent of its values at the earlier times then P{Xn 5 t n j xn — 1, — 1 r · · - f Xl r t l ) — Pi (Xn; tn) . (5.19) On inserting this in (5.18) and on repeating the argument, it follows that Pn(xn,tn; ■■■]Xl,t1) =Pl(xn,tn) ' ' ’ Pi ( xi, t i ). (5.20) This shows that there is no correlation between the values of the random function at different times. However, if £(t) is a continuous function of time, we expect its value at a time to be correlated with its value at a time at least infinitesimally close to it in the past. Bearing aforementioned arguments in mind, next in the scheme of classifi cation based on the conditional probabilities would be the processes in which the value of the random function at a time t is correlated only with its value at the time immediately preceding t. Such a process, called a Markov process, is characterized by the relation P(Xri i | Xn — l:t n—l,...',X l,t i ) p ( x n , t n jXn— 1, tji — 1)5 (5-21) tn > ■ ■ ■ > ti. On substituting this in (5.18) we find that P„({xi,t}) = p(xn,tn\xn- i,t n-i)pn~i{{xi,t}) (5.22) On using (5.21) again and on repeating the argument, we obtain Pn({Xi,U}) = p(xn,tn\xn^ i,t n-i)---p(x2,t2\xi,ti)pi(xi,ti). (5.23) This implies that a Markov process is determined completely by its single time probability distribution pi (x, t) and the conditional probability p(xi, t i\xj,t j ) called the transition probability. The transition probability of a Markov pro cess obeys the condition (i3 > t 2 > t\) p(x 3,t 3\x i,t i ) = J p ( x 3,t 3\x2,t 2) p( x2,t 2\x i,t i ) d x 2 (5.24) called the Chapman-Kolmogorov equation [64]-[67]. The rate of transition from Xj to Xi for Xi φ Xj is defined by 104 5. Theory of Stochastic Processes w(Xi\Xj) = Lt At ^ o p(xj,t + At\xj,t) - p(xj,t\xj,t) At (5.25) p(xi, t\xj, t) = S(xi — Xj). Markov processes describe a variety of physical phenomena. In practice, a process may be specified in terms of its transition rates. If it is a Markov process, then its probability density can be shown to evolve according to the master equation [6 6 ] This may be interpreted as a probability balancing equation. For, the first term on the right hand side of (5.26) represents the rate at which probability of assuming a value x is gained at the cost of the probabilities of the other values whereas the second term gives the rate at which the probabilities of the other values gain at the cost of the probability of the value x. The differential equation form of this equation is the Kramers-Moyal expansion [65] The equation (5.27) determines the evolution of the probability in terms of the moments of the process. Its formal solution yields Verify by combining this with (5.25) that the rate of transition may be ex pressed as Now, the quantities of practical interest are the averages of functions of x. The equation for the average of a function /(x) can be found by multiplying (5.27) by /( x) and integrating with respect to x to get with h(x) and g(x) obeying appropriate boundary conditions. Verify by re peated partial integration that if the functions in question and their deriva tives vanish at the boundaries then (5.26) p(xt) = L(x,t) p(xt), (5.27) (5.28) ξ ( ί ) = χ (5.29) (5.30) w(x\x') = L (x, t) δ (x — x'). (5.31) (5.32) wher e Lt (x ) is the adjoint of L(x) defined by the relation (5.33) 5.3 Detailed Balance 105 / d x g ^ ^ h^ = ^ m J d x h ^ - ^ 9 ( x ) · (5.34) On comparing this with (5.33) we see that .Jm \ t j m = Η ” = · (5-35) dxmJ dx In the foregoing, we assumed that the random function assumes contin uously distributed real values. We will have occasions to deal with random functions which assume only integral values. The master equation (5.26) then reads d — p( m, t ) = 5 3 [w{m\n)p( n, t) — w( n\m) p( m, t ) }, (5.36) n the p ( m,t ) being the probability that the value of the random function at time t is m and w( n\m) is the rate with which its value m makes a transition to the value n. Th e e q u a t i o n f or s i ngl e t i me p r o b a b i l i t y of a Mar kov pr oces s e nabl es us t o d e t e r mi n e i t s mul t i - t i me cor r e l a t i ons as wel l. Th i s p r o p e r t y i s unde r l i ne d by t h e regression theorem. It states that if £i ( t ),... ,£jv(t) are N random functions of time described by a Markov process such that their averages evolve according to d N ^ & ( ί ) ) = Σ αΰ·&(*)>. (5·37) j=i where are independent of time then the evolution of their two-time cor relations is governed by d N ~^(£i(t)£k(to)) = (5.38) j = 1 This equation is the same as (5.37) for single time average. See [65, 6 8 ] for its proof. 5.3 Detailed Balance If L( x ) is time-independent then (5.27) may be solved by the method of eigenfunction expansion discussed in Chap. 10. Let zero be a non-degenerate eigenvalue of L( x ) and let the real part of all its other eigenvalues be negative. Consequently, as t —> oo, p( xt ) —> pSs(x) where pss(x), called the steady st at e distribution function, is the eigenfunction corresponding to the eigenvalue zero: pss(x) = L( x ) pss(x) = 0. (5.39) 106 5. Theory of Stochastic Processes The solution of the steady state equation is greatly facilitated if the process obeys the condition of detailed balance discussed next. Consider a stationary Markov process described by N random functions £1 (£)>··· >£iv(i)· Let u>({a^}|{a:j}) be the rate of transition from the values {xi} to {x'j} and let pss({a:i}) be the corresponding steady state. Under the operation of time-reversal t —> —t, let Xi —»· eix, {μ*} —>· {είμί} where {μ*} denotes a set of external variables and = ±1. The process is said to be in detailed balance if pss({^i},{Mi}) = Pssde^i}, {eiMi}), (5.40a) u>({e^'}|{ei^})pss d ^ }, {μ*}) = ^ ( { ^ i }| } ) i > ss({ari}, {μ*}). (5.40b) The first of the conditions above is the statement of invariance of the steady state distribution under the time reversal. The second condition states that the number of transition from a state described by the values {a:*} to another state described by the values {a?'} is the same as the number of transition from {x^} to { x ^ in the reverse direction of time. Using (5.31) to express u>({?/}|{z}) in terms of L({y}), and after a little algebra, (5.40b) may be reduced to the form [6 6 ] Pss ({χί}, {μ*}) ({tiXi}, { m } ) F ({a?J) = L ( M ) Pss ({Zi}, {μ*}) F ({a;*}), (5.41) the F being an arbitrary function. This is the detailed balance condition in terms of the operator governing the evolution of the probability density of a Markov process. We demonstrate the usefulness of this condition in the next section for finding pss. Th e d e t a i l e d ba l a nc e c ondi t i on for t h e cas e of di s c r e t e l y var yi ng r a n d o m va r i a bl e d e s c r i be d by t h e ma s t e r e q u a t i o n ( 5.36) r e a ds w(m\n)ps* = w(n\m)ps^l. (5.42) This states that the number of transitions in unit time from the value n to the value to in the steady state is the same as that from m to n. 5.4 L i o u v i l l e a n d F o k k e r - P l a n c k E q u a t i o n s An a d v a n t a g e of wor ki ng wi t h t h e Kr a me r s - Mo y a l expa ns i on (5.27) l i es i n t h e pos s i bi l i t y of t e r mi n a t i n g i t a t s ome f i ni t e s t e p. However, a t h e o r e m due t o Pa wu l a (s ee [6 6 ]) asserts that the solution of (5.27) is everywhere non negative if it is truncated at the first or at the second term but not if it is truncated at any other finite step. Since the admissible solutions of (5.27) should be non-negative, it follows that it can be terminated up to the second step or else all its terms should be retained. This theorem holds also for a multivariate system. 5.4.1 Liouville Equation The expansion (5.28), generalized to N variables and truncated at the first step, reduces (5.27) to the Liouville equation p({xi },t ) = (5-43) i w r - T> ( m + At) - Ut ) } ) (,u ) Ai {{xi }) — Lt At ^o τ - · (5.44) l i t £i(t)=Xi To find the solution of (5.43), consider the equation zt(t) = At({Zl(t}), (5.45) with Zi(to) = Vi as the initial condition. Let Zi({yi},t) be the corresponding solution of (5.45). Let p({xi},to) — δ({χί — Vi})· Verify by direct substitution that (5.43) is solved by = &{{xi ~ Zi({yi},t)})· (5-46) This means that, for given initial conditions, the system traverses a definite path determined by the solution of (5.45). For details, see [65]. 5.4.2 The Fokker-Planck Equation The multivariate form of (5.28) terminated at the second term reduces (5.27) to the Fokker-Planck equation p( {xk},t) = LFP({xk})p({xk},t), (5-47) where the Fokker-Planck operator is given by lmm) - Σ [- ·') +1Σ (M*> i j= 1 The column of t he elements {Ai ({xi })}, defined in (5.44), constitutes the drift vector whereas 5.4 Liouville and Fokker-Planck Equations 107 DM n } ) - ( 5.4 9 ) ik(t)=xk c o n s t i t u t e t h e diffusion matrix. The diffusion matrix is positive. The Fokker- Planck equation is a continuity equation for the flow of the probability as is revealed by writing it in the form £({*<}> 0 + Σ S T = 0 (5·50) . , dxi 1=1 wher e 108 5. Theory of Stochastic Processes 1 d J i = Ai ({xi },t ) - - Υ ^ — Οί:ί {{χ},ί ) p({x},t). i = 1 3 3 = 1 (5.51) is the probability current. Now, i f Lpp({xi }) is time-independent, then the steady state pss({^i}) of the Fokker-Planck equation is obtained by solving For a univariate process, this implies J = constant, i.e. (5.53) where C is a constant to be determined by the boundary conditions. If J is assumed to vanish at the boundaries then (7 = 0. In this case (5.54) shows that pss is peaked at the maxima of Φ(χ). However, i f t h e pr oces s i s mu l t i v a r i a t e t h e n ( 5.52) does no t neces s ar i l y i mpl y Ji = constant for each i. Solving (5.52) is generally a formidable task. This task is simplified considerably if the process satisfies the condition (5.41) of the detailed balance. In order to see that, substitute the Fokker-Planck operator Lpp of (5.48) for L in (5.41). Note that, by virtue of its defini tion, D({eiXi: CjXj}) = ei€jD({xi,Xj}). Now, equate to zero separately the coefficients of F ({a^}) and of dF( {xi ])/dxi to show that Pssd^i}) solves The task of finding pss for a detailed-balanced process is thus reduced to solving first-order equations. The equation to be solved in the absence of the detailed balance is a multivariate second-order partial differential equation. We have outlined above the way of finding the probability density of a process described in terms of its moments. In many practical situations, a system is described by equations of evolution of its dynamical variables. In the next section we discuss how to extract the stochastic properties of a system from its dynamical equations. (5.54) the Φ(χ) being the potential function defined by (5.55) i (5.56) 5.5 Stochastic Differential Equations 109 5.5 Stochastic Differential Equations An equation of motion of a dynamical variable subject to the influence of a random force is named a stochastic differential equation (s.d.e.). We consider the s.d.e. written in the form where the first term on the right hand side is a deterministic function of time and of {£i(t)} whereas τ^(ί) in the second term are random functions of time. Given the probability distributions of ηι(ί), our aim is to derive those for Solution of a stochastic differential equation involves integration over ran dom functions of time. For details of the theory of integration of stochastic differential equations, see [65]. Of several ways of defining integration involv ing random functions, it turns out that the definition of Stratonovich enables one to use the rules of the theory of the calculus of ordinary functions [65], In what follows, we treat the stochastic integration in the same way as the integration of ordinary functions. In other words, we implicitly follow the approach of Stratonovich. It is known that if η{(ί) are delta correlated, i.e. if {(ηι(ίι) ■ ■ · ηη(ίη))) = D( t i ) S( t i —12) ■ ■ ■ S( t i — t n ) then ξ(ί) is Markovian [64]. The equation govern ing the evolution of a probability density of the process described by (5.57) can be derived analytically in general if {r]i(t)} is a stationary delta-correlated Gaussian process characterized by Note that the spectral density S(uj), defined in (6.91), is independent of ω if the process is delta-correlated. Hence a delta-correlated process is also called a white noise. The processes for which S( uj) exhibits dependence on ω are said to constitute coloured noise. The process described by (5.58) is the Gaussian white noise. Now, t h e def i ni t i on (5.1) of s i ngl e t i me p r o b a b i l i t y d e n s i t y o f a uni v a r i a t e pr oc es s gene r al i ze d t o n random functions £i ( i ),..., £zv(i) reads (5.57) 3 {&}■ (m(t)) = 0, (Vi(t)%(t')) = Di5ijS(t - t ’) ■ ■ -Vi(tn))) = 0, n > 3. (5.58) N (5.59) i—1 Differentiate p with respect to time and substitute (5.57) for ξι. Use next the property (A.7) of the delta-function to show that 110 5. Theory of Stochastic Processes Derivation of the equation for the probability density p({xi},t) is facilitated by the following theorem: Theorem 5.1: Let y({x}, (t)) be a vector function of variables ({cc}) obey ing y( {x},t ) = A + J 2 B ^ k(t)^y({x},t), (5.61) k= 1 where ^4({a;}) and Β^({χ}) are matrices whose elements are differential op erators of {x}. If {qk{t)} is a Gaussian white noise characterized by (5.58) then (y({x},t )) obeys <y(M, t)) = U- + - DkBl 1 (y({x}, t)) k= 1 (5.62) where the average is over the distribution of {ηk{t)}· We refer to [69] for its proof. On applying this theorem to (5.60) and on recalling from (5.59) that {p) = p({xi },t ), we find that p({xi],t) obeys the Fokker-Planck equation d N 1 x β Q η + 2 Σ Dk^m({xi},t)-^gjk({xi},t)\p({xi},t). (5.63) I f p({xi },t o) = i ( xi — yi) ■ ■ ■ 6(xn — Vn ) then the solution of (5.63) gives the transition probability p({xi}, t\{yi}, to). Since delta-correlation of {r]k{t)} ensures that {£,(£)} is Markovian [64], the transition and single time proba bilities determine all its multi-time correlation functions. In the next two sections we discuss some analytically solvable cases of (5.57). To that end, it is useful to classify the equations according to how the system is coupled to the random influence (%(ί)}. If it is so coupled that {g^} are independent of {£i(t)} then the noise induced by (?jfc(i)} is called additive. Else it is termed a multiplicative noise. There are, however, situations which permit transformation between the two types of noises [6 6, 67]. 5.6 Linear Equations with Additive Noise In this section we solve the linear s.d.e. = /(<)£(*) + 9(t)v(t)· (5·64) If η(ί) is a Gaussian white noise then ξ(ί) is said to describe a Wiener process if f ( t ) = 0, and an Omstein-Uhlenbeck process (OU) if f ( t ) φ 0. The formal solution of (5.64) is 5.6 Linear Equations with Additive Noise 111 rt a(t, to) = exp ^ d r/( r ) Hi>r) = d(T)a(t,T)· (5-66) We insert (5.65) in the definition (5.9) of the characteristic function to obtain Ci (u) = (exp(i<(i))) = exp [iua(t, io)^o] ^ exp d r b(t, τ)η(τ) Appl y ( 5.13) t o e xpr es s t h i s e q u a t i o n i n t h e f or m M! ml Ί= 1 Ci(u) = exp [iua(t, io)^o] exp Σ Kmitj to) (5.67) (5.68) Km(t, to) = [ dTm · · · f άτ1( ( η( τ ι ) · · · η( τ πι) ) ) δ( ί,η) · · · δ( ί,τ πι). (5.69) Jto J ίο Since C\(u) constructed above is under the condition that ξ(ίο) = x0 ) its substitution in (5.8) yields the conditional probability p(x, t\xo, to)· The in tegral in (5.8) can be evaluated analytically exactly if Ci(u) is a Gaussian, i.e. if K m(t) = 0 for m > 2. This, in turn, requires that η(ί) be Gaussian. By assuming that to be the case and with (τ?(ί)) = 0 we get (x - a(t,t0)x o) p(x,t\x0,t0) 2π K2(t,t0) exp 2K2(t,to) (5.70) Verify by direct substitution that p(x, t\xo,t Q) obeys the Chapman-Kolmogoro-* equation (5.24) for a Markov process only if if η(ί) is delta-correlated. If η(ί) is not delta-correlated then we need to construct multi time probability den sities for a complete characterization of the process described by ξ(ί). The multi time probabilities may be derived in a manner similar to the one fol lowed above for deriving the single time probability. Let η(ί) be a Gaussian white noise characterized by (5.58). Also, let f ( t ) = 0 so that ξ(ί) describes a Wiener process. If, in addition, <7 = 1 then a(t, to) = 1, b(t,r) = 1, K 2(t,to) = D(t — to). The equation (5.70) then assumes the form (x - Xq) p(xt\xQto) 2π D(t — to) exp 2 D{t - 10) ( 5.71) Ne xt, l e t f ( t ) = —7 ( 7 > 0), g(t) = 1 so that ξ(ί) describes an OU process. On inserting these values in (5.66) and (5.69) we find that a(t, t0) = exp{—7 (i - t 0)}, b(t, r ) = a(t, r ) K2(t,t0) = ^ | l - e x p ( - 27( i - i 0))}· (5.72) The expression (5.70) then assumes the form p( x,t\xQ,to) = 112 5. The o r y o f St o c ha s t i c Proc es s es 7 ■kD(\. — exp(— x exp 7 :p(—2 7 (i - i 0))) — exp(—7 (i — io)^o)| D 1 - exp(-27(i - t 0)) I n t h e l i mi t t —> oo it yields p(xt\x0t0) -S· J ^ exP ( - ^ χ2)> (7 > 0). (5.73) (5.74) The asymptotic distribution is independent of the initial distribution. It is straightforward to verify that (5.73) obeys the Fokker-Planck equa tion _ [_ a , ρ { χ Λ Μ _ (5J5) dt d x 2 dx2 Thi s i s t h e s a me as (5.63) s peci a l i z ed t o t h e p r e s e n t s i t u a t i o n. Ne xt, us e ( 5.65) t o show t h a t (Φι)ξ&)) = e x p ( -7( i i +h)) D pt i pt Jo Jo d r i d r 2 e x p (7 (ri + τ2))δ(τι - r 2).(5.76) The 5-function reduces the double integral above to single integral in which the upper limit is the smaller of t\ and ί 2· Assume also that 7 ^1,7 ^ 1- The expression (5.76) then reduces to (£(ίι)£(ί2)) = 7T~ exP{—7^1 ~ h I}· 2 7 (5.77) This is t he two-time correlation of an OU process in t he long time limit. Generalization of t hese results to mul ti var i at e systems is straightforward. 5.7 Linear Equat i ons wi t h Mul t i pl i c a t i v e Noi s e In t hi s section we discuss t he problem of solving k = [F(i) + G(i)T?(i)]f (5.78) Here ξ is a column vector constituted by the elements (ξι(ί ),... ,&ν(ί)); F(t ) and G(t) are the N x N matrices independent of {£(£)}. We discuss separately the univariate and the multivariate cases. 5.7 Linear Equations with Multiplicative Noise 113 5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations The univariate form of (5.78) reads £ = Its formal solution is given by (xq = £(0)) ξ(ί) = xq a(t) exp Γ /·* / d7-77(1 Jo )g(r) a{t) = exp [ f(r)di Jo (5.79) ( 5.80) On r a i s i n g t h i s t o power m and on invoking (5.13) we obtain r ( i ) ) = < « m(i)ex p Σ n= 1 t Kn(t) I d-n---J Jt 0 J to dτη((η(τι) · · ■τ?(τ„)))5 (τ1) ■ · ■ g(rn). (5.81) If η(ϊ) is a Gaussian with zero mean then <r(t)> = < a m( t ) e x p [ — t f 2(t) Substitute this in (5.9) to obtain 00 2 rn C\(u) = 5 3 —^ iuxoo(i)] exP (5.82) (5.83) Now, rewrite the exponential in (5.83) recalling (A.20) (with β = m, a = 1/2 K 2(t)) so that I J ΛΟΟ r 0 0 1 Γ Ί 1 °'{U) = ^ [ i w W e x p M j x e x p ( - y 2/2i^2(i))] · (5.84) The summation over m can now be carried. Substitute the resulting expres sion in (5.8). The integration over u is a delta function in x — xoa(t) exp(y). This enables us to integrate over y and obtain the probability distribution as p(x,t\x0,0) = 1 2 n K2(t)x2 exp 1 In 2 I. 2 K 2(t) Ιχοα(ί) }]■ (5.85) For an application of this result in laser physics, see [6 8 ]. 114 5. Theory of Stochastic Processes 5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations Now we examine (5.78) when ξ is an TV-dimensional column vector. We as sume F and G to be independent of time. Define £'(t) = exp (~t F) i ( t ) so that £'(t) = v (t)G(t)?{t), 0 = exp(-iF)Gexp(iF). Its formal solution, averaged over the distribution of η(ί), is exp (€'(*)) [ dτ η( τ ) 0( τ ) Jtn m - (5.86) ( 5.87) (5.88) Recal l f r om Sect. 1.1 t h a t i f G(h), G(h) 0 f or al l t\,t 2 then the time- ordered exponential integral is simply the exponential of the integral. This condition will be satisfied if F, G = 0. Assuming this to be the case, we invoke (5.13) to perform the cumulant expansion of the average of the exponential of the integral and transform back to ξ to get (£(t)) = exp (tF) ( 5.89) x exp 1 Λ Σ ^ ί t*! n! λ ■ f άτη{{η{τλ) · · · η{ τ η) ) )0η Jt o m - I f F, G Φ 0 then (5.88) can be evaluated if η(ϊ) is a Gaussian white noise. For such an η(ί), we can apply theorem 5.1 to (5.78) to get (i)= [F(i) + | (?2 ( φ ). (5.90) The problem of solving (5.78) for a multivariate system driven by coloured η(ί) is generally analytically formidable one. A case of coloured noise of com mon interest is when η(ί) is a Gaussian with zero mean and the two-time correlation given by D-y (v(ti)v(t2)) = — exp( -7 |i! - t 21). (5.91) The form of the coefficient of the exponential above ensures that, in the limit 7 —> oo it reduces, by virtue of (A.8 ), to DS(ti —t 2). The correlation in (5.91) is of the form (5.77) which arises in the long-time limit of the OU process. We, therefore, assume η(ί) to be an OU process governed by V(t) = -ΊΉ + «(O: (5.92) where a(t ) is Gaussian white noise with zero mean and the two-time corre lation given by 5.8 The Poisson Process 115 (a(t)a(t')) = DY6 ( t - 1'). (5.93) The problem now boils down to solving the coupled equations: equation (5.78) for £(i) and (5.93) for Tj(t). The noise in these equations is a Gaussian white noise contributed by a(t). These equations may be combined to resemble (5.57) whose probability distribution obeys (5.63) if the noise is Gaussian white noise. To that end, we let £jv+i = v(t) so that the set of equations JV ζί ^ ^ \Fij (t) -(- G{j 1 (1 ^iJV+l)^' 3 = 1 +a(t)SiN+i, (5.94) i = 1, 2,..., TV + 1, is equivalent to (5.78) and (5.92). If x i,..., x n and y stand for realizable values of ξι ( ί ),... ,ξη(ί) and £jv+i then the probability density p( x,y,t ) obeys N r οι ί p(*,y,t)= Σ [ - +Gijy)xj\p{x,y,t) dt I,j = 1 d , v - r d ■ ( ^ i + ( a,· 1 2 (595) d D-y2 d2 7-s-I/·' Our interest is in evaluating the averages (6 ) = J Xfe p(x, y, i)dxi ■ · ■ dxjydy ξ J zkdy. (5.96) where zk = J x kP(x, y,t )dxi •••dxjv. (5.97) On multiplying (5.95) by x k and by integrating over x\,..., x„ we find that Zfc obeys the equation N p η r f) D'V2 β2 1 = [iFkj + Gkj y)zj + 7 — H 2 ~dy^ Zk' (5.98) d dtZk J = 1 This equation is solvable exactly for a univariate system. We have anyway already solved the univariate problem in Sect. 5.7.1 for most general noise. For the details of the methods of approximate analytical and numerical solution of (5.98), see [6 6 ]. In the context of quantum optics, this finds application in the study of the dynamics of a two-level atom in a fluctuating field in which noise is characterized by (5.91) [70]. 5.8 The Poisson Process We have so far considered continuously varying random functions of time. In this section we consider a discrete random process, the so called Poisson 116 5. Theory of Stochastic Processes Process. This process is described by a random variable ξ (t) whose value increases randomly by one as a function of time without correlation between any two values. The numbers m = 0,1,... are the realizations of the values of £(£). Let p(m, t) be the probability that ξ(ί) = m at time t. Let R be the rate of increase of ξ{ί) so that w(n\m) = ϋ δ η:Τη+ι. The master equation (5.36) for p(m, t) then reduces to p(m, t) = R p(m — 1, t) — p(m, t) . I f p( m, 0) = Smo then this is evidently solved by m p(m,t) = ml exp (—Rt). By us i ng t h e def i ni ng r e l a t i on OO (/( £ ( * ) ) ) = Σ /( m ) p( m >i ) m —0 f or a n aver age, ver i f y t h a t ( 5.99) (5.100) (5.101) Ci(u) = (exp(iu£(i))) = exp Rt ^ exp(iw) — 1^ Compare this with the cumulant expansion (5.12) of C\(u) and verify that a e m = m. (5.1 0 3 ) This states that all single time cumulants of the Poisson process are equal. Recall from (5.14) that ((£(£))) is the mean of ξ(ί). Hence all single time cumulants of a Poisson process are equal to its mean. (5.102) 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise In this section we outline a method of solving m = M ( v ( t ) ) m (5.104) where the matrix Μ(η(ί )) is a function of the telegraph noise η(ί). The func tion η(ί) is such that (i) it jumps between discretely spaced values a,b,· ■ ■ at a rate R and remains constant between the jumps, (ii) the jumps occur randomly without any correlation, (iii) the process is stationary. Now, let us denote by χ( μ,ί ) the value of £(i) at time ί in a realization in which the value of η(ί) is μ. The value of £(i) in different realizations in which η(ί) = μ need not be same. This is because evolution to time t depends on the values of η(ί) at earlier times which are random. Let (χ(μ, t)) denote the average of £(t) under the condition that r)(t) = μ. Clearly m = Σ (5·105) μ=α, 6... 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise 117 Let us consider a realization in which η(ί) = μ. We assume that η(ί) does not change during the evolution of ξ(ί) till time t + di. The solution of (5.104) gives £(i + di) = ( 1 + Μ( μ)άί )χ(μ,ί ). The η(ί) may or may not change its value at ί + di. Let η(ί + dt) = u so that the value of £(i + di) is denoted by x(v, di + t). The probable value of η(ί + di) is found by introducing the probability that η(ί) jumps from its value μ to v in time dt. It is given by ιν(ι>\μ)(άί) = ϋρ(ν\μ)άί (μ φ ν), R being the rate of transition. The probability of transition p{v|μ) obeys Σ Ρ(ν\μ) = 1· (5-106) μ=α,6,... Hence, the probability that the value of the variable does not change in time di is 1 — Rdt. It then follows that (x{v,t + di)) = (1 — i?dt)(l + M(v)dt )(x(u, t)) + i?^ p ( ^ ^ ) d i ^1 + M ( μ ) d ί ) (χ(μ,ί )). (5.107) μ In the limit dt —> 0, (5.107) reduces to {x(y,t)) = (M(v) - R){x(v,t)) + R y p ( y |μ)(*(μ,ί)). (5.108) μφν If the number of values assumed by η(ί) are N then (5.108) constitutes a set of N equations whose elements are operators. The simplest telegraph process is the two-state one assuming values a, b with pab = pba = 1· As an example, see [71] where two-state telegraph noise is used as a model of fluctuations in the phase of a laser interacting with a two-level atom. 6. The Electromagnetic Field In this chapter we recapitulate the classical and the quantum theories of the electromagnetic (e.m.) field. This is followed by a discussion of the concept of the field correlation functions and their role in characterizing the statistical and spectral properties of the field. We highlight some signatures of non- classical features of the field carried by the correlation functions. 6.1 Free Classical Field The e.m. field may be described by the vector potential A( r, t) and the scalar potential </>(/*, i) related with the electric and the magnetic fields E( r,t ) and B( r,t ) by the relations B(r,t) = v x A( r,t ), E( r,t ) = ~ ^ ~ A ( r,t ) - V</>(M)· (6-1) The potential functions corresponding to particular electric and magnetic field vectors are not unique. They are arbitrary up to the gauge transforma tion A A - VX> Φ —>■ Φ + c ^ d x/d t. We may use the freedom available within the gauge transformation to work with potentials having desired prop erties. The gauge convenient for developing the quantum theory of the e.m. field is the Coulomb gauge characterized by the transversality condition on A: V - A ( r,t ) = 0. (6.2) The potentials in a region of charge density σ and current density j under this condition obey the equations V2 0 ( r,i ) = - σ, (6.3a) 1 Λ2 1 V2 A ( r,i ) - A ( r,i ) = - - j T. (6.3b) Here, j = J l + 3 t with the longitudinal and the transverse currents, j l and j r defined by v x 3l = 0 and V - J r = 0. Consider a region free of charges and currents. In such a region, the in- homogenous terms in (6.3a) and (6.3b) vanish. We may exploit the Gauge freedom to choose φ = 0 as the solution of (6.3a) corresponding to σ = 0. In order to solve (6.3b), we note that, for the boundary conditions of interest, V 2 is a hermitian operator. Hence, its eigenfunctions constitute a complete orthonormal set. Let (iifc(r)} be the eigenfunctions of v 2> called the spatial mode functions. They solve the eigenvalue equation \/2uk(r) + \k\2uk(r) = 0, (6.4) along with appropriate boundary conditions. Their orthonormality relation reads [ ■Ufc(r).Mi (r)d3r = 4; (6.5) Jv where the integration is over the volume V of the region containing the field. We may then express A( r, t) as A{r,t) = - ^ = { u k{r)ak(t) + u*k{r)a*k(t)}. (6.6 ) The form of the expansion coefficients above is chosen for later convenience. Note that the mode functions, by virtue of the transversality condition (6.2) on A( r,t ), should obey also the transversality condition S7.uk(r) = 0. (6.7) For example, in a cubic region of side L = ( V) 1^3 with periodic boundary conditions, the mode functions are the plane-waves u k(r) = u k<\( r) = exp(ifc · r), (6.8 ) where e\(k) is called the polarization vector. As a result of (6.7), e\( k) - k = 0. Hence, for a given k, £\(k) may be along any direction orthogonal to k. Let €χ(fe) and e2(k) be the directions orthogonal to k and to each other such that (ei(fe), e2 (fe), fe/|fe|) forms a right-handed triad of mutually orthogonal directions. The eigenvector index k on u k in this case stands for the set k, X. We d e t e r mi n e t h e expa ns i on coef f i ci ent s a k(t) in (6.6 ) by substituting it in (6.3b) (with j = 0). We find that the a k(t) obey the equation of evolution of a harmonic oscillator: d2 — a k(t) + J l a k{t) = 0. Lok = c\k\. (6.9) We work with its solution ak(t) = exp(-i LOkt ) ak (6.1 0 ) so that, with u k given by (6.8 ), A is a function of k ■ r — ujkt describing a travelling plane wave: Uk(r)ak exp(—iwfci) + u k(r)ak exp(iwkt) . (6.1 1 ) 120 6. The Electromagnetic Field k \Z2uJk The expression for the electric field, obtained by combining this with (6.1), reads 6.2 Field Quantization 121 E(r,t) = E^+\r,t) + E^\r,t), (6.12) EW (r,t) = J2E[+)(r,t), E' ^.+) (r,t ) = \^J^-Uk{r)atk exp(—io^i) = E[ ]* (r,t ) (6.13) The E ^ ( r,t ) are the so called positive and negative frequency parts of the field. We will see that separation of the electric field into positive and negative frequency parts plays a central role in the theory of the e.m. field detection. Now, we recall that the hamiltonian of the e.m. field is given by H = 1- 2 \E[ dJr. (6.14) This may be evaluated using the expression (6.13) for E and that for B, obtained by combining (6.11) and (6.1). Using the orthogonality properties of Uk and the boundary conditions, it may be shown that Refer to [73] for details. Introduce real variables Qk,Pk defined by Oik wkQk + iPk ν/2 ω^ Consequently, the hamiltonian (6.15) assumes the form Η= ο Σ Η 2 2 ω Ik ( 6.1 5 ) ( 6.1 6 ) ( 6.1 7 ) B y t r e a t i n g t h e m a s c a n o n i c a l l y c o n j u g a t e g e n e r a l i z e d d y n a m i c a l v a r i a b l e s, w e n o t e t h a t t h e c l a s s i c a l H a m i l t o n ’ s e q u a t i o n o f qk,Pk are, in accordance with their definition, the equations of a harmonic oscillator. The form of its hamiltonian (6.17) suggests that the free field may be regarded as a collection of harmonic oscillators. It affords an easy passage to the quantum theory of the radiation. 6.2 Field Quantization The quantum theory of the e.m. field can be formulated by following the standard method of quantizing a classical system, namely, by treating the canonically conjugate variables qu,Pk for each mode as operators qk,Pk which obey the commutation relation [<7fc,.Pi] = iMfcj. (6.18) The quantum analog of the classical free-field hamiltonian then is 122 6. The Electromagnetic Field Define the creation and the annihilation operators ^kQk "l· lPk ^kQk ~ lPk ία on^ = V E S T' = (6·201 obeying, by virtue of (6.18), the commutation relation Ski- (6.21) H — 2 Σ + ω · (6.19) a,k, aj We may rewrite the Hamiltonian (6.19), after shifting the zero of the energy of the fcth oscillator to Tux>k/2, as H = h'ywkaldk· (6.22) k On comparing this with (6.15), we note quantum classical correspondence ak -> Vhak, a*k -> Vha[. (6.23) The electric field is similarly quantized by replacing the c-number dynamical variables in (6.12) by operators. On invoking also (6.20), the electric field operator reads E(r,t) = E {+\r,t ) + E (~\r,t), (6.24) E(+) (r,t)=J2E[+) (r,t ), ~ ( + ) Πωΐο - i-H E k (r, t) = i y —^ ~u k{r )flfe exp(-iwfci) = E k (r, t) (6.25) Now, recall that the eigenstates of a\.a,k are the number states \nk) (rik = 0,1, · · ·. Hence energy of the field in the state |rife) is nkhoJk· Since the energy in a field mode is an integral multiple of TkUk, it is postulated that each unit of energy in a field mode is carried by an indestructible particle, called photon. Accordingly, an eigenstate |{n/c}} ξ | n i,7i2 · · · ) of (6.22) represents a collection of photons with rik photons in mode k. The ground state |{0fc}}, which describes a field having no photon in any of the modes accessible to it, is called the vacuum state of the field. Invoke the properties of the h.o. operators given in Chap. 3 to verify that ^Jlk\Ek\rik^ — 0, ( n k\Ek ■ Ek\rikJ - ( n k\Ek\n k j. ^nfc| l;fc|nfc^> = ( n k + ~ j . (6.26) This shows that the expectation value of electric field in the number state is zero but its fluctuations are finite even when it is in the vacuum state. These are the so called vacuum fluctuations. 6.3 Statistical Properties of Classical Field 123 Some particle-like properties of a photon are (see [73] for details): • By invoking the expression p = )- J ( E x B)d3r, (6.27) it may be shown that the momentum of a photon corresponding to the plane wave (6.8 ) of wave vector k is Tik. • On us i ng t h e r e l a t i v i s t i c r e l a t i o n be t we en mas s m and energy E, namely, me2 = y/E2 — \p\2c2 along with already noted facts that the momentum and energy of a photon of wavevector k are p = hk and E = hiOk = c%\k\, it follows that the mass of a photon is zero. • Besides energy and momentum, a photon state is characterized also by the polarization vector e\. Since e\ transforms like a vector, a photon is viewed as a particle whose rotational properties are those of an angular momentum quantum number L = 1. Any component of such an angular momentum has eigenvalues ±1,0. However, due to the transversality condition, the eigenvalue zero is excluded. The quantum theory thus assigns particle-like character to what is classi cally perceived as a wave. Recall that the quantum theory predicts wave-like behaviour for what are perceived classically as particles. The quantum charac teristics of the e.m. field are identified by comparing its statistical properties predicted by the quantum and classical theories outlined in the next two sections. 6.3 Statistical Properties of Classical Field The classical electric field is expressible in terms of the spatial mode functions as in (6.13). The functions ak(t) for free fields are harmonic. The same expan sion holds also in the presence of sources except that the time-dependence of ak(t) is determined by that of the source through (6.3b). The e.m. field may thus be generally characterized by a set of complex numbers {α^(ί)}. For the sake of simplicity, we assume that the field is polarized in one direction and treat it as a scalar. Detailed considerations based on the theory of atom-field interaction show that the optical detectors are square law detectors, i.e. they respond to the square of the field amplitude E(r, t) and that their time of resolution is orders of magnitude longer than the time period ω- 1 of optical oscillations [74]. An optical detector, therefore, measures E 2(r,t ) averaged over many cycles of oscillations. The detectors can be so arranged as to measure an nth order correlation function [74] n G^ ^ (^{τίίί}η; ^ {yiti) ( rn+it n+i) i = 1 = ^ ( r i t U . . . , 7*71^725 * · * ί f'2n^2n) * (6.28) 124 6. The Electromagnetic Field The first order correlation function G ^ (rt.rt ) is a measure of intensity: I (rt) = |£(+> (rt) | 2 = G ^ ( r t r t ). (6.29) In many situations of practical interest, field is a random function of time. Recall that the theory of random functions predicts only the averages over its many realizations. Now, the response time of optical detectors is usually much longer compared with the time over which the field changes. Consequently, the output of a detector at a given time is a result of an average of the function of the field being measured over the response time of the detector. In such a situation we can invoke the ergodic theorem, according to which, the average of a function of a random variable over a long period is the same as its average over all possible realizations of the values of the random variable. As explained in Chap. 5, the latter averages are characterized by the multi- time distribution functions. Accordingly, the correlation function (6.28) is determined by the average G(n) ({ri U}n; {ri t i }n) = ^ 1S(-) (rfa) E {+) (r n+it n+i)^ . (6.30) over an appropriate probability density characterizing the field. Note that, if ^*n+z — then G(n) ({rit i }n·, {ri t i }n) = ^ I (r i t i ) ^ > 0. (6.31) This function determines the correlation between intensities at different space-time points. Non-negativity of the probabilities leads to a number of inequalities be tween the correlation functions. As we will see below, inequalities play impor tant role in identifying signatures of purely quantum effects. To that end, note that since probability densities are positive, (|Φ({χ})|2) > 0 for any Φ({χ}). Let Φ(χ) = Fi({a:}) + AF2 ({x}) and show, following the method outlined in Sect. 1.1 for the derivation the Schwarz inequality that \(F;F2)\2 < ( |F i |2 )(|F2|2}|· (6.32) Now, let Fx = £ ( + ) ( n i O • • • £ (+) (rnt n), F2 = E {+) ( r n+1t n+1) ■ ■ · £ (+) (r 2nt 2n) and apply (6.32) to show that 2 ) ({Vjij}n5 {7’j_|_rii£j_|_rii}-n) < £?("> ({ri t ^n, ( r ^ } „ ) G<"> ({r n+it i+n}n; { r i+nt i+n}n). (6.33) It turns out to be useful to introduce normalized correlation functions j'» 1 ( { r,i,} »: {r i +nt,+„}„) = <6-34> Π » = ι v G W ( r i t i i n t i ) 6.3 Statistical Properties of Classical Field 125 The correlation functions of frequent occurrence in quantum optics are the first and the second-order ones. In the following subsections we summarize briefly some of their important properties. 6.3.1 First-Order Correlation Function The first order correlation function characterizes the interference effects in wave amplitudes like in the Young’s double-slit and many other interfer- ometric experiments. In order to see it, recall that in Young’s double-slit experiment, the wave amplitude is divided in to two parts at two pinholes at positions r i and r 2 (see Fig. 6.1). The intensity of the field is measured Fig. 6.1. Schematic diagram of Young’s two-slit interference experiment. at points r on another screen at a time t. The field E (rt) at (r, t) is due to superposition of the fields E(ri,t i ) (i = 1,2 ) that propagated from the two pinholes at earlier times i*. Hence, the field intensity I (r,t ) at the point of observation is I (r, t) = |/,£ ( + ) (η*,) + /2£ (+) ( r 2t 2) |2 = \fi\2I (rih) + |/2|2/ (r2t 2) + 2 R e [/ί/2GW ( n i i; r2 i 2) ] , ( 6.3 5 ) t h e fi being the constants inversely proportional to \r — ri\. An important characteristic of the interference pattern is its visibility defined by = m W h W G ^ j r x h - r ^ l |/i | 2|/( n i i ) + |/2| 2/( r 2i 2) · V = ~ I n + la ( 6.36) On i nvoki ng ( 6.33) a n d on combi ni ng i t wi t h t h e de f i ni t i on (6.34) of i t fol l ows t h a t t h e vi s i bi l i t y i s ma xi mum whe n | 5 (1) (ri t i -,r2t 2) \ = 1. (6.37) 126 6. The Electromagnetic Field The fields for which (6.37) holds are said to be first order coherent. The field is incoherent if (r*iti; r 2t 2) = 0. The visibility with incoherent fields is zero, i.e. such fields do not produce any interference pattern. The field is said to be partially coherent if (t*i*i ; ^*2 *2 ))I Φ 0 ) 1 · It may be proved that (6.37) would hold for all space-time points if [74] G(1) ( r i i i;r 2 i 2) = ε* (rt f i ) ε (r2t 2) (6.38) If, in addition, the field is stationary then G ( r i t i;r i t 2) will be a func tion only of t\ — t 2 = t. The condition (6.38) for such a process implies that £*{r\t i)£(r2t 2) be a function of t\—t 2. This can hold only if ε(ί) ~ exp(—iwi) implying thereby that a stationary first-order coherent field is monochro matic. 6.3.2 Second-Order Correlation Function We have seen that the first-order correlation function characterizes interfer ence effects between the field amplitudes at two space-time points. The equa tion (6.31) shows that G<'2^ ( r i,t i,r 2,t 2] r 2,t 2,r\,t\), a second-order correla tion function, is a measure of intensity correlations at two space-time points. The forerunner of the experiments for measuring intensity correlations is the one performed by Hanbury Brown and Twiss [75]. Consider the measurement of G^2\r t,r t + T;rt + T,rt) assuming the field to be stationary. The correlation function in question is then a function only of t. We denote it by G ^ ( t ). Its normalized form is (2) ( £ ( - ) ( t ) £ i - ) ( t + r ) P ( 1 + r )#) ( t ) ) 9 1 ’ (E(-)(t)E(+)(t) ) 2 = » Ι » > _ ο. (,3£1) The inequality above is a result of the fact that the intensity and the prob ability density are non-negative. Furthermore, the inequality (6.32) with Fi = I(t) and F2 = I( t + r) implies that gi2)(r) < g ^ ( 0). (6.40) Next, let = I(t) and F2 = 1 in (6.32) to obtain <7(2 )(0) > 1. (6.41) This will turn out to be a useful inequality for distinguishing the quantum nature of the field. 6.3.3 Higher-Order Correlations The higher-order correlation functions can be similarly interpreted. Of par ticular interest is the generalization of the definition of first-order coherence. A field is said to be nth order coherent if [74] 6.3 Statistical Properties of Classical Field 127 5 (m) ({ri t i }n·, { r i+nt i+n}n) = 1 (6.42) for all m < n. It may be shown that this would hold if [74] m 2m G(m) ({ri t i }n·, { r i+nt i+n}n) = J J e * ( r j i i ) e^i U). (6.43) i—1 i=m-\-1 As a consequence of (6.31), we note that if the field is nth order coherent then g(m) ({ri t i }n· {ri t i }n) = 1, (m < n). (6.44) A field which is coherent to all orders is said to be coherent. Ne xt, we e xa mi ne t wo commonl y e n c o u n t e r e d exa mpl e s of f i el ds for t h e i r coher ence p r o p e r t i e s. 6.3.4 S t a b l e a n d C h a o t i c F i e l d s The s i mpl e s t e xa mpl e i s of a s i ngl e- mode fi el d wi t h c o n s t a n t a mpl i t ude Eq and phase φ: E^+\t ) = E0 exp[i(fc.r — ωί + φ)\. (6.45) Such a stable classical field, describes approximately the output from a laser operating far above its threshold. For this field, the condition (6.42) of co herence holds for all n. Cons i de r now t h e fi el d o b t a i n e d by s u p e r p o s i t i o n of r a d i a t i o n f r om N radiators. For the sake of definiteness, let the radiators be atoms each emitting spontaneously at frequency ujq. The atoms may be colliding with each other as well. The radiation from each atom is of constant amplitude eg but its phase is a random function of time. The positive frequency part of the total field at a point of observation is N N Ei+)(t) = e0 ]Pexp(-iwo£ + i &(<)) = Σ Ε<ϊ+\ί ) (6·46) i—1 i— 1 The phase 0j(i) is distributed uniformly between 0 and 2π. The corresponding distribution function being ■ (6.47) We further assume that each φί(ί) undergoes a sudden change at randomly distributed times but remains constant between two consecutive interruptions and that there is no correlation between the phase of radiation from different atoms. These characteristics are adequately described by the equation ^ j ( t ) = μ^ί), (6.48) where μj(t ) is a Gaussian white noise with zero mean and {μί(ί + τ)μ^(ί)) = 2-ySij. (6.49) The rate 7 is related with the rate of spontaneous decay and the rate of collisions. Note that (6.48) and stationarity imply Jjl exp [ - i {φί - 0j(±r)}] = ±ϊμ,(τ) exp [ - i {φί - φί ( ±τ ) }\. (6.50) On recalling the deliberations of Chap. 5, we find that, with r > 0, (exp [ - i {φί - 0i(±r)}]) = - 7 (exp [ - i {φί - φί (±τ) }]). (6.51) This proves useful in evaluating two-time averages. For, N = e^exp(-iu;or) (exp [ - i{φί (ή - φό(ί + r)}]) i,j = 1 N = eoexp(-iwor ) ^ ( e x p [ - i {0 j(i) ~ φί(ί + r)}\) i —1 = Nel exp(-iwoT-) (exp [ - i {<&(*) - 0,(ί + r)}]) = Nel exp(—ίω0τ — 7 |r|). (6.52) In writing the second line above we have invoked the fact that the atoms are not correlated enabling us to write the average of a product involving different atoms as a product of the averages. Also, by virtue of (6.47), the average of exp(i0j) is zero. The property of equivalence of the radiators has been applied in writing the third line. The final result is obtained by inserting the solution of (6.51). Verify also that, due to (6.47), {E(-\t)E(~\t + T)} = ( E ^\t ) E ^ ( t + T)) = 0. (6.53) We recall again the considerations of Chap. 5 now to construct single time probability density P(E,E*) by evaluating P (E, E*,t) — (^j J d ^ e x p [ - i (Εξ + Ε*ξ*)} x ( exp [ ί ( £ ( +) ( ί ) ξ + ). ( 6.54) Wi t h £?(+)(£) gi ven by ( 6.46), we f i nd t h a t ( e x p [ i ^ + ^ + f ^ W O ] ) 128 6. Th e El e c t r omagne t i c Fi e l d / \ιη Σ \ ) 2mi <r|2m (m!)2 |ξ| m= 0 ' ' N 6.3 St a t i s t i c a l Pr ope r t i e s o f Cl as s i cal Fi el d 129 « exp [—i Ve0| £| ] . (6.55) The l a s t l i ne above i s t h e r e s u l t of r e t a i n i n g onl y t h e f i r s t t wo t e r ms i n t h e ser i es i n t h e s econd l i ne a n d by i nvoki ng t h e de f i ni t i on of a n e xpone nt i a l as s umi ng TV ^ 1. I t i s cal l ed t he Gaussian approximation. On combining (6.54) and (6.55) and on applying (A.28), follows the chaotic field distribution p { E ’ ΕΊ = ύ.N exp · (6-56) 112 /AT„ 2 ^0 The average of an ar bi t r ar y product of field for t hi s di st ri buti on is ^ = f d2E E nE*mP (Ε, E*) = m! ( Ne2) m 5mn. (6.57) The multi-time probabilities may similarly be approximated as Gaussians. Consequently, all the cumulants except the first two vanish. We exploit this property to evaluate ( E ^ ( t ) E ^ (t + T)E^-+\t + r)E^+\t ) ) by equating the cumulant {{{E^~\t ) E^~\t + r ) E ^ ( t + ^ E ^'1 (t))) to zero. We use the ex pression for the cumulant of four variables in terms of the moments (see, for example [65]). Since all odd-order moments are zero, we find that ( E ^\t ) E {-\t + r)E^+){t + r)E^+\t ) ) = ( E^- ){t )E{+\t ) ) ( E ^ -\t + T)E(+)(t + r)> + (£<-> ( t ) E ^ ( t + T ) ) ( E ^\t + T ) E ^ ( t ) ). (6.58) This relation holds for any Gaussian with zero mean. On recalling (6.52), (6.58) yields ( E ^ ( t ) E ^ ( t + r ) E ^\t + T)E^+\t)) = N2eQ [1 + exp(-27 |r|)] .(6.59) Now, the normalized first-order correlation function for a stationary field at a fixed point of observation with ti = t, t 2 = t + τ is given, using (6.52) and (6.59), by " ' 1 (Εί->(ί)βί+>(ί)) = exp(— \ω0τ — 7 |r|). (6.60) The normalized second-order correlation function defined in (6.39), is given by ff(2)(r ) = 1 + e xp(—2 7 | t | ). (6.61) Consider first the properties of g ^ ( r ). • For 7 ^ 0, 5 (1*ίτ)| φ 0,1. Hence the chaotic field is partially coherent. 130 6. The Electromagnetic Field • For r <C 7 _1, \ί/1}(τ)\ « 1, which is its value for a coherent field whereas for r 2> 7 ^ 1, <7^(r ) “ ^ 0 which is its value for an incoherent field. Hence 7 _ 1 is identified as the coherence time of the field. • ί?^ (0 ) = 1 - Hence <7^ ( 0 ) can not distinguish between a coherent and a chaotic field. Consider now g ^ ( r ) given by (6.61). Note that which is in contrast with the value <7^ ( 0 ) = 1 for a coherent field. The chaotic and the coherent fields can thus be distinguished by a measurement of <7^2^(0 ). However, note from the expression (6.61) for the chaotic field that Hence, on a time scale much longer than the coherence time, the chaotic and the coherent fields are indistinguishable. Let us now express the field as in (6.13). Since we are restricting our attention to a fixed point (r = 0 ) in space, we find that the C being a constant. This shows that (a( uk)) = 0. Following the steps in (6.89), and on redefining the constant C, we obtain This shows that the frequency in a chaotic field has Lorentzian distribution. In view of (6.56), the set {ak,a*k} also has a Gaussian distribution. Their distribution function is, therefore, given by We will use this result in the next section to construct the quantum analog of the classical chaotic field. 6.4 Statistical Properties of Quantized Field <?(2) ( 0) = 2, (6.62) </(2 )(r) ->■ 1, τ » 7 J. (6.63) (6.64) (.a(u>k)a(uji)) = —S(uk - ui )S(uk), ( 6.65) (6.6 6) P( ak, a*k) = [ J exP {~\a k\2/(H^/c)!2}) · (6.67) The observable statistical properties of the e.m. field in the quantum theory are determined by the correlation function 6.4 Statistical Properties of Quantized Field 131 G{n) ({ri t i }n; { r i+nti+n}) = ( e ^ (ritx) ■ (r nt n) x M+) ( E {+) (r2nt 2n) Υ (6 -6 8 ) Bear in mind that & +\r t ) = M ^ ( r t ) and that the average is the quan tum mechanical expectation value with the density operator of the field. The expression (6.6 8 ) reduces to the corresponding classical one if operators are replaced by corresponding c-numbers and the averages are identified as over the distribution of those c-numbers. Since E ^ is a linear combination of field annihilation operators for different modes and E = E^+^ is that of field creation operators, (6.6 8 ) shows that the process of photo detection measures the field operators in normal ordering. Recall that it is this fact that forms the basis for classifying the field states in Chap. 4 as classical or quantum. Now, purely quantum effects may be isolated by comparing the bounds on correlation functions derived in the last section for the classical theory with those predicted by the quantum theory, to be derived below. To that end, recall from Sect. 1.1 that & C for any C is a positive operator. Since density operator is also positive, it follows that TrfC’^C'/)] > 0. Let C = A+XB. Follow the method outlined in Sect. 1.1 for the derivation the Schwarz inequality and show that Ι(1+β) | 2 < ( A'A) ( B'B). (6.69) The choice η n A = J [ M +\r iti), B = J[E<-+\ri+nti+n), (6.70) leads to the same inequality (6.33) as derived for a classical field. However, we show below that there are situations in which the quantum predictions differ from the predictions based on classical inequalities. 6.4.1 First-Order Correlation The normalized first-order correlation function for a quantized field at a fixed point of observation and for a stationary process is defined as in (6.60) with the c-numbers replaced by operators. For a single-mode field it assumes the form 9 { } <at(f )o(t)> : (bJi) Q u a n t u m t h e o r e t i c a l g ^ ( r ) does not lead to any inequality which is in con flict with the classical theory. This amounts to saying that the first-order correlation function does not carry any signature of field quantization. 132 6. The Electromagnetic Field 6.4.2 Second-Order Correlation Now we examine the properties of the quantized version of the second-order correlation function defined in (6.39): On applying (6.69) with A = ^Ε^+\ B = 1 we see that the first term on the right hand side above is greater than one. However, non-zero positive values of the second term may lead to Notice that (6.75) can hold only if the commutator in the second term in (6.74) is non-zero. On recalling the classical prediction (6.74) viz. g(2\0 ) > 1, it follows that (6.75) is a signature of the quantum nature of the field. Furthermore, using the P-representation introduced in Chap. 3, we leave it function is non-classical. A field for which (6.75) holds is called antibunched. An example of antibunched field is the number state |m). For this state, it is easy to see using (6.73) that <?(2)(0) = 1 for to = 0, and <7^(0) = 1 — 1/m for to > 1 . Now, for a single-mode field, (6.73) and (6.75) imply The left hand side above is the variance and the right hand side the mean of photon number distribution. Recall that variance is equal to mean if the distribution is Poissonian. The inequality in (6.76) indicates sub-Poissonian photon number distribution. Hence, sub-Poissonian photon number distribu tion is another signature of the quantum nature of the e.m. field. 6.4.3 Quantized Coherent and Thermal Fields (2) = ( E ^ ( t ) M - ) ( t + T ) E W ( t + T ) E ( + ) ( t ) ) ( £ ( - ) ( ί ) £ ( + ) ( ί) ) 2 ( 6.7 2 ) F o r a s i n g l e - mo d e f i e l d i t r e d u c e s t o (2) / \ = ( ά ^ ) ά ^ + τ ) ά ( ί + τ ) ά ( ί ) ) 9 {T) (at(i)a(i))2 Let r = 0 and reexpress (6.72) as (6.73) (6.74) <?(2 )(0 ) < 1. (6.75) to the reader to confirm that g(2 )(0) < 1 implies that the corresponding P- {(a^a)2) — {a)a)2 < a^a. ( 6.76) I n t h i s s u b s e c t i o n we i dent i f y t h e q u a n t u m a na l ogs of cl as s i cal s t a b l e a n d c ha ot i c fi el ds. 6.4 Statistical Properties of Quantized Field 133 Like in the classical description, a quantum field is said to be nth order coherent if (6.42) holds. It implies the factorization property (6.43). On us ing the operator representation of the field, it follows that the condition for coherence to all orders can be satisfied if the field is in an eigenstate of the annihilation operator ak of every mode k (see [74] for details). Since the op erators corresponding to different modes commute, it follows that the said eigenstate is a direct product of the eigenstates of each of the annihilation operator ak. An eigenstate |a) of the field annihilation operator a is, there fore, called a coherent state of the field. This provides physics basis for the concept of a coherent state introduced in Chap. 3 based on mainly mathe matical considerations. Recall also from Chap. 3 that a) does not admit right eigenstates. Hence the states |a) are the only ones for which the correlation function factorizes to all orders. The expansion (3.28) of a coherent state in terms of the number states and the interpretation of the number state |m) as the state having m photons shows that the photon number distribution in the coherent state is Poissonian. Verify using (6.24) that the expectation value of single-mode electric field in the coherent state |a), (a = |a| exp(i^)) is given by This expression is similar to the one for stable classical field. Also, like the stable classical field, the field in a coherent state is coherent to all orders. The field in a coherent state is therefore analogous to a stable classical field. Next, we construct the quantum analog of the classical chaotic field. To that end, we invoke the correspondence (6.23) and note that the quantum density operator analogous to the classical distribution function (6.67) of the chaotic field may be written as where βι~ and the normalization constant A k are determined as follows. • Working in the basis of the eigenstates | m) of a) a we obtain (6.77) p = Π Ak exp(~Pka[ak) ( 6.78) k oo ( 6.79) Hence, t h e n o r ma l i z a t i o n c o n s t a n t A k is given by Ak = Π I1 - βχρ(-&)1 (6.80) k • The expectation value of (a^a)m is similarly found to be given by Tr ^(ά^ ) ”1 exp(—/3a t a)j 134 6. The Electromagnetic Field = ( - r ^ T r [ e x p ( ^ a ) = ^ d/3™ 1 - exp(—/?)' ^6'81^ Using this, it is straightforward to see that <4 “«> = nkSkh nk = 1 ^ (^ fc)- (6-82) The correspondence (6.23) implies that ^(ά^ά;) = (a^ai). Use this relation, the expressions (6.65) and (6.82) to show that C hnk = — S^fe). (6.83) uk We may now evaluate normalized first order correlation function defined in (6.60) by inserting in it the expression (6.25) for the field. Bearing (6.82) and (6.83) in mind, verify that (iu_) Efc u knk exp(—Sukr) J2k S( uk) e x p ( - i ^ r ) Y^,k uknk Y^k S(u)k) Convert the sum to an integral over ω from 0 to oo. As is the case with optical fields, the width 7 of the Lorentzian is small compared with the op tical frequencies. Hence, we can extend lower limit of integration to —0 0. On carrying the integration by standard contour integration, we recover the classical result (6.60). We leave it to the reader to evaluate g ^ ( r ) in the same way and confirm that it is same as (6.61) for the classical chaotic field. In particular, g ^ ( 0) = 2. The value g(2\0 ) = 2 may now be interpreted to indicate increased correlation or bunching of photons at a given space-time point over what is found in a coherent state. 6.5 Homodyned Detection In the last section we noted that a direct measurement on the field at a given space-time point yields information only about the quantities which are products of equal number of positive and negative frequency parts of the field operator. The observables which are not of the said form can be measured by the method of homodyned detection. In this method, the signal field is mixed with a strong coherent field called the local oscillator. Photo detection of the field so mixed can give information about the statistical properties of certain correlation functions of the kind in question. See [76] for details. As an illustration, let the signal be a single mode field described by the operators ά,ά^. Let frequency of the local oscillator field be the same as that of the signal. We treat the local oscillator classically. The positive frequency part of the mixed fields is 6.6 Spectrum 135 Ε = [ά + \Ei\ exp(—i0)] exp(-iwi). (6.85) Here \E[\ and φ are the amplitude and phase of the local oscillator. Perform a measurements of the variance in the photon number distribution in the homodyned field. Assume the local oscillator field to be very strong compared with the signal and retain the highest non-zero power of \Ei\ to show that Δη 2 « 2\Ei (Α2Φ) ~ (ΑφΫ (6.8 6) Aj, — V2 [exp(i0 )a + exp(— ΐφ)αϊ ( 6.87) A me a s u r e me n t of Δ η 2 in homodyned detection of a single-mode field is a measure of the variance in Αφ. Th e me t h o d o u t l i n e d above for a s i ngl e - mode fi el d ca n be e xt e nde d t o mul t i - mode fi el ds by means of a p p r o p r i a t e choi ce of f r equency of t h e l ocal os ci l l at or. As a n e xa mpl e, see (55). 6.6 S p e c t r u m A c h a r a c t e r i s t i c of t h e l i ght fi el ds of u t mo s t i n t e r e s t has t r a d i t i o n a l l y be e n i t s s p e c t r u m. By s p e c t r u m we commonl y u n d e r s t a n d t h e i n t e n s i t y of t h e Four i e r c ompone nt s of t h e fi el d. Let Ε(ω) = j / exp(io;i)£;^+^(i)di. (6.8 8 ) v 2 π J —oo Assume that the field is stationary. Now, (Ε*(ωι)Ε(ω2)) -i p O O p O O = — / dh di2 exp{—i (^iii - u)2t 2) ( E(-~'1 ( t ^ E ^ ( t 2)) ^ J — OO J — o o 1 z*0 0 = — / dT exp{—iΤ(ωχ - ω2)} J — O O / OO d r e x p { i r ( w i +ω2)/2 } { E ^ Ε^+\τ ) ) -OO = δ(ωι — u;2 )<S(u;i). (6.89) The second line above is obtained by changing the integration variables to T = (ti + t 2)/2, τ — t2 — t\ along with the use of the stationarity, (£(-)(*!)£(+)(i2)) = (£(-)£*+)(t2 - tj)), and / OO d r Β χ ρ ( ϊ ω τ ) ( Ε ^ Ε ^ (r)) -OO p o o — 2 Re I d r βχρ(ίωτ)(^~^Ε ^\τ ) ). (6.90) Jo The last line above is obtained by using the relation ( E ^ E ^ +\—t )) = (Ζ?(“ )(τ)£'(+)) = (t ))*■ The function S( uj) is called the spectral density or the spectrum of the field. The equation (6.90), relating spectral density with the two-time correlation function of a stationary process, is the content of the Wiener-Khintchine theorem. For a plane-wave field, the mean power per unit area carried by the field past a point of observation is proportional to ( E ^ E ^ ), it follows that S( ω) is indeed a measure of the power distribution among different frequency components of the field|77]. The definition (6.90) may be extended to quantized fields by treating the field amplitude as an operator so that 136 6. The Electromagnetic Field Note that the definitions of spectrum given above are based on mathematical considerations. The mathematical soundness of these definitions, their rela tionship with what is actually measured as also the question of generalization to a non-stationary process have been greatly debated. We refer to [77, 78] for a detailed account of the theory of the spectrum of classical and quantized e.m. field. The definition (6.91), however, is adequate for most commonly encountered situations. (6.91) 7. Atom-Field Interaction Hamiltonians In this chapter we introduce the hamiltonian governing the interaction of quantized electromagnetic (e.m.) field with free atoms. We show how the facts of the physics of the problem can be used to reduce that hamiltonian to mathematically manageable forms. Simple prescriptions for constructing models conforming to frequently encountered conditions are given. 7.1 Dipole Interaction Consider an atom consisting of Z electrons surrounding its nucleus which is assumed to be at rest. Let the atom be irradiated by the e.m. field. Express the hamiltonian of the combined system of the atom and the field as Η = Ά y ' iVk^ak + Ha + -ffint· (7-1) k The first term on the right hand side above is the free field hamiltonian, given by (6.22), Ha and Hint are, respectively, the free-atom and atom-field interaction hamiltonians derived below. If the wavelength of the field is large compared with the size of the atom, which is typically of the order of 1 0 ~ 8 e.m., then the variation of the e.m. field over the atom can be ignored. This would be the case if the field frequency is less than about 1018 Hz. It can then be shown that the dominant part of the atom-field interaction arises by treating the atom as a dipole constituted by Z atomic electrons bound to the nucleus [72], If r, (i = 1, · · ·, Z) denotes the position of the ith electron then the quantized atom-field interaction hamiltonian in the electric dipole approximation reads z Hint = -d- E(r0) = - e ^ T i ■ E(r0). (7.2) i= 1 Here e is the electronic charge, d is the atomic dipole moment, and E ( r 0) is the electric field operator at the position ro of the nucleus: 138 7. Atom Field Interaction Hamiltonians We assume that all but the valence electrons constitute a hard core and that the e.m. field changes the state of only the valence electrons. We consider atoms having single valence electron and restrict the summation in (7.2) to Z = l. N o w, l e t { |φί)}, i = 1, · · ·, N be the complete set of orthonormal energy eigenstates accessible to the electron and let {Ei } be the corresponding energy eigenvalues. As a result, the relevant part of the free-atom hamiltonian reads N H* = J 2 E iAii, (7.4) 2=1 Aij = \Φί)(Φί\, (i,3 = 1,··· ,N). (7.5) The orthogonality and completeness of the states implies that N Aij-Aki = AiiSjk, ^ ^ Ag = 1. ( ^ - 6) i— 1 The dipole moment operator in t he basis of t he energy eigenstates reads N d = di j\Φί)(Φ] I; (7.7) i,j = 1 where = β(φί\τ\φ^) is the dipole matrix element between the levels |φί) and \φί). In the representation spanned by the eigenstates |r) of the position operator f, obeying r |r) = r | r ), I d3r r t f ( r ^ - i r ), φ ^ ν ) = ( r ^ k). (7.8) Verify that if the electronic states are of definite parity, i.e. if Φη{—r) = then dij = 0 if the parity of |φί) is same as that of |φj). For such states da = 0. Each state of a free atom has a definite parity. On combining (7.2), (7.3), and (7.7), the interaction hamiltonian assumes the form N Hint — h ^ ] y ]Ajj gijkcik + > (7-9) i,j = 1 k t h e gijk being the atom-field coupling constant given by dij — 6 9ijk = - i y ^ 7 di j.e\ exp(ifc.r0) ξ dij.exek. (7.10) If the matrix element of the dipole operator between two levels vanishes then the transition between those levels is said to be electric dipole forbidden. Else it is an electric dipole allowed transition. The selection rules for identifying electric dipole allowed transitions are well known. The hamiltonian for an interacting system of an atom and the e.m. field in dipole approximation thus assumes the form 7.1 Dipole Interaction 139 N N H = h Uka\ak + ^ Ei Aa + h ΣΣ Aij [gijkCik + gtj ka{\ . (7.11) fc= 0 i= 1 i,j= 1 k I t t r e a t s t h e a t o m as wel l as t h e fi el d q u a n t u m mechani cal l y. I f we wi sh t o i dent i f y t h e c h a r a c t e r i s t i c ef f ect s of fi el d q u a n t i z a t i o n t h e n we need t o com p a r e t h e r e s u l t s of t h e q ua nt i z e d h a mi l t o n i a n ( 7.11) wi t h t hos e pr e d i c t e d by t r e a t i n g t h e fi el d cl assi cal l y. On r e c a l l i ng t h e qua nt um- c l a s s i c a l c or r e s pondenc e ( 6.23), we n o t e t h a t t h e cl as s i cal fi el d ver s i on of t h e at om- f i el d i n t e r a c t i o n i s o b t a i n e d by t h e r e pl a c e me nt ak —> a.k/yf%. The equations of the field variables are then the Hamilton’s equations for the pairs of con jugate variables { ak,a^}. Those equations are the same as the Maxwell’s equations. The approximation which treats the e.m. field classically but the atoms quantum mechanically is known as the semiclassical approximation. It is equivalent with replacing the quantum averages (a^manA), where A is an atomic operator, by (a)m) (an) (A) in the Heisenberg equations. The field in the semiclassical approximation is a classical dynamical vari able. Further simplification is obtained by treating it as an external variable. To see this, consider the Heisenberg equation N i-Q’k ivkak ^ ^ g^j^Aij (7.12) i,j = l where ’dot’ over a quantity denotes derivative with respect to time. The formal solution of (7.12) reads r f t n „ ak(t) = exp(-iwfci) | a fe(0 ) - i / drexp(iwfer ) ^ 9i jk^i j{T)\· (7·13) First term on the right hand side of this equation describes contribution due to free evolution of the field. The second term is the reaction on the field from atomic transitions induced by it. If the applied field is sufficiently intense compared with what can be contributed by the atomic transitions then the second term in (7.13) can be ignored so that ak(t) « exp ( - iu kt) ak(0). (7-14) The field then is no longer a dynamical variable. The field operators at time t may then be replaced by their averages in the initial state. We assume the field to be in the coherent state |{c*fc}) (|α&| 2> 1), replace the operator Sfc(O) in (7.14) by a k and substitute it in the hamiltonian to reduce it to N Η = Ύ2 Ei Aa + h y Aij I gijk exp(-iwfei ) a fc + c.c.l. (7-15) i=l k The atomic operators are now the only dynamical variables. 140 7. Atom Field Interaction Hamiltonians The solution of the dynamical problem associated with even the semiclas sical hamiltonians is generally a formidable task. However, there are numerous realistic situations in which the atom-field interaction can be approximated by manageable forms. The approximations central to quantum optics are the rotating wave and the resonance approximations, discussed next. 7.2 Rotating Wave and Resonance Approximations To understand the conceptual basis of these approximation, consider a sim plified version of the hamiltonian (7.15) in which the field is assumed to be prescribed externally. Assume that it is a single mode field of frequency ω and that it couples only the levels |g) and \e) (Ee > Eg). The interaction part in (7.15) may be written, assuming a and g to be real, as = 2gha[S+ + <S_] cos (cot), S+ = \e)(g\, S - = \g)(e\, Sz = ^ (\e ) { e\- ( 7.16) (7.17) T h e compl e t e ne s s r e l a t i o n f or t h e t wo s t a t e s r e a ds IS'Kfi'l + | e ) ( e | = 1. ( 7.18) Th e o p e r a t o r s §μ are the same as the spin-1/2 operators introduced in Sect. 1.6. On using the definition of Sz along with the completeness rela tion (7.18), and on shifting the zero of energy to (Ee + Eg)/2, the free-atom Hamiltonian (7.4) may be rewritten as Ha = hiv0S z, ω0 = (Ee - Eg)/h. (7.19) The hamiltonian of a two-level atom in a monochromatic external field thus reads H(t) = hu)oSz + 2 hga S+ + S - cos( o;i ) = h,u)Sz + Άδ£>ζ + 2hga S+ + 5_ cos(wi) (7.20) δ = u>o — u> being the detuning between the atomic and the field frequencies. The state \ψ(ϊ)) of the atom under this hamiltonian evolves according to rt IV’(O))· Apply (1.51), with λ in it identified as —iu>Sz, to rewrite (7.21) as / t d t H i ( t ) \m h ( 7.2 1 ) (7.2 2) 7.2 Rotating Wave and Resonance Approximations 141 Hi (t ) = exp ^iwiSz) [H - hu>Sz\ exp ^-iu;iSz) Hr, hga exp(2 iw£)S+ + exp(—2 iwi)S_ (7.23) The second line above is obtained by using (2.44), and Hrwa, h SSZ + ga ^S+ + 51- ) (7.24) On applying (1-51) again, with A = —i i i rwa/7j., (7.22) assumes the form I1p(t)) = exp ( - i UJtSz^ exp f-^-Hrwa x 1 exp - h i dTH^ T\ 1-0 (0 )), Hn(t) = hga exp(2iu>f)S+(f) + exp(—2ίωί)5'_(ί) 5M(i) = exp ( ^ i r w a ) 5Mexp ii ft' Hr, (7.25) (7.26) (7.27) The <ί?μ(ί) may be evaluated as in Sect. 12.2 by transforming to new set of spin operators R„ related with £>μ by (12.59a). In the new representation, -f^rwa = hQRz, where Ω = y/4 g2a2 + δ2. Recalling the results of Sect. 12.2 we note that S+(t) = 2 - ^ R z + i ( l + ^ exp ( i m) R+ + K 1 “ ^ ) exp(_ii2i)^ - } ’ (7.28) Sz (t) = (exp(ii2 t ) R + - h.c.) . ίga Ω (7.29) (7.30) Co mbi na t i on of t h e s e r e s u l t s i n (7.26) d e t e r mi n e s Hn( t ). Th e t i me - o r d e r e d i nt e gr a l (7.25), r e a d wi t h ( 7.26) a n d ( 7.29) c a nnot be pe r f or me d a n a l y t i c a l l y e xa c t l y even f or δ = 0. Its approximate value of practi cal interest is obtained by examining the expansion (1.43) of a time-ordered integral. It shows that each time integration contributes the factors 1 /ω, 1/(2ω ± Ω). Hence, (7.25) is expressible as an expansion in powers of ga/ω, ga/(2w ± Ω) and their products. We consider transitions between bound state far below the ionisation limit. The strength of optical fields inducing such transitions and the atomic dipole moments are such that Ω ω. (7.31) 142 7. Atom-Field Interaction Hamiltonians Note from (7.30) that Ω is the frequency with which the atomic popualation is exchanged between the two levels. It is known as the Rabi frequency. The condition (7.31) states that the Rabi frequency be much smaller than the frequency of the field. For a simple argument revealing that |<7||a:| of the order of optical frequencies would ionize an atom, see [79]. Now, due to (7.31), the terms containing ga/ω and ga/(2oj + Ω) are negligibly small. The terms proportional to ga/^Ιω — Ω) are also small except when ω « ω0/3 in which case ga/( 2ω — Ω) ~ ω/ga which is large. However, an inspection of (7.29) reveals that the coefficient of exp(±2z(u> — Ω)ί), which is responsible for such terms, contributes additional factor which is of the order of g2a2/ω2. Hence the overall contribution even from the terms like ga/(u> — Ω) is of the order of ga/ω. I n vi ew of t h e pr ecee di ng d e l i b e r a t i o n s, we c a r r y t h e t i me - or de r e d i n t e g r a t i o n i n ( 7.25) t o s econd- or der. Now, t h e t i me scal es of obs e r va t i on ar e or de r s of ma g n i t u d e l onger t h a n ω- 1. Hence, wh a t i s obs er ved i s a n ave r age over ma ny cycl es of os ci l l at i ons a t f r e que nc y ω. Consequently, the contribu tion from a term oscillating at frequency of the order of ω averages to zero. Bearing this in mind, we find that non-zero contribution in second order of perturbation comes only from the terms containing exp(±ifJ?). It follows that exp J dτΗπ(τ)^ « 1 - 2i9'^ Sz « exp ( - i SBs t Sz^ , (7.32) where, for the reasons elaborated below, S b s = ^ (7.33) ωΩ describes shift in the atomic frequency α>ο· It is known as the Bloch-Siegert, shift. The expression (7.32) may alternatively be derived by evaluating (1.108) to second order. We substitute (7.32) in (7.25) and invoke the smallness of ga/ω to write the product of the exponential containing HIWa and (7.32) as exp{—it{HIvja/Ti + Sb s Sz)}· The expression (7.25) then reduces to |ip(t)) = exp exp{-it (HTwa/h + £bs5z)}|V>(0)) exp d τΗ{τ)/% 1-0(0)), ( 7.34) Η = %{ωo + 5b s )Sz + hga ^exp(—iut ) S+ + exp(iwi)5_) . (7.35) The reader may verify that the second line in (7.34) reduces to the first if (1.51) is applied with A = —iu)SZ. The hamiltonian (7.35) shows that in the process of making approximations, the atomic transition frequency is shifted by Sb s ■ This shift is negligibly small for optical frequencies. In what follows we ignore Sb s - We note that the terms ignored in the approximate form (7.35) of (7.16) are exp(iwi)5’+ and exp(—ϊωί)5_. It is known as the rotating 7.2 Rotating Wave and Resonance Approximations 143 wave approximation (RWA). Dropped terms are the so called counter rotating terms. For an explanation of this nomenclature, see, for example [79]. The interaction hamiltonian is simplified still further if the detuning be tween the field and the atomic transition frequency is large. To that end, note that an atomic operator βμ, under the action of (7.35), evolves to 5μ(ί) = *T exp i Η(τ)άτ §μΫ exp h j 0 H { T ) d T i - i - HTwa,t rt 5 μ e x p — Hrwat rt = exp(—i Szujt) exp x exp(i5zwi). The exact analytic expressions for βμ under the action of Hr, been evaluated in (7.29). We examine them under the condition \δ\ » ga. ( 7.36) have a l r e a dy (7.37) Thi s s t a t e s t h a t on- r es ona nce Ra bi f r equenc y b e much s mal l e r t h a n t h e de t u n i n g be t we e n t h e a t o m a n d t h e fi el d. A p e r t u r b a t i v e expa ns i on of t h e e x a c t 5 μ( ί ) shows t h a t, t o z e r o t h or de r i n ga/δ, Sz (t) « ^(Ο)· Hence, if the atom is initially in its ground state then the expectation value of Sz (t) is —1/2 to zeroth order in ga/δ. To this order, we may let S z(t) ~ —1/2. The commu tation relation [5+, 5_] = 2SZ then reduces to [5_, 5+] « l. (7.38) This suggests that the atomic operators may be approximated by the har monic oscillator operators: b\ [Mf] = 1. S- -> b, (7.39) Th e f or egoi ng c ons i de r a t i ons a ppl y al s o whe n t h e fi el d i s t r e a t e d q u a n t u m mechani cal l y. Th e at om- f i e l d s ys t em i n t h i s cas e, i n t e r ms of s pi n o p e r a t o r s i n t r o d u c e d above, i s de s c r i be d by t h e h a mi l t o n i a n H = TuxiqSz + Τιωα'α + Τι (§+ + s J j (^ga + g*a^. (7-40) By following the method outlined above and by using the solution (11.22) for a two-level atom in a single-mode quantized field we find that, if (7.41) (a^a) -C ω, t h e n ( 7.40) r e duces t o ( i gnor i ng t h e Bl o c h - Si e g e r t s hi f t ) H = hwoSz + Ηωά^ά + h (]gS+a + g*a^S-^J . Th i s i s known as t h e Jaynes-Cummings model [80]. Reducing (7.40) to (7.42) constitutes the RWA for a quantized field. In this form, we see that RWA amounts to ignoring energy non-conserving terms (7.42) 144 7. Atom-Field Interaction Hamiltonians in which emission (absorption) of a photon is accompanied by the transition of the atom from its lower (upper) to its upper (lower) state. In other words, RWA ignores terms in which the atomic raising (lowering) operator multiplies the field creation (annihilation) operator. Similarly, following the method outlined above for a classical field, it may be shown that if | j | » Isly^ata) (7.4 3 ) then the atomic evolution is adequately described by harmonic oscillator approximation of two-level operators. A pair of levels for which (7.43) holds are said to be far off-resonant with the field mode in question. Else they are said to be nearly-resonant with it. The deliberations above show that if a level is coupled far off-resonantly with one and nearly resonantly with another then the off-resonant coupling can be discarded in comparison with the nearly resonant one. We refer to it as the resonance approximation. I n wh a t fol l ows we as s ume t h e RWA t o hol d t o r e wr i t e (7.11) as Η = ^ E[m)Au + y uk &l &k + Σ Σ [dijkO-kAji + h.c. . (7.44) i = 1 k k E i < E j Bas ed on t h e r e s onance a ppr oxi ma t i on, we o b t a i n i t s si mpl i f i ed f or ms for s ome s i t u a t i o n s of p r a c t i c a l i nt e r e s t. T h e i s s ue of t h e i r s ol ut i on i s a ddr e s s e d i n Cha p. 11. 7.3 Two - Le v e l A t o m Cons i de r a n a t o m i n a s t a t e s |<7) i r r a d i a t e d by a mul t i c hr oma t i c fi el d. Le t t h e r e be a f r e quenc y ω in the field for which an allowed transition from |(/) to another level |e) is nearly resonant. Let this transition be far off-resonant with all the other frequencies present in the field. Also, let all the allowed transi tions from \e) or from |(/) to any other level be far off-resonant with all the frequencies present in the field. As discussed above, the atom then interacts dominantly only with the mode of frequency ω and undergoes transitions only between |g) and \e). The atom can then be considered as a two-level atom and the field as monochromatic of frequency ω. The corresponding hamiltonian in RWA is obtained from (7.44) by retaining in it the free-field term corre sponding only to the mode of frequency ω, the free-atom term corresponding only to the two levels and the interaction term only between those levels and the single mode. It is the Jaynes-Cummings hamiltonian given by (7.42). Its generalization to N identical two-level atoms is described by N N Η = Άω0 ^ + Uuja'a + h ^ [siS+^ά + 9*, (7-45) i=l i= 1 7.4 Three-Level Atom 145 where S±\Sz*^ are the spin operators for the itb atom and gt is its coupling constant. The dependence of the coupling constant on the position of the atom arises owing to the variation of the field in space. If it is assumed that that variation is negligible over the region containing the atoms then ss g for all i. It reduces (7.45) to the form Η = Τιωο Sz + Ηωά^α + [(/θ+ά + g *^ ^ - ], (7.46) where the collective spin operators are as in (3.73). 7.4 Three-Level Atom Consider now the situation in which only a frequency ωα of a multichromatic field is nearly resonant with a pair of levels |<7) and |i). Let the frequency ujb be the only one nearly resonant with the pair of levels |i) and \e). If the states have definite parity then a direct transition between \g) and \e) is dipole forbidden. Let all transitions from any of the three levels in question to any other level be far off-resonance with any frequency in the field. The atom can then be visualized as having three levels and the field as having only two frequencies if ωα φ LOb- It acts as a monochromatic field if ωα = u>b· Note that, there are three different configurations possible for transitions between the three levels depending upon the position of the intermediate level |i). Those are shown in Fig. 7.1 along with the allowed transitions. In the configuration of the first diagrams in Fig. 7.1, the energy of |i) is inter mediate between the energies of the other two levels. It is called the ladder configuration. The energy of |i) in the Λ or Raman configuration is higher than that of the other two levels (second diagram in Fig. 7.1) whereas the level |i) in the V-system is below the other two levels (third diagram in Fig. 7.1). If (ά, ά^) and (b, tf ) are the e.m. field mode operators corresponding to the t C0b -L τ' C0a Fig. 7.1. Three configurations of a three level atom in a two-mode field each acting on only one transition. 146 7. Atom-Field Interaction Hamiltonians frequencies ωα and Wb respectively then it is straightforward to see that the hamiltonians for the ladder, Raman and V configurations in the RWA are given respectively by hl = h 0 + n i j j t = Hq + h Hy = Hq + h A gi + gaA iga + h g f f i A ie + gbA eib 9 a ^ A g i g a Ai gCl g*aa} A ig + gaA gi a^ + h I n t h e e q u a t i o n s above, H$ = Hf + H& where Hf = Τιωαά^ά + TiuJb&b, H & = EmA gb&Aei -\- g b & Ai e + gbAeib m=g,i,e (7.47) (7.48) (7.49) (7.50) are the free field and the free atom hamiltonians. The problem of solving these hamiltonians is addressed in Chap. 11. 7.5 Effective Two-Level Atom In the processes described above, a photon mediates transitions between a pair of levels coupled by an allowed dipole transition. This is the lowest order process in the interaction of radiation with atoms. If there is no pair of direct dipole-coupled levels available close to an applied frequency, higher order processes come in to play. The process next in line is a two-photon process. As the name suggests, it describes a situation in which transition between two levels is mediated by simultaneous action of two photons. The action in question may consist in simultaneous absorption or emission of two photons, or absorption of one accompnaied by simultaneous emission of the other. The former process takes place if the sum of two applied frequencies is close to the frequency of transition between a pair of levels having a non zero dipole matrix element for the said transition. The latter process applies when the difference of two applied frequencies is close to the frequency of transition between a pair of levels having a non zero dipole matrix element for the said transition. For the sake of definiteness, let a and b be the field annihilation operators for the modes of frequencies ωα and Wb■ If the levels \e) and |g) are such that huiQ = Ee — -Eg ~ h(oja + u>b) then the interaction between the field and the atom is goverened by the hamiltonians HL = n diVeaebab\e)(g\ + h.c. (7.51) Here, deg* is the dipole matrix element for the process in question and ea,eb are as in (7.10). 7.5 Effective Two-Level Atom 147 In case Ee — Ez ss %{ωα — uib) then H r eael&& \e) (g\ +h.c. (7.52) the dig ^ being the dipole matrix element for the process of absorption of one and emission of another photon. For further details see [81]—[83]. These considerations can be generalized to a multiphoton process involv ing m emissions and n absorptions. Such a process is described by the hamil tonian of the form (7.57). The strength of a multiphoton process is enhanced if there is an inter mediate transition close to a field frequency participating in the multiphoton process but still sufficiently far away enabling that transitions to be termed far-off resonant. We elaborate this in the following. To that end, in the three-level models introduced in the last section, let the detunings δη = I Ei — E„ 5b = I Ee Ei | uib (7.53) h ““ n of ea ch p a i r of l evel s f r om t h e f r equency of t h e i r r e s pe c t i ve coupl i ng fi el ds be such t h a t \ga\\/fi^ |#Q|, \gb\y/rib ■C Sb- Here na,nb are the average number of photons of frequency ωα and uib· In this case, the transition between |(/) and |i) as well as that between \e) and |i) are far off-resonant. We can then solve the Heisenberg equations perturbatively with ga/5ig and gb/Sie as the per turbation parameters. However, exact solutions for the three configurations of a three-level atom are also available (see Chap. 11) under the condition Ee — Eg = Τι{ωα +Wft), ladder configuration, (7.54a) Ee — Eg = Η\ωα — Raman and V configurations. (7.54b) Starting with the exact solution we have carried the said perturbation ex pansion in Chap. 11. We find that, to the lowest order of perturbation, the intermediate level remains unoccupied if it is unoccupied initially. The three- level atom is thus reduced to an effective two-level atom. The dynamics of the three configurations is found to be governed by the following effective two-level hamiltonians HL = H f + [ E q + \9 a - a ^ a 9 9 + E e + 19b\: + 9 a 9 b abAeg + h.c. (7.55) HK = Hf+[E, + Ee + ^ t f b ) A, ^jr^&aAeg + h.c. (7.56) The hamiltonian in the V-configuration is similar to that in the Raman con figuration. Each term in the expressions above has a simple physical meaning: 148 7. Atom-Field Interaction Hamiltonians • The operator Aegab in (7.55) describes the process of the atom making transition from its lower state \g) to its upper state |e) by absorbing simul taneously one photon of frequency ωα and another of frequency uib· The operator a^t f Age describes the reverse process. In the same way, aw Aeg in (7.56) describes transition from \g) to \e) by simultaneous absorption of a photon of frequency ωα and emission of a photon of frequency Wb- The strength of these processes is evidently enhanced by decreasing the detun ing of the intermediate level to the extent that it still qualifies to be called far off resonance. • The shift in the energy levels in (7.55)-(7.56) is the Stark shift. Note that it depends on the intensity of the field. It arises due to virtual transitions to the intermediate level. The atom in the lower state in the ladder configu ration makes virtual transitions to the intermediate state by simultaneous absorption and emission of a photon of frequency ωα. Note that in this case atom must absorb a photon first and then emit. This process is represented by a)a. The atom in this process remains in the state |(/). It is, therefore, described by the part of the Stark shift term reading a)aAgg. The anal ogous process for the atom in the upper state |e) is, however, performed by emission followed by simultaneous absorption of a photon of mode b. This is represented by bw. Note the order of the operators. The atom, of course, remains in the state \e). The process of virtual transition from the upper state is thus represented by bwAee. The Stark shift terms in the other configurations can be similarly interpreted. A process in which ωα φ uib is termed non-degenerate. Else it is a degen erate process. The hamiltonians for the degenerate processes corresponding to (7.55), (7.56) are obtained by the replacements —► uia, b —► a. T h e i n t e r p r e t a t i o n of t h e ef f ect i ve t wo- l evel model s out l i ne d above p r o vi des a r e c i pe f or e l i mi na t i ng one or mor e of f - r e s ona nt l evel s. I f t wo l evel s a r e coupl e d by n off-resonant dipole transitions between them which involve absorption of m photons and emission of n in going from |(/) to \e) then the hamiltonian, in the interaction picture generated by the free hamiltonian, will be of the form Hi(t) = gexp(-iAt) αχ ■ ■ ■ amb\ ■ ■ ■ ^|e)(c/| + h.c. (7.57) Here A = ω\ + ■ ■ ■ ujm — uJm+\ + ■ ■ ■ ωη — ωο· This describes a coherent mul tiphoton process. The problem of solution of such a hamiltonian is addressed in Chap. 11. 7.6 Multi-channel Models 149 7.6 Multi-channel Models The atomic levels configurations and the fields in the models above are such that only one mode couples a pair of levels. Several interesting and useful phenomena are observed when a pair of levels is coupled by two or more frequencies. A phenomenon of particular interest, possible with multi-channel models, is the wave mixing. It refers to the phenomenon of generation of a new frequency by algebraic addition of frequencies or by a division of a frequency. In order to see that, let the hamiltonian in the interaction picture be Hi(t) = exp(—iZ\i)F|e)(<7| + exp(iZii)Ft |c/){e|. (7.58) Notice that the combination of the field operators responsible for the tran sition that starts and ends up at |(/) is F^F. Comparing (7.58) with the hamiltonian (7.57) describing a single-channel multi-photon process, we see that F for such a process is a product of annihilation and creation operators for various modes. The operator F^F is, therefore, a product of number op erators for the modes in question. Hence, at the end of a cycle, the energy of every mode is the same as at the beginning. Exchange of energy between modes at the completion of a cycle may take place if F is a sum of products of field operators. As an example, consider the process depicted in Fig. 7.2. In it, the atom can go from level |<7) to |e) by absorbing two photons, one of frequency ωα and another of frequency It can make that transition alternatively by absorbing a photon of frequency u>c. Notice that parity considerations would prevent simultaneous occurrence of these two processes if the states involved are of definite parity. This pro cess can therefore occur only if the states are not of definite parity. Since parity conservation is violated in the absence of space-inversion symmetry, for the process in question to be possible, atom must be in a space-inversion symmetry breaking environment like in a crystal. The process of Fig. 7.2 is e> (Ob j_§a |l> C0c (pa J 1----------------- |g> Fig. 7.2. Effective two-level atom in which levels are coupled by two channels. 150 7. Atom -Field Interaction Hamiltonians described by (7.58) with F = gxab + g2c, Δ = ωα ~ ω0, wc = ωα + (7.59) In this case F^F involves terms like a)wc which do not commute with the number operator of any of the fields. There is thus an exchange of energy between the modes at the completion of a cycle. We discuss the process of wave mixing arising out of this hamiltonian in the next section. As another example, consider the process depicted in Fig. 7.3. It describes generation of Stokes and anti-Stokes fields of frequency ω3 and ωα on interac tion with a pump of frequency ωρ with a three-level atom in Raman configu ration in which the intermediate level is off-resonant with all the transitions. The corresponding interaction hamiltonian is (7.58) with [84] F = g3a\ap + gaal&A. (7.60) This is the so called two-channel Raman-coupled model [84]. For some other two-channel models, see [85[. |i> Fig. 7.3. Two-channel Raman-coupled model. 7.7 Parametric Processes In the processes introduced above, the state of the atom as well as that of the field changes. Parametric processes refer to the class of processes in which an atom in a level makes only virtual transitions to other levels. Consider the interaction Hamiltonian (7.58). Let |Z\| | | F | | so that the transitions between |<?) and \e) are off-resonant. Let the atom be initially in state \g). The transitions to |e) are then only virtual. The procedure outlined in Sect. 7.5 leads to the following hamiltonian for the fields: HP = Hf + -^FiF. (7.61) 7.8 Cavity QED 151 As discussed in the last section, wave-mixing can occur if F is a sum of products of field operators. As an example, (7.61) for F given by (7.59) (with 9i = 9 2,G = \g\\/A) reads HP = H{ + G c^ab + WcJc (7.62) Now, we assume that the mode c is initially in an intense coherent state |a), [a| 1. On account of the procedure outlined in Sect. 7.1, we treat the mode c classically by replacing c by aexp(—iuict) so that (7.62) reduces to Hp = ωαά^ά + + ξ* abexp(iu)ct) + h.c. (7.63) ξ = Ga. The dynamics generated by this Hamiltonian may be investigated by recalling from (2.71) that it is an element of SU( 1,1). An application of the similarity transformation (2.75) shows that the modes a and b grow in time. The Hamiltonian (7.63) thus transfers energy of the mode c, called the pump to the modes a and b. The Hamiltonian (7.63) thus describes the process of down conversion of a frequency. 7.8 Cavity QED It should be clear that the models introduced above, and their likes, rely on the availability of a set of well-separated discrete modes. This may be achieved only inside an appropriately constructed cavity. Recall that the possible val ues of wavelength inside a cavity of finite volume V are discretely spaced and that the density of modes at frequency v is proportional to (12 + m2 -I- n 2) where Ι,τη,η are integers characterizing the order of a mode. This implies that lower order modes are better suited to meet the requirement of well- separation of frequencies. For cavities of the size of a few millimeters, lower order frequencies fall in the microwave region. However, the field in a cavity loses energy due to leakage at the walls of the cavity. The methods for accounting for such losses are outlined in Chap. 8. The models introduced above ignore such loses. Hence, in order to realize these models experimentally, the time of interaction iint should be such that iint < k^ 1 where κ is the rate of leakage of the field at frequency ω. That rate is usually expressed as κ = ω/Q where Q is called the cavity Q-factor. We consider a microwave cavity with high enough Q factor and find out what kind of atoms are suitable for realizing the said models. To that end, note that in order to enable the atoms to pass through it, the cavity has to be open ended. The modes travelling parallel to the axis of the cavity are, of course, discretely spaced. However, the modes entering from the open sides form a continuum. We will see in Chap. 8 that interaction with a continuum of modes causes an atom to decay spontaneously. It is obviously an undesirable hurdle in the way of realization of the models in question. We, therefore, need to restrict the time of interaction to a time scale much shorter than the time 152 7. Atom-Field Interaction Hamiltonians scale of spontaneous decay. The time of interaction, however, should be long enough to allow for appreciable exchange of energy between the atom and the field. Recall that the rate of exchange of energy between a pair of atomic levels coupled resonantly with a field mode is given by the Rabi frequency which is proportional to the dipole moment of the transition in question. Hence, the dipole moment of the operative transition should be high enough to make its Rabi frequency g much larger than the spontaneous decay rate 7 from any of the operative levels. These considerations imply that the experimental conditions should be such that g~l < iint < 7 - 1,«T1. (7-64) For experimentally meaningful interaction times in microwave cavities, such a condition can be realized with transitions between states of high principal quantum number n called the Rydberg states. An atom in such a state is called a Rydberg atom. In the following we enumerate some properties of Rydberg atoms to bring out their usefulness for the purpose in question. 1. The energy of a level of principal quantum number n is proportional to n~2. Hence, the frequency ω of transition between the levels of neigh bouring principal quantum number ~ n~3 if n is high. For n ~ 30, ω lies in the microwave range. 2. The dipole moment d for transitions between nearby levels is proportional to n 2. Clearly, the dipole moment for n ~ 30 is almost two orders of magnitude higher than normal optical transitions corresponding to, say, n ~ 3. The choice of high n is thus in consonance with the aforementioned requirement of strong atom-field coupling. 3. We will see in Chap. 8 that the rate of spontaneous emission 7 between levels having dipole matrix element d and transition frequency ω is pro portional to d2u!3. Hence, in view of the properties 1 and 2 above, 7 ~ n ~ 5 for transition between two Rydberg levels. The rate of spontaneous decay between two Rydberg levels is thus reduced considerably compared with that in optical domain. For n ~ 30, 7 ~ 100 sec- 1. 4. The spontaneous emission in 3 above is for transitions between two oper ative Rydberg levels. An atom may decay spontaneously to other lower energy levels. The total rate Γ of such an emission may be shown to be ~ n ~ 3 [8 6 ]. However, emission to states other than the ones in question does not have any bearing on the model since the probability of an atom returning to an operative level from a non-operative one is negligible. These considerations show that the conditions appropriate to realizing the models in question can be met by working with Rydberg atoms on a time scale of few milliseconds. The problem of interaction of isolated atomic transitions with isolated e.m. field modes is thus a problem of cavity quantum electrodynamics (QED). See [ 8 6, 87] for details of applications of cavity QED in probing fundamental aspects of quantum mechanics and atom-field interaction. 7.9 Moving Atom 153 7.9 Moving Atom In our treatment so far of atom-field interaction, we have tacitly assumed that the atom is fixed at one position or, if the atom is moving, there is no spatial variation of the field over the distance that it traverses during the duration of its interaction with it. These conditions exclude many important situations of interest. For example, in cavity QED, an atom enters a cavity and interacts with a spatially varying field while in motion. An atom oscillating in a laser cooled trap is another example which is of current interest. The atomic motion may be included in the description of atom-field inter action by making the position vector r in the mode function in the interaction hamiltonian a dynamical variable so that Here P is the momentum conjugate to r, m is the atomic mass, V(r) is the external potential and Ha^-f(r) is the atom-field interaction hamiltonian. Now, the momentum of an atom changes due to the influence of V (r) and due to the fact that it recoils to compensate for the momentum of radiation emitted or absorbed by it. Each process of emission or absorption of radiation of wave vector k changes the atomic momentum by 7i|fe|. If this is negligibly small compared with the momentum of the atom then we can ignore it and assume that the atomic momentum changes solely due to V(r). Hence, if rm(t) is the position vector at time t due to the evolution under the mechan- - 2 ical part P /2m + V (r) of the hamiltonian then the effect of atomic motion on atom-field interaction is adequately described by replacing r in H (r) by (i'm(t)) so that The position is thus no longer a dynamic variable. Many a times it is adequate to describe the mechanical part classically. In this case, and for an atom evolving freely (V = 0) with fixed momentum p, {r m(t)) -> r*o +pt/m. A s ys t e m of c u r r e n t i n t e r e s t i n whi ch t h e a t o mi c mo t i o n i s coupl ed t o i t s e l e c t r oni c t r a n s i t i o n s i s t h a t of an a t o m i n a l a s e r cool ed t r a p. Th e mot i on i n t h e t r a p i s wel l a ppr o x i ma t e d as ha r moni c. For a de t a i l e d r evi ew of t h e q u a n t u m o p t i c s of a n a t o m i n a l as er cool ed h a r mo n i c t r a p, see [88], Η = - * - P 2 + V (r) + Ha + H{ + H&- i ( f ). (7.65) H — Ha + Hf + Ha-{ {{vm(^))) · (7.66) 8. Quantum Theory of Damping The evolutions governed by the hamiltonians we have introduced are re versible in time. However, irrevesible motions are facts of life. Spontaneous emission from an excited atomic level, absorption of radiation etc. are some examples of interest to us. How irreversibility arises is the issue addressed in this chapter. We will see that irreversible evolution is generally an outcome of a system’s interaction with the environment having nondenumerably in finite number of degrees of freedom. Such an evolution is described by a so called master equation. We derive the master equation under the conditions frequently encountered in quantum optics and specialize it to various model systems. For further reading, see [89]—[91]. 8.1 The Master Equation Consider an isolated system composed of subsystems named S and R. We are interested in the dynamical behaviour of the system S alone. In what follows we refer to S as the ‘system’ and R as the ‘reservoir’. Let the hamiltonian of the combined system S + R be expressible as H — Hs + Hr + Hrs = H0 + Hr s - ( 8 . 1 ) Her e, H$ and H r describe free evolution respectively of the system and the reservoir, and H rs is their interaction hamiltonian. We know from (1.121) that the density operator p(t) of the combined system at time t is given in terms of p(0) by the relation p{t) = exp - n m p( 0) exp nHt exp Lt p( 0), (8.2) where Lp = (—i/h)[H, p\. On using (8.1) for H and on applying (1.51), this expression may be rewritten as r t p(t) exp exp ^H0t I *T exp ί ά τ H\( t ) Jo P( 0) ^ Jo d r f l i ( r ) exp | - H0t (8.3) 156 8. Quantum Theory of Damping Hi(t) = exp ( -Hrsexp H0t ) . (8.4) The behaviour of S alone is determined by the density operator ps(t) = TrR[p(i)] obtained by performing the operation of trace over the reservoir operators in the combined density operator p(t). Let S and R be decoupled at t = 0 so that p(0) = ps(0) <8> P r ( 0 ), Trs/3s(0) = TrRpR(0) = 1. (8.5) Substitute this in (8.3) along with the definition H$ = Hg + i i R and perform trace over R. Since Hs does not contain any reservoir operator, the exponen tial containing it can be brought out of the trace. The cyclic property of the trace can be applied to the reservoir operators. As a result we find that Psi(t) = D(t)ps(Q), pSi(t) = exp | - H st Ps (t) exp -Hst n (8.6) (8.7) D(t)ps(0) = TrR| ^ e x p J drfl'i(r)j pR(0)ps(0) x?exp[ i/dT",(T)]} - - / d t L/( t ) pR(0) >/5s (0) The superoperator Lj(t ) in the equation above is defined by Li ( t ) p= Hi {t), p . Expr e s s t h e t i me - o r d e r e d i nt e gr a l as i n ( 1.43) t o r e wr i t e (8.8) as D(t) = 1 + x, oo rt X (8.9) (8.1 0) d τη J dr„_i ''' J d r 1TrR| L/( r „ ) - - - L/( r i ) p R( 0 ) |. (8.11) Owing to the reasons outlined circa (1.108), we reexpress D(t) as OO D(t) = exp ln(l + i ) J = exp [ £ M fc(t)j, (8.12) m=1 8.1 The Master Equation 157 Mi(t) M2(t) U T / d^ R L i ( t ) p r (0) dr, TrR L i (t2)L i (t 1) pr (0) (8.13) and so on. In writing the second step in the equation above, we have (i) ex panded ln(l + x) in powers of x, and (ii) grouped together the terms having the same number of L/. The density operator ps(t) of the system S may now be determined by substituting (8.12) for D{t) in (8.6). However, exact evalu ation of D(t) is generally a formidable task. Approximate expressions may be derived by exploiting realistic conditions. A practical situation of widespread interest is the one in which the interaction hamiltonian is much weaker than the free hamiltonian. In this case, it is adequate to retain in (8.12) terms up to the second order in the interaction. It is called the Bom approximation. Moreover, we assume that /3r(0) is such that TrR[iJi(i)pR(0)] = 0 so that M i = 0. Hence, in the Born approximation, Psi{t) = exp (M 2(i)) As(0). (8.14) To derive the equation obeyed by psi ( t ), differentiate (8.14) with respect to t using the identity (2.3). In order to be consistent with the Born approx imation, the terms only up to the second order in the interaction need be retained. The operation of differentiation of the exponential operator is then equivalent with that of a c-number exponential. On recalling the definition (8.13) read with (8.9) we obtain drTrt H\(t), H\(t — t ), pn(0)psi(t) (8.15) Now, let the system-reservoir interaction hamiltonian be expressible as N k=1 HnS = h y ( s l F k + Flsk (8.16) Fk being a reservoir operator and Sk a system operator. Insert this in (8.4) to obtain N Hl(t) = (sUt)FkI(t) + F U t ) S kI(t)), ( 8.17) k=1 Fki(t) = exp ( iHr t/Tij Fk exp t/Tij . We a s s ume t h a t t h e s y s t e m o p e r a t o r Sk is such that (8.18) 158 8. Quantum Theory of Damping Ski{t) = exp ( ^ Hs ί j Sk exp exp ( - i Jl kt ) S k- ( 8.1 9 ) On c ombi ni ng ( 8.17) a nd ( 8.15) a n d on t r a n s f o r mi n g psi (t ) back to ps(t ) using (8.7) we find that (8.15) reduces to the master equation d . d t Ps h Hs, ps + LsoPs = LsPs, ( 8.2 0) LsoPs Σ k,l + + [ ( ^ p s ^ - p s ^ ) ^ 1 ^SkpsS} - PsS}Sk) w ^ + [sl/i ss] - ps S} s l ) wl ( S k Ps S i - P s S i S k) w [ 4) ( 3 ) Ik + h.c. (8.2 1) Th e rates W(t )'s are related with the two-time averages of the reservoir operators by W, (i) Ik dre xp(i/V )T rR F ]u (t - r)Fki (i )pK(0) = / Jo = / d r e x p ( - i i 7 i r ) T r R Fu(t Jo 1 r°° / d r e x p ( - i i?i r ) T r R Fu(t Jo L poo _ / d r e x p ( i ^ r ) T r R F^( t - τ ) ^ 7(ί)ρκ (0) Jo L d r, r ) 4 t/( i )PR(°) r)Fki(t)pR( 0) d r, d r, d r. (8.22) In writing the expressions above, the upper limit of time-integration is re placed by oo by assuming that the reservoir correlation time r c is very small compared with the time t of observation. This usually requires the reservoir to have non-denumerably infinite number of degrees of freedom (see Sect. 8.4). On the said time scale of observation, the system loses all the memory of its past. It is, therefore, called the Markov approximation. The time of observa tion should, however, be short compared with the time scale on which the system evolves. The Liouvillean L$ in (8.20) describing the evolution of S consists of two parts. The part involving the commutator of Hs describes its free evolution whereas the influence of the reservoir, contained in Lso, describes irreversible motion. Now, let the subsystem S, while interacting with the reservoir, be driven also by an external field whose action on the system is described by Hexi so that total hamiltonian of the combined system is H = Hs + Hext(t) + + H r s - Le t us def i ne ( 8.2 3 ) 8.1 The Master Equation 159 t t y ) = 2 * exp | i/s + ^ R + ^ e x t ( r ) | Verify that psi(i) obeys the equation d r . (8.24) (8.25) (8.26) (8.27) The form of H\{t) is as in (8.17) with Fkj{t) as in (8.18) but Ski(t) are now given by rt r ■> ' Sk Ski{t) = 2* exp ^ J | i i s +ffext(T)}dr (8.28) The formal solution of (8.26) is the same as in (8.6). We can follow the steps leading from (8.6) to the master equation (8.20) by first rearranging H\(t) in the form: N „ f * Hi(t) = Τ ι ^ ( exP (i^fe ^ S kF kI +exp ( - i i 7 fe t j F kISk). (8.29) k=l It then follows that the master equation in the present case would read d „ i d t PS h Hoxt (t), ps + LsoPs, ( 8.30) wi t h Lso given by (8.21) but with Sk -> Sk, Fkj —> F ki, and Qk —> Dk. How ever, if HiTsll >> j | J | then the time evolution in (8.28) may be assumed to be solely due to Hs, i.e. Hext(tj in the exponentials in that expression may be ignored in comparison with Hs- This is the weak external field approxi mation. It reduces (8.28) to (8.19). The Liouvillean Lso in (8.30) is then the same as Lso in (8.21). In other words, the Liouvillean of the master equa tion of a system in a weak external field is the sum of the Liouvillean of its evolution in that field alone and that of its evolution in its absence. Now, compare the exact formal expression for ps(t) = Ττβ[ρ(ί)], where p(t) is given by (8.2), with the formal solution of the master equation (8.20) which is in the Born-Markov approximation. Assuming, for the sake of sim plicity, Ls to be time-independent we infer that, in the Born-Markov approx imation, TrR [exp ( Lt j p(0)j « exp ( Lst j TrR [p(0) ]. (8.31) This relation will prove useful in finding multi-time averages in Sect. 8.3. 160 8. Quantum Theory of Damping 8.2 Solving a Master Equation Consider a master equation = Lp. (8.32) It may be solved by converting it in to a c-number equation by taking its matrix elements in an appropriately chosen basis. If L is independent of time then the resulting equation may be solved by the method of eigenvectors exapansion outlined in the Chap. 10. In the problems of interest to us, one of the eigenvalues of L is zero whereas the real part of all its other eigenvalues is negative. Then, in the limit t —> oo, the system approaches the state pss, called its steady state. It solves Lpss = 0. (8.33) We assume that the solution of (8.33) is unique. The task of solving this equation is greatly facilitated if L obeys the condition (5.41) of detailed bal ance. Along with the requirement pss = pss (where A is time-reversed form of A), that condition in the superoperator language may be rewritten as PssLF = LtTpssF, (8.34) F being an arbitrary operator, Ltr is the time-reversed form of L whereas L is related with L by Ύτ[λίΒ} = Tr[BLA} (8.35) for all A and B. See Chap. 12 for an application of the condition (8.34). Alternatively, we may work with the equations for the expectation values of the operators. These equations are obtained by multiplying the master equation by the operator whose expectation value equation is desired, fol lowed by performing the operation of trace. As an illustration, consider d „ dtP B, + 2 i p i f - A ^ A p - p A ^ A. (8.36) Notice that the evolution operator of any master equation is a sum of terms of the kind appearing on the right hand side of the equation above. The equation of motion for the expectation value (O) = Tr[Op] of an operator O, obtained by multiplying (8.36) by 0 and on using the cyclic property of the trace reads d (O) = - i T r { 0 [β, p ] } + T r [ { 2 i t o i _ 0 i t i _ J4tJ4 o J/5] d t = —iTr {p [θ, 6] } + i v [ { [ i t,0 ] i - A'[A,0]}p] = —i ( [ 0, b ] ) + ( [ AK0} A- A^ [ A,0} ). (8.37) 8.2 Solving a Master Equation 161 The operator in the expectation value sign on the right hand side in the equation above may or may not be a linear combination of a constant and a constant multiple of O. If it is not, then we need to derive equation for new operators arising therein. This process is carried till a closed set of equations is obtained. The equation of motion of an operator may be put in a form suitable for applying to multi time averages. To that end, and for the sake of simplicity, let Ls be time-independent. On using the formal solution of (8.20) and the definition (8.35), note that the expectation value of an operator acting in the space of S alone may be written as (A^(t)) = Trs [ i (S)/5s (i)] = Trs [ i's ) exp ( l st ) ps (0) Trs Ps(0) exp ( Lst ) A(S) (8.38) Note that L acting on a system operator yields a linear combination of the system operators. We may, therefore, write A(S)(t) ee exp ( l st \ A(S) = (8.39) ' ' m t h e f m(t) being c-number functions of time. On combining this with (8.38) we find that (A(s\t)) = J 2 f m ( t ) { A ^ ), ( i L S)) = Trs [/5s(0)iLS)]· (8-40) m Thus, the average of any system operator is determined by solving a closed set of equations for Αχ, A2,... Whi ch of t h e a ppr oa c he s o ut l i ne d above s houl d b e pr e f e r r e d de pe nds on t h e s i t u a t i o n a t h a n d. For exampl e, see Cha p. 12. However, s ol vi ng a ma s t e r e q u a t i o n a n a l y t i c a l l y e xa c t l y i s gene r a l l y a f or mi dabl e t a s k. We o u t l i ne t h e a p p r o x i ma t i o n me t h o d s for sol vi ng ma s t e r e q u a t i o n s for t h e s i t u a t i o n s enc ount e r e d ge ne r a l l y i n q u a n t u m opt i cs. Con s i der f i r s t a s ys t e m i n i n t e r a c t i o n wi t h a r e s er voi r a n d s u b j e c t al s o t o an e x t e r n a l fi el d. T h e s ys t e m i s t h e n de s c r i be d by ( 8.30). I f t h e i n t e r a c t i o n of t h e s ys t e m wi t h t h e e x t e r n a l fi el d i s weak c o mp a r e d wi t h i t s i n t e r a c t i o n wi t h t h e r e s e r voi r, t h e n we c a n c a r r y a p e r t u r b a t i o n e xpa ns i on of i t s s ol ut i on i n power s of | | i Zext | | · Th e d e t a i l s of t h i s cas e ar e p r e s e n t e d i n t h e Chap. 9. Cons i de r n e x t t h e cas e when, i n a n a p p r o p r i a t e i n t e r a c t i o n pi c t ur e, t h e h a mi l t o n i a n p a r t of t h e ma s t e r e qua t i on ( 8.20) i s i n d e p e n d e n t of t i me. Tr a n s f or m t h e d e n s i t y o p e r a t o r ps to pi(t) = exp jiiTsi/Tij ps exp j-iiTsi/Ti} . (8.41) The pi obeys the equation 162 8. Quantum Theory of Damping Li(t) = exp jiiTgi/Ti j Lso e*P i Hs t/h^ . (8.43) The Li ( t ) will be of the form h(t ) = A0 + A(t), (8.44) the Ao being independent of time. The time-dependent part A(t) would in volve non-zero differences of the eigenvalues Λ * of Hs- The contribution to the solution of (8.42) from A(t) part would, therefore, be of the order of ||£so||/|Ai — Ajj, Ai φ Xj. Hence, if the differences in the eigenvalues is suf ficiently large compared with ||£so|| then we can ignore A(t) in comparison with Ao. The approximation wherein only the Aq in (8.44) is retained is re ferred to as the secular approximation. As examples of its applications, see Sect. 12.2 and Sect. 14.2. —pi = L i { t ) p i, (8.42) 8.3 Multi-Time Average of System Operators So far we have discussed the problem of solving a master equation for deriving operators averages at a single time t. The multi-time averages are also of experimental interest like in characterizing the spectrum and determining correlations between the dynamical variables at diferent times. Consider the two-time average of operators A ^ and B ^ of S. By definition, (ASs\t + r)B^s\t ) ) = I W [ ^ (S)(i + r)B^s\t )p(0) ( 8.45) Th e o p e r a t o r s a r e i n t h e Hei s enber g p i c t u r e g e n e r a t e d by t h e combi ned h a mi l t o n i a n H. On invoking (1.77), and on applying the cyclic property of trace, (8.45) leads to i (S)(i + r ) B {s\t ) = Trfi+S Tr R+S i (s)(T)S(s) exp ( — —Ht^jp(0) exp exp ( — ^ H r j B ^ p ( t ) exp ( - H r ) Ττβ+s exp ( Lr ) (l?(S)p(i))j 3.46) The last line above is by virtue of the definition (8.2). Use (8.31) to reduces (8.46) in the Born-Markov approximation to 8.4 Bath of Harmonic Oscillators 163 A ^ ( t + r)B^(t)J = Trs = Trc exp ( Ls t ) | s (,s)ps(i)} B {S)ps {t)exp ( Lst ) i (s) (8.47) In writing the last line above, we have invoked (8.39). The two-time average is thus expressed in terms of the functions f m(t) determined by single time averages. This is the quantum regression theorem. It is the quantum analog of the regression theorem (5.38) for a classical Markov process. Similar results may be derived also for multi-time averages [68]. Next, we specialise the master equation by fixing the choice of a reservoir. 8.4 Bath of Harmonic Oscillators A reservoir of common interest is the one constituted by an infinite number of harmonic oscillators. An example of such a reservoir is the e.m. field in free space. Its interaction with an atom leads to the phenomenon of sponta neous emission from an excited atomic level. The infinite system of harmonic oscillators may even be used to model the motion of atoms in the walls of a cavity. The system S then is a mode of the cavity field. The reservoir in this case causes decay of the cavity modes at the walls of the cavity. Let SJ., Sfc be the creation and annihilation operators of a harmonic oscilla tors of frequency u>k constituting the reservoir. The free reservoir hamiltonian is then given by HR = h J 2 “ki>lbk. (8.48) k The hamiltonian of the combined system of the reservoir and the system S interacting with it then reads H = Hs + h Σ “kbth + hJ2 [ i i s; + F}St] . (8.49) k i We assume that the reservoir operator appearing in the interaction hamil tonian above is a linear function of bk and b\. Its form depends on whether or not we choose to work in the rotating wave approximation (RWA) introduced in Chap. 7. Assuming St to be a lowering operator, the form of Ft in the RWA is F = ^ ^ tj/ k bk - (8.50) k the gjk being the system-reservoir coupling constant. Without the RWA, Fi = Σ [9ik^k + 9^ l ■ (8·51) k The Fki defined in (8.18) are found by noting that hi(t) = exp (i HR t/h j bk exp (-i-ffR t/h j = exp(-iu>kt)bk. (8.52) Now, in order to evaluate the rates defined in (8.22), we need to fix the state of the reservoir. The reservoirs of common interest are: (i) a t hermal reservoi r, and (ii) a squeezed reservoir. 8.4.1 T h e r m a l R e s e r v o i r Th e r ma l r e s e r voi r i s c o n s t i t u t e d by h a r mo n i c os c i l l a t or s i n t h e s t a t e of equi l i br i um a t t h e i n p e r a t u r e T. It is described by the density operator f a = l [ ( l - e x p ( - 0 k ) ) e Xp ~Pkb\bk ], (8.53) fc —1 = hwk/k-QT, k-Q being the Boltzmann constant. The correlation functions needed for evaluating the rates (8.22) are easily found by applying the for mulae (6.79)-(6.81). Verify that (b(wk) ξ bk) (b(wk)) = (b\u>k)) = 0, (tf {u)k)b(u)i)) = n(u>k)S (wk - ωι), n(ojk) = 1, (6(w/)6t (u;fe)) = (n(wfc) + 1) δ (wk - ωι), (b(u}k)b(u}i)) = (tf (u}k)tf (ωι)) = 0. (8.54) Consider first the case of interaction in the RWA. On using (8.18) read with (8.50) and (8.52) we find that (F]i(t - r)Fi/(i)> = 5 3 s ^ f c e x p ( - i w fcT)n(w*), k (Fj i ( t - r) F}j ( t ) ) = ^ g ^ g i k ^ V ^ k T ^ n ^ k ) + 1), k (FjjQ - T)Fu (t)) = ( F j ^ F ^ t - r)> = 0. (8.55) Now, let the frequencies be continuously distributed and let Η(ω) be the density of oscillators at frequency ω. We can then convert a sum over k to an integral over ω by means of the correspondence /»oo 5 3 f k ->■ / du) h( w) f ( u) ). (8.56) k J o _ Derivation of involves evaluation of terms of the kind, 164 8. Quantum Theory of Damping ΛΟΟ I± = / dT y exp (j=i (uk ~ Ω) t ) f k J o k f i O C /* o c —> I dr I άωΗ(ω) exp {±i (ω — Ω) τ }/(ω). (8.57) Jo Jo On recalling (A.3), this reduces to I± = J dw h(w) /( ω) | π ό (ω — Ω) ± iΡ ^ ------— = πΙι(Ω)/(Ω)±ίΡ ί άωΗ^ ^\ (8.58) J u) — ί ΐ On a p p l y i n g t he s e r e s ul t s, ( 8.22) yi el ds W^p = = 0 and WV = 1ι,(Ωι)η(Ωί) + ϊΩ^\ = Ίί] (Ωi) ( n ( ^ ) + 1) - \Ω(2), (8.59) 'Yij (Ωί) Tr ^ ’' gjf,gjkS(ujk ~ Ω^, k = ( 8.60) 13 ^ t t i - Wk Th e Ω ^ is obtained by replacing η(ω^) in Ω ^ by n(u)k) +1- Substitution of these rates in (8.21) gives the master equation for a system interacting with a reservoir of harmonic oscillators in thermal equilibrium at temperature T in RWA. Without the RWA, the reservoir operator is given by (8.51). The rates W^3,4·* are now non-zero. Note that in the interaction picture defined in (8.7), the terms multiplying these rates in the equation for psi(t) oscillate at Ωί + Ω^. The time t of observation is usually very large compared with 1/J?fc. Hence, contribution of these terms averages to zero on the time scale of ob servation. The terms multiplying the rates may, therefore, be ignored. This amounts to making the RWA on the master equation. The resulting master equation, like the one derived by making the RWA on the hamilto nian, does not contain terms corresponding to However, verify that W{k' now involve in addition to 1+ defined in (8.57), the terms of the kind 8.4 Bath of Harmonic Oscillators 165 ΛΟΟ J±= dT exp (±i (wfc Jo k —>■ j du) Η(ω) f(uj) β ) τ ) Λ duj h(ui) f (w) ^ πδ (ω + Ω) ± iP ! ^ K to -f- f2 * τ ι f* i = ±,p J Ή τ τ τ τ - <8 ·β1> 166 8. Quantum Theory of Damping These terms contribute to the principal part in W ^'2>. We will see in the examples below that the principal part contributes to reversible part of the evolution. Thus, the master equation obtained by employing the hamiltonian without the RWA but making it on the master equation is the same as the one obtained by using the hamiltonian in the RWA except in the modification of frequencies of the rversible evolution. 8.4.2 Squeezed Reservoir A squeezed reservoir is characterized by the density operator psq = S ( 0pt h&{ 0 (8.62) where pth is given by (8.53), and s'(i) = Π exp (ξ*ί(Ωρ - ujk)b(Qp + wk) - h.c.) (8.63) k is the so called two-mode squeezing operator. It is an element of S i7 ( l,l ) group. The correlation functions needed for evaluating the rates may be de rived by noting that, for any A, Tr[/5sqA] = Tr[/5t h 5 t (ξ) AS(£)]. Use the results of Chap. 2 to show that 5t(C)S(i2p ±u;fc)5(C) = cosh(|£|)S(i2p ± wk) + exp(i0) 8Ϊη1ι(|ξ|)^(.βρ =f wfc), (8.64) ξ = |ξ| exp(iφ). It is now straightforward to verify that (S(u;fc)) = (fr(u)k)) = 0, (tt(u}k)b(u}i)) = N ^ k)6 (wfc - ωι ), Φ(ωι)&(u>k)) = (N{u)k) + l) δ (u>k - ωι ), Φ(ωφ(ωι)) = Μ(ω^δ (wfc +ωι - 2Ωρ), (8.65) Ν = n(cosh2(|C|) + sinh2(|C|)) + sinh2(|£|), Μ = (2η + 1) sinh(|£|)cosh(|£|) exp(i0). (8.66) Verify that N(N + 1) = |M|2 + n(n + 1). (8.67) For ξ = 0 the relations (8.66) reduce to those of a thermal bath. On the other hand if n = 0 then pth = |{0})({0}|, i.e. pt h then describes the vacuum state. The squeezed bath is consequently in a pure state, S^KO}), called the squeezed vacuum . We e v a l u a t e t h e r a t e s by t r e a t i n g t h e i n t e r a c t i o n i n t h e RWA. Th e r a t e s a r e gi ven by ( 8.59) wi t h η —» N: Wj p = 7i j N M ) + ifiijK W.f - 7 y ( N ( n t) + 1) - iflg). (8.68) 8.4 Bath of Harmonic Oscillators 167 Ignoring the principal part, WS] = wij]* = 7 y ( ^ ) ^ ( A ) e x p ( - 2 i i V ), (8.69) ''fij ^ ^ fjj ^fc)^(^fc ^i )· (8.70) k Th e ma s t e r e q u a t i o n of a s ys t e m i n t e r a c t i n g wi t h a s queezed b a t h of ha r moni c os c i l l at or s i s d e t e r mi n e d by s u b s t i t u t i n g t h e r a t e s der i ve d above i n (8.21). 8.4.3 R e s e r v o i r o f t h e E l e c t r o m a g n e t i c F i e l d As s t a t e d bef or e, a r e a l i z a t i on of a h a r moni c os c i l l a t or b a t h of p a r t i c u l a r i nt e r e s t t o us i s t h e one c o n s t i t u t e d by t h e e.m. fi el d i n f r ee space. The p r o p er t i e s of t h e e.m. fi el d a r e of p a r a mo u n t i n t e r e s t. For t h e s ake of i l l u s t r a t i o n, we cons i der a n os c i l l a t or of f r equency ωο at interacting with the e.m. field in free space. Corresponding interaction hamiltonian in the RWA, by virtue of (7.44), may be written as The hamiltonian (8.71) is of the form (8.49) read with (8.50). Now, the solu tion of the Heisenberg equation for bk is The field at any position R = r — Tq may be found by inserting (8.74) in (6.25). Recall that the summation over k in that expression stands for summation over the wave vector and the polarization directions: The last expression above is the integral equivalent of the summation over k in the continuum limit. The summation over λ and integration over k yields, to order | ϋ | - 1, (8.71) k (8.72) the d being the electric dipole moment of the oscillator. The free evolution of the oscillator is described by S(t ) = exp(—iujQt)S. (8.73) bk{t) = exp (—iwfc t )bk( 0) (8.74) (8.75) £?(+)(Λ,ί) = E (0+){R,t)-k20\Rx {Rxd)\R\-3 xe x p( - i k or.r o/\r\) S( t — \r\/c) . (8.76) Here ko = loq/c. We refer the reader to [93] for the details of the derivation. See also [89, 94]. The expression (8.76) for the field in the radiation zone determines the field in terms of the oscillator dynamics. The dynamical prperties of the oscillator are determined, of course, by solving the corresponding master equation. As an application of (8.76), we note that the two-time correlation function of the field in the far-field zone is (E(~\R, t).E (+> {R, t + t )) ~ (5*(t - \r\/c)S(t - \r\/c + r)). (8.77) This, on substitution in (6.90) shows that the spectrum of the radiation is determined by /»oo S(w) ~ Re I dr exp(iwr)(S't 5'(r)). (8.78) Jo In following two sections we derive the master equation for some atomic systems in contact with the reservoir of the e.m. field. 168 8. Quantum Theory of Damping 8.5 Master Equation for a Harmonic Oscillator The system treated in this section is that of a harmonic oscillator of frequency ωο described by the hamiltonian Hs = Ηωοά^ά. (8.79) Let its interaction with the reservoir be described by H rs = h 5 3 (tffefyU + Qk ^ h j. (8.80) k The hamiltonian of the combined system S + R is then given by (8.49) with Si —> a and Fi given by (8.50). The correspondibg rates are given by (8.6 8 ) and (8.69). We assume that Ωρ κ ωο and invoke the fact that gu varies little around ωο· We, therefore, have 7(ω0) = 7(^o)- (8.81) The master equation (8.20) then assumes the form p(t) = -ιω [a)a, p\ + 7 (N + l) [2άρά) - pa)a - a)ap\ + 7 -/V [2a) pa, — pad) — aa) p] +7 j-Mexp(2 iQpt) [2apa — pa2 — a2p] + h.c.j. (8.82) Here ω is renormalized to include contribution from the principal parts of W/fc1,2) and N ξ Ν(ω0), Μ = Μ(ω0). 8.5 Master Equation for a Harmonic Oscillator 169 The equation (8.82) is obtained by eliminating the bath variables in the Born-Markov approximation. However, the bath variables in this model can be eliminated exactly. In order to see it, note that the Heisenberg equations for the system and the bath operators reading \a = ω0ά + ^ 2 g k h, \bk=ubk +g*ka (8.83) k are linear. These may be solved by the method of Laplace transformation. Let F(z) denote the Laplace transform of F(t) defined by ΛΟΟ F(z) = I exp(—zt)F(t)dt. (8.84) Jo Then -j /*7+ioo F(t) = ■— I exp(zt)F(z)dz, (8.85) 2πι J wher e 7 i s s uch t h a t t h e s i ngul ar i t i e s of f ( z ) lie on the left of the line at a distance 7 parallel to the imaginary axis. Use the property ΛΟΟ / exp( - zt ) F( t ) dt = - F { 0) + zF(z) (8.8 6 ) Jo to transform (8.83) to (z+ ιω0)α = a(0)-i^gkbk, (z + iu>k)bk = ojkbk(0) - \g*ka. (8.87) k Eliminate bk from these equations to express a in terms of a(0) and (6fc(0)}. The inverse transform of the resulting expression for a gives “ (i) = 0 ) + 5 3 <f>k(t)h{0) (8·88) k where 1 P m - ™ L Λζ- ^ +i o° e x p ( ^ ) A us 9k Γ + j exp(zi) οηλ 4>k{t) = - T r / —Γ Τ Τ ’ 8·89) ioo (z + 1 Uk)v{z) f 7 — 1 0 0 N ^ ) = z + iu,0 + £ - M -. ( 8.90) f ^ Z + lUJk Cl ear l y, a ny f u n c t i o n of a(t), a)(t) may be evaluated using (8.8 8 ). For an exact evaluation of the density operator see [92], The nature of evolution is determined by the nature of the time-dependence of f ( t ) and <j>k{t) which, in turn, is dictated by the roots of η(ζ). If the number N of oscillators in the reservoir is finite then η(z) = 0 reduces to a polynomial of degree N + 170 8. Quantum Theory of Damping 1 in z. Its roots are isolated poles on the imaginary axis. The functions f (t ) and 4>k(t) are then sums of oscillatory functions of time. As a result, S executes a reversible motion. However, in the limit of continuous distribution of frequencies, it may be shown that /( f ) ~ exp( -iωι - j t ). (8.91) The φ/ί (t), obtained by substituting (8.91) in (8.89) also acquires a damping part. The motion is thus irrevrsible in the limit TV —> oo. In this limit, the density operator ps{t) evolves according to (8.82). For details, refer to [92], 8.6 Master Equation for Two-Level Atoms Consider a system of N identical two-level atoms each of frequency ωο in teracting with the radiation field. The evolution of the combined system is governed by the hamiltonian N N Η = Τιω + h J 2 “ka{ak + ^ ' (8'92) i=l k i=l Here S'a r e the raising and lowering operators for the ith atom and F* is given by (8.50). By identifying Si in (8.16) as S [ i t turns out that Ωk in (8.19) are ωο· Invoking also the (8.6 8 ) and (8.69), the master equation (8.20) assumes the form h = ^ p] i = 1 i,j = 1 + Σ ^ [(Ar + 1) ( 2S ^ )pS+ - p s ^ s ^ - S ^ S ^ p J i,j =1 +N(2S%)pS® - p S ^ S ^ - + {Mexp( 2i npt )(2S{?)pS{_i) - p S ^ S ^ - +h.c.}]. (8.93) The rates 7 ij, Ω^ = Ω ^ — Ω ^ are as in (8.60). The gij in those equations are as in (8.72) (with vq there replaced by the position of the *th atom and 9 k —> 9tk) · For the details of evaluation of the rates see, for example, [89]. For a single two-level atom, 7 = i M 2^, (8.94) <5 Cr t h e | d i2 | be i n g t h e di pol e moment be t we en t h e a t o mi c l evel s i n ques t i on. T h e i ndi ces i,j in 7 y and indicate their dependence on the atomic position through the factor exp(fe.(ri — rj )). Due to the presence of the 8.6 Master Equation for Two-Level Atoms 171 delta function in the expression for 7 y, the magnitude of k is restricted to |fc0| = wo/c- Hence, if fc.(rj — Tj) -C 1 for all i and j, i.e. if the atomic sample is confined to a region of dimensions small compared with the wavelength corresponding to the atomic transition frequency, then 7 y become indepen dent of space. The master equation (8.93) in the small atomic sample size approximation reduces to We have thus at hand the equation governing the evolution of a system of two-level atoms interacting with the reservoir of the e.m. field. Now, let the atoms interact also with an external field whose action is described by i?ext. As discussed in Sect. 8.1, if the field is weak then the master equations if the field is strong then the master equation is derived by following the procedure outlined in Sect. 8.1. We illustrate that procedure by treating the example of a single two-level atom in a monochromatic external field. 8.6.1 Two-Level Atom in a Monochromatic Field We consider a two-level atom in contact with the reservoir of the e.m. field and driven also by a monochromatic field of frequency ω. This interaction descries the phenomenon of resonance fluorescence. Let the interaction with the applied field be described by We derive the atomic master equation assuming, for the sake of simplicity, ω = ujq. To that end, we need to evaluate (8.27): N } = -iL>& + Σ p i,j = 1 + 7 [(N + 1 ) ( 2S- pS+ - pS+S~ - S+S- β) + { M e x p (2 i J?p£) ( 2S- pS- - p S - S _ - + h.c.} . (8.95) Here έ>μ are collective atomic operators defined in (3.73). derived above acquires an additional term reading —i[Hext/h,p]. However, -Hext = 7i(a:exp(—iu;t)S+ + a* exp(iwi)6'_]. (8.96) (8.97) 172 8. Quantum Theory of Damping Hext = h[aS+ + a*S_] (8.98) In writing the second line in (8.97) we have applied (1.51). For simplicity, we let a to be real. The similarity transformation in (8.97) is a special case of (7.29) corresponding to δ = 0. Using that expression (or by evaluating (8.97) directly by applying (2.42)) we find that m ) — 2 F+(t)S+ + F-(t )cos(2at )S- + iF_ (t) sia(2at)Sz + h.c., ( 8.99) F±(t) = F}(t) exp(—ίωοί) ± h.c.. (8.100) The H\(t) is of the form (8.17). Assuming that the reservoir is in the state of vacuum, the Liouvillean (8.21), ignoring the frequency shifts, assumes the form LsoP = ~ ^ [#ext(i),/5 +7+ (2S-pS+ - S+S - p - pS+S ^ j + {7 - (s+pS+ - S+S+p^j + 7 ( s z/s£+ - s +s z/s ) + h.c.}, (8.101) 7 ± = - [2 7 (010) ± {7 (ω0 + 2a) + η(ω0 - 2α)}] 7 = ^ {7(ωο + 2α) - 7(ω0 - 2α)} . ( 8.102) Note that if |α| <C ωο (i.e. if ||fls|| S> 11-f^ext 11) then the non-commutator part in (8.101) reduces to the vacuum field version ( N = M = 0) of (8.95) without the frequency shift. It is in accordance with the assertion that the Liouvillean of the master equation of a system in a weak external field is a sum of the Liouvillean of its evolution in that field alone and that of its evolution in the absence of the field. 8.6.2 Collisional Damping The master equation (8.95) describes radiative decay of an excited atomic level. It is caused by the resrvoir of the e.m. field. There are, however, other mechanisms responsible for the decay of an atomic level. One such mecha nism of considerable interest is collisions of an atom with other atoms. Let us consider the collisions that alter the phase of the atomic state but not its population. Such collisions may be incorporated by assuming that the evolution of the atom is governed by the hamiltonian Η = Κ(ω ο + μ(ί)) Sz. (8.103) 8.7 Master Equation for a Three-Level Atom 173 The equation of motion of the atomic density operator then reads d di pit) = -iw0 Sz, p(t) + L 0p(t) - ίμ(ί) [ s z, p(t) ( 8.104) Her e Lq accounts for any other interaction that the atom may be involved in. On assuming that μ(ί) is a delta correlated Gaussian process with zero mean, it follows by applying the methods of Chap. 5 that the density operator averaged over collisional fluctuations obeys —p(f) = -iu; Sz, p(t) + L0p(t) + Lcp(t) where Lcp(t) = - 7 C[SZ, [Sz,p]] = 7 C 2 SzpSz - Sz p - pSz is the Liouvillean of collisional damping. (8.105) (8.106) 8.7 Master Equation for a Three-Level Atom Consider a three-level atom described by one of the hamiltonians (7.47)- (7.49) depending upon the configuration of the three levels. For the sake of definiteness, let the levels be in the ladder configuration. The hamiltonian of such an atom in a multimode field reads Η = ^ ' EpApp + h ^ + AieP), +h.c. (8.107) p=g,i,e k where (F2) are given by (8.50) with g\k (g2k) as the coupling constant between the mode k and the levels |g) and |i) (|i) and |e)). The interaction in (8.107) is of the form (8.16) with S\ = Agi, S2 = Aie, Ωγ = Wig, Ω2 = uiei, (8.108) Wjg = (Ei — Eg)/h, ωβί = (Ee — Ei )/h. Assuming the bath to be thermal, the master equation then assumes the form (ignoring frequency shifts) P = k=g,i,e ^ ^ ^ ^k^-kk-i P k=g,i,e + Σ l i\{rn + l ) ( 2S3pSt - pSt Sj - S\SjP = 1,2 +fn (2sjpSt - pSiSj - . (8.109) Here 7 * ξ 7 (i?*), ήi = ΰ{Ωι). Now, if |wjg — wej|£ 1 then i Φ j terms in (8.109) can be ignored as, in the interaction picture, those terms contribute factors oscillating at the frequency \wig — ωΡΛ | which average to zero on the time scale of observation. Hence, the master equation for nearly degenerate sets of level separations will have additional terms as compared with widely separated sets of levels. 174 8. Quantum Theory of Damping 8.8 Master Equation for Field Interacting with a Reservoir of Atoms So far we have considered a system of atoms interacting with a reservoir of the e.m. field. The field frequencies are continuously distributed whereas the atomic system is characterized by a denumerable set of frequencies. Consider now the situation in which the atomic frequencies are almost continuously distributed interacting with field modes described by a set of denumerable frequencies. Such a situation is realized, for example, by well-separated field inodes in a cavity. The atoms in the walls of the cavity then act as a reservoir whereas each cavity mode is a small system. A simple model to describe the atomic oscillators in the walls of a cavity is in terms of harmonic oscillators. The hamiltonian for the combined system of a field mode, described by the operators a, a\ and the atomic oscillators described by the operators {6 fc, 6 ^.} is then given by (8.80). The corresponding master equation is (8.82) with N —¥ ή, M = 0 and 7 is now the damping constant of the field. There are, however, situations when the reservoir of atoms can not be modelled as a collection of harmonic oscillators. In the following we derive master equation for the evolution of one or two field modes interacting with a reservoir of two level or effective two level atoms. • Let a single mode field of frequency ω interact with a reservoir of two-level atoms of frequencies ωχ, ω2, ■ · · described by H = fiiooa)a + Ti ^ 5 3 5 * ^ + ^ + ^.c. (8.110) I I The interaction hamiltonian is of the form (8.16) with Si = ά, Α = 5 3 ^ - }· (8.1 1 1 ) I We assume that the atomic reservoir is in a state of thermal equilibrium at temperature T characterized by the density operator Patoms = Π [exp(—/3j/2) + exp(/3i/2) ] _ 1 exp(-/3iS'W), (8.112) i—1 Pi = haii/hsT. In this state, ^ • ’> = 0' ( «'’> = - 5 ρ ϋ Τ Π ) · fl- - i!t p ( - A ) · <8'I13> We see that in the same approximation as used for deriving (8.59), and on ignoring the principal parts, 8.8 Master Equation for Field Interacting with a Reservoir of Atoms 175 where ή = ή(ωο) and κ is the same as 7 (^0 ) of (8.60). The summation there is converted in to an integral by applying (8.56) with h(u>) there being the atomic lineshape function. The master equation for the field mode in question is then the same as (8.82) with N + 1 —»· (ή + l)/(2n + 1), iV —^ ή/(2 n + 1), M = 0. Consider a single-inode field causing two-photon transitions in a system of two-level atoms in the ladder configuration. It is described by (7.51) rewritten in the form #RS = y ^ f l i S y a S + h.c. (8.115) This is of the form (8.16) with S\ = ab, and Fx is as in (8.111). Assuming the state of the reservoir of atoms to be (8.112), verify that the master equation for the density operator of the field modes is ρ = —ϊωα [a)a,p] - iw;, b]k ιη ,P + κ (n + 1 ) (2f i + l ) κη 2 bapa)bi — a)tfabp — pa^Dab 2 Pa)pab — aba)h) p — paba)W + ^ 3 —j—j-y 2 b'a'pab — aba'b'p — paba'b' . (8.116) The η in the equation above comes from the principal part of the integrals. The equation for the degenerate process is obtained by replacing b by a. • For the levels in the Lambda configuration, the interaction is governed by (7.52), rewritten in the form Hrs = Y'GiSWtfb + h.c. (8.117) For the atomic reservoir in the state (8.112) the master equation reads ρ = —ίωα [ά^α,ρ] — ΐω^ &b,p — \η | ά ^ ά έ,p + τ ^ ~ ~ τ τ \2a)bpi)a — b^aa^bp — pDaa^b +- ( 2 n + 1 ) Kfi — a^btfap — pa^Wa (8.118) ( 2 n + 1 ) The equation for the degenerate process is obtained by replacing b by a. The method of the solution of these equations is outlined in Chap. 13. 9. Linear and Nonlinear Response of a System in an External Field We have seen in the preceeding chapters that the problem of studying an optical process reduces to that of solving an appropriate master equation. However, barring the harmonic oscillator model of atoms, the master equa tions in question can seldom be tackled analytically exactly. This is particu larly so when there are more than one frequencies coupling a transition. We identify in Chaps. 12 and 14 analytically exactly solvable atomic systems in which a transition is driven by only one frequency. In this chapter we discuss a perturbative approach to solving the master equation of an atomic system in which a transition may be driven by more than one frequency. 9.1 Steady State of a System in an External Field Consider a system described by a density operator p whose evolution is gov erned by the time-independent master equation p(t)=L 0 p(t). (9.1) We assume that the eigenvalues and the eigenvectors of Lq are known and that one of the eigenvalues of Lq is zero whereas the real part of all its other eigenvalues is negative. Hence, as t —> oo, the atomic system approaches the steady state pss satisfying Lopss = 0· (9-2) We a s s ume t h a t t h e s ol ut i on of (9.2) i s uni que. Le t t h e s ys t e m b e s u b j e c t t o a n e x t e r n a l i nf l uence a f t e r i t ha s a t t a i n e d t h e s t e a d y s t a t e pss. We wish to study the properties of the system as t —> oo after the application of an external field. To that end, let Hext(t) be the hamiltonian of interaction between the sys tem and the externally applied field which is treated classically. The density operator p, after the application of the field, evolves according to }{t) = [ l a (t) + i 0] p(t), l i ( t ) p = [tfext(i), pj ■ (9.3) 178 9. Linear and Nonlinear Response Let the external field be such that ||Iq|| <C Lq. Hence, (9.3) may be solved perturbatively in powers of L\{t). This task is facilitated by applying (1.51) to rewrite the formal solution of (9.3) as <'t Ps·. p(t) = exp (^L0t j ^ exp \J d r L/( r ) | OO = + n = l Li(t) = exp ( - L 0t j Li(t) exp (Lot j ■ (9.4) (9.5) Th e p(n\t ) in (9.4) is the nth order perturbative contribution to the density operator. On carrying the time-ordered expansion in the first line in (9.4) and on using (9.5) and (9.2), we find that ο(")(ί) = ^ Υ\ dTi exp | ( r i + 1 - n) L0} M ^ ) i = ί L-'O Pss, (9.6) with τη + 1 = t > τη - 1 > · · · >T\. On transforming the integration variables successively as t — rn rn,t — rn — r„_! —»· r n_i, · · ·, and on letting t —► oo, (9.6) reduces to n r° p{n\oo) = ^Ily dri exp(L0Ti)Li ~ Στ·? I pss· (9.7) Now, l e t t h e e x t e r n a l fi el d be a l i ne a r c o mb i n a t i o n of M quasiinonochromatic fields whose frequencies are centered at νλ,..., vm ■ Let the field-system cou pling be described by M Σ Σ £x{vj,t)eKY>(ivj t )Q\{vj ), (9.8) λ j = - M v~j = —Vj, e\( —Vj,t) = e*x(vj,t ) are c-numbers, and Q\{vj ) is a system operator such that Q\{vj ) = Q\( ~ vj)· On inserting (9.8) in the definition (9.3) of L\ and substituting it, in turn, in (9.7) we obtain M p<">(oo)= Σ Σ exp{i(^i +··· + ^η)ί} {Afc}{ifc = -M} n /» o o X f t « ο ’ U dr,· vij> t ~ J 2 Tk \ k=j 9.2 Optical Susceptibility 179 L\(v)p = - i [<3λΜ, P ( 9.10) Le t ea c h of t h e c o mpone nt i n (9.8) b e p e r f e c t l y monoc hr oma t i c so t h a t £\(vi,t) are independent of time and let £\(vi,t ) —¥ £\(vi). The integration in (9.9) can then be performed formally to give M Λη (°o) = 5 > Ρ { ^ 1 + · * ■ + Vi }t {ik = — M} {Afe} j '} Π ε^ κ ) 3 = 1 X ΐ ο - i E' k = 1 ( Uh ) Pss- (9.11) The left arrow on t he product denotes t h a t t he product index number in creases from right t o left. Note t h a t t he exponential t erm and t he product of t he field amplitudes in (9.11) are invariant under t he exchange of t he sets (Xj, and (A*,, Uik) (i, j = 1,..., n) but not the operator part in it. Hence, the contribution to a particular value of + ■ ■ ■ + Vin and a particular prod uct of the field amplitudes to (9.11) arises from a sum of the operators in it obtained by exchange of n sets of indices (Ai, i/^), · · ·, (Xn, Vin). Keeping this in mind, the nth order contribution A ξ Tr(.Ap(n)(oo)) to the expectation value Tr(Ap) of a system operator A may be expressed as M exp(i(i/il Η 1- uin)t) χ θ'Αλ ι,-,λ η (n, n ■.* η ) Γ Κ Κ ) ] > 3 = 1 (9.12) where the so called response function of the operator A is defined by r e/ \ - 1 5Αλι,···,λ„ (vi! ,· · ■, Vin) = —:Sym n Tr i U o - i E ( i o 11Λ k = 1 “ Ι Λ ti I I J\1(Ui1')pa: (9.13) with Sym standing for the operation of symmetrization of the product on its right in the indices (Ai, v^), ■ ■ ■, (An, Vin)· If A = Q\ then the response function is called a susceptibility of the system. We apply the results of this section to the problem of the response of an atomic system to an externally applied e.m. field. 9.2 Optical Susceptibility Let us express the hamiltonian of interaction between an atom and the field in the form 180 9. Linear and Nonlinear Response M flext(i) = ft Σ Σ dx(vj)e\(uj)exp(iuit). (9.14) λ i = - M If the interaction is without the RWA, then d\{v) = d\ where d\ = d ■ eχ is the component of the atomic dipole operator in the polarization direction ε λ of the field. Recall that N d = Y ^ dij\i)(j I, (9-15) i j = l where |z) (z = 1,2,..., N) is an atomic state of energy Ei and dij is the electric dipole moment component between the states |i) and |j ) in the direction e. In the RWA, and for v > 0, d{v)= Σ d(-u)= Σ (9·16) Ei<Ej Ei>Ej The label v on d\(v) thus enables us to incorporate compactly the RWA in the formalism. The hamiltonian (9.14) is in the form (9.8). The nth order contribution to (άχ(—ι/)) in the asymptotic limit t —> oo is, therefore, given by (9.12), with A -> dx(-v)·. M d{x\- v ) = Σ exP (i(^i H------- i k ~ - M n χ Σ χ λλ1,...,λη κ.· · ·.ι/ί η } ) Π ε^ Κ ) » (9·17) {λ,} 3= 1 where Χλλι,···,λ„ Κ."·,\) = ^ j S y m T r { d A( - i/) Lxn(viri) x · · · ( l 0 - Zon^ijpss}] (9.18) is the optical susceptibility of the atomic medium and Lx(v)p = [dx(v), p] · (9.19) If rij is the density per unit volume of the atoms then the dipole moment per unit volume of such a collection of atoms is given by rid (d) ■ Not i c e t h a t i f t h e s ys t e m i s s y mme t r i c u n d e r s pa c e i nver s i on t h e n, due t o t h e f a c t t h a t t h e di pol e o p e r a t o r i s o d d u n d e r s pa c e i nver s i on, i t fol l ows t h a t Χλ"λ)1,.,λ2„ ( ^ 1> · · ·,^ „ ) = °, (9-20) i.e. even-order susceptibilities vanish if the system is space-inversion symmet ric. This implies in particualr that even-order susceptibilities of a free atomic system are zero. Details of the relationships between susceptibilites arising as a result of the symmetries of the system may be found, for example, in [96, 97], Further simplification of the expression (9.18) for the susceptibility is achieved by noting that |ε| is usually small compared with the optical fre quencies. As a result, we may ignore those denominators in (9.18) which contain terms of the kind Γ + ί(ω + Ω), (Ω > 0). It leads essentially to the same results as are obtained by making the rotating wave approximation (RWA) on the hamiltonian along with the use of (9.16). The nth order susceptibility determines the contribution to the dipole moment induced by the applied fields in the nth order of perturbation. The induced moment is linear in the amplitudes of the applied fields in the first order and non-linear in the higher orders. The induced moment oscillates at linear combinations of the applied frequencies. The nth order contribution to the dipole moment consists of all the combinations of the n v[s selected from the set of applied frequencies v±\, v±2, · · ·, v±m- The oscillations in the first order are at the frequencies of the applied fields whereas new oscillation frequencies appear in the higher orders of the perturbation due to linear algebraic combination of frequencies. This is the process of wave mixing. Note that a particular frequency may arise not in one but in several orders of perturbation. The dominant contribution to a particular frequency comes from the lowest order in perturbation in which it appears. Let n be the lowest order of perturbation in which a combination Ωη of the frequencies appears. Let ά)η\Ω η) be the lowest order component of the dipole oscillating at Ωη. The radiation at frequency Ωη may be visualized as coming from a dipole of moment ά)η\Ω η). Note that the field in the radiation zone from an oscillator is given by (8.76). It shows that the intensity of field from an oscillator of dipole moment d is proportional to |d|2. Hence the intensity of the field at Ωη produced in the wave mixing is proportional to \ά^η\Ω η)\2. Since ά(Ωη) is proportional to ... λ ■ ■ ■, ν%η) where + · · · + i/*n = Ωη, the intensity of the radiation of frequency Ωη from a driven atomic system is proportional to ΙΧλλΙ,-,λη K > ‘ ‘ ‘ I2· We may write explicit expression for susceptibilities of various wave- mixing processes. We have derived such an expression in Chap. 12 for in vestigating the process of four-wave mixing in a bichromatic field. For the present, we derive in the next section the rate of absorption of energy by the system from the field in the first order. 9.3 Rate of Absorption of Energy Consider a system subject to an external field of frequency v so that the interaction hamiltonian is given by (9.8) with M = 1 and = v. Now, the 9.3 Rate of Absorption of Energy 181 182 9. Linear and Nonlinear Response rate of change of the mean internal energy U(t) of a system described by the hamiltonian H is given by [96] In the present case, the explicit time-dependence of the hamiltonian arises only from the externally applied field. It then follows that W* = [ελ(*')(<3λ(ϊ')>βχρ(ϊΐ'ί) - c.c. . (9.22) λ Substitute for the average in the equation above the expression (9.11) to the first-order. Note that the resulting expression contains exponentials like exp(±ii4), exp(±i2i/f) and the terms without any f-dependence. If the time of observation is very long compared with ν~λ then the observed values are averages over several periods of oscillations. Such an average of the oscillating terms gives vanishing contribution. Hence, keeping only the time-independent terms, we get Wa = i/5 ] e Ae^T r|Q A ( L0 + ii/) Qp( - v), pss } + c.c. Χ,β r o c = v / ό τ 5 3 ε λε^Ίτ|<5Αβχρ ((L0 + ι ν) τ\ Qp{-v), pss }+c.c. λ,/3 = v £Χεβ f d re xp(i i/r )( Qx (t,u), Qp(- v) ] ) + h.c., (9.23) i r Jo J' Q{t) = exp y L0t j Q. (9.24) This determines the rate at which the system absorbs energy from the field in terms of the two-time correlation function of the system operators. Variation of Wa(y) as a function of v gives the absorption spectrum of the field. The first equation in (9.23) shows that the resonances in the absorption spectrum are located at the imaginary parts of the eigenvalues of Lq. For t h e cas e of a t wo- l evel a t o m i n a si ngl e c ompone nt monoc hr oma t i c fi el d, t h e h a mi l t o n i a n i n t h e RWA i s gi ven by ( 9.14) wi t h M = 1, λ = 1, and by virtue of (9.16), d(v) = dge\g)(e| ξ dgeS -, d( - v) = d*geS+. (9.25) The expression (9.23) for the absorption spectrum then reads Wa(v) = 2 i/\s\2Re J dr exp(ii/r) ^ 5 _ ( r ), 5+] ^ . = 2 i/|e|2 Re[Tr j s _ ( l 0 + iv) ' [s+, pss] j ]. (9.26) This determines the absorption characteristics of a two-level atom. 9.4 Response in a Fluctuating Field 183 9.4 Response in a Fluctuating Field We have thus far confined our attention to the asymptotic response of a sys tem to a linear combination of discretely spaced monochromatic fields, though we have at hand also the expression (eq.(9.9)) for the density operator when the fields are quasimonochromatic. A situation of considerable interest con cerns the fields fluctuating around a mean frequency. The expectation value of an observable A in the nth order in this case is found by first evaluating Tr[j4/5(n)(oo)] and then averaging it over the fluctuations: ip(")(oc) . (9.27) AW = Tr The bar denoting average over the fluctuations. This may be evaluated by finding /3(n)(oo). Now, consider the signal S(u) at the frequency v resulting from mixing of the applied frequencie. If domonant contribution to S(v) arises in the nth order then the signal averaged over the field fluctuations is S(v) ~ |d(nV ) | 2, (9.28) where d^n\v ) is the average of d(u) found by applying (9.27). Clearly, S(y) is not linear in p and hence the signal average over fluctuations can not be derived by averaging the density operator over fluctuations. Carrying the average in (9.28) for non-linear response is generally an involved task. Cal culations for the second and the third order response for some models of fluctuations may be found in [98]. The reference [99] draws attention to er roneous conclusions arrived at by first averaging dW over fluctuations and then squaring it. 10. Solution of Linear Equations: Method of Eigenvector Expansion We have seen in previous chapters that a variety of dynamical problems reduce to solving a set of coupled linear first order equations expressible in the form ” Ι<Κί)> = *#(*)>> ί 10·1) where \φ{ί)) is a vector in an Η,-diinensional space and X an operator acting on the vectors in that space. The operator X may or may not be time- dependent. In this chapter we assume X to be independent of time. The formal solution of (1 0.1 ) then is | φ(ί)) = exp(Xi)|V>(0 )). (1 0.2 ) In principle, we can evaluate (10.2) by expanding the exponential in powers of X and by evaluating X n|^>(0)). This procedure may be simplified by express ing the exponential as products of exponentials of operators whose action on IV’(O)) is simpler to evaluate. The problem of disentangling an exponential has been addressed in Chap. 2. Alternatively, we may express |ϊ/?(0)) in a basis which is such that the action of X on a basis state results in a linear combina tion of fewer number of vectors than the dimension n of the space in question. In other words, we would like to choose the basis vectors which are reducible to a sum of subspaces each of which is invariant under the action of X. Smaller the dimension of such subspaces to which a basis can be reduced, simpler it is to handle. The most desirable choice then is the basis which can be reduced to a sum of one-dimensional invariant subspaces. A one-dimensional subspace invariant under the action of an operator is constituted by the eigenvector of the operator. Hence, the set of all the eigenvectors of X is the most desirable basis for evaluating (10.2). However, the set of eigenvectors of an arbitrary X need not be complete. We then have to discover additional vectors to make the set complete. These additional vectors are constructed by introducing the concept of generalized eigenvectors. In this chapter we discuss the eigenvalue problem of an operator to evaluate (1 0.2 ). 186 10. Method of Eigenvector Expansion 10.1 Eigenvalues and Eigenvectors Recall from (1.32) that A is an eigenvalue of X and [Φχ) the corresponding eigenvector if Χ- Χ ΐ }\ψ χ)=0. (10.3) If [X — A/] - 1 exists, then we can operate (10.3) on the left with [X — A/] - 1 leading to the conclusion that \ψχ) = 0. Thus (10.3) admits non-trivial solu tion only for those values of A for which X — XI is singular. Recall from Chap. 1 that a vector in an n-dimensional space may be rep resented by a column of n rows and an operator by an n x n matrix. We recall also the theorem that necessary and sufficient condition for a finite- dimensional matrix A to be singular is that det(A) be zero [1 0 0 ]. Hence, the necessary and sufficient condition for (10.3) to admit a non-trivial solution is det(X - XI) = 0. (10.4) This equation determines the eigenvalues A. In the following we enumerate some properties of eigenvalues and eigenvectors. Some results are stated with out proof. Their proofs may be found in [3, 100]. 1. The equation(10.4) is a polynomial of degree n and hence admits n roots Ai, · · ·, An. However, not all the roots need be distinct. The number of times a root is repeated is called its multiplicity. Let Ai,---,Am be m distinct roots. Let r* be the multiplicity of Aj so that (10.4) may be written as m (A - Xi )ri ■ ■ ■ (A - Xm)rm = 0, Ύ η = η. (10.5) i= 1 Corresponding t o each A j of multiplicity r*, (10.3) admits a number rii < ri of non-trivial solutions. The eigenvector corresponding to an eigenvalue of multipicity one is unique. The multiplicity of a root is also called the geometrical degeneracy of the eigenvalue and the number of independent eigenvectors corresponding to a multiple root its dynamical degenemcy. Since a polynomial has at least one root, (10.3) admits at least one non trivial solution. 2. The eigenvalues of the adjoint of an operator are complex conjugates of the eigenvalues of the operator. To prove this, let χϊ\φμ)=μ\φμ). ( 1 0.6) This implies (φμ\Χ = μ*(φμ\. (10.7) The eigenvalue μ of is a solution of det(X^ —μ/) = 0. Since det(^t) = (det(A))*, it follows that μ = A* where A is an eigenvalue of X. 10.1 Eigenvalues and Eigenvectors 187 3. Property 2 implies that the eigenvalues of a hermitian operator are real. (10.7) it reduces to (μ* — \){φμ\ψχ) = 0. This implies that {φμ\ψχ} = 0 if Χ φ μ *. 5. Th e e i genve c t or s cor r e s pondi ng t o d i s t i n c t ei genval ues a r e l i near l y i n d e p e n d e n t. To see t hi s, o p e r a t e t h e e q u a t i o n a i |"0 i ) + · · · + amI'i/’m) = 0 involving linear combination of m eigenvectors of X belonging to distinct eigenvalues Ai, · · · ,Am by ( X- Ai ) · · · ( X- Ai _ i ) ( X- Ai+i) · · · (X - X m)· It leads to the equation αί\ψί) = 0 which implies ai — 0 for all i. Hence, by the definition of linear independence, I'i/’i), · · · > I'i/’m) are linearly inde pendent. 6. We infer from 5 that if the eigenvalues of an n-dimensional operator X are distinct then it would admit n linearly independent eigenvectors. These eigenvectors may be employed to serve as a basis for the vector space of X. The set of vectors orthonormal to the set of eigenvectors of X is con stituted, in view of 4, by the eigenvectors of X ^. If |V>Ai )> · · ·, \Ψ\η) are the eigenvectors of X corresponding to distinct eigenvalues Ai,... ,An, and I^Aj), · · ·, |φ\η) are the eigenvectors of corresponding to the eigen values A J,..., A* then the orthonormality and completeness relations in volving these vectors read By setting f ( X ) = ( X - Ai) · · · ( X - Ai - i ) ( X - Ai+i) ■ ■ ■ ( X - Xn) it We exploit this result in the Appendix C to write general solution of a set of two and three coupled linear equations. 4. The eigenvectors of X corresponding to an eigenvalue A are orthogonal to the eigenvectors of X t corresponding to the eigenvalues μ φ A*. To prove it, take the scalar product of the eigenvalue equation (10.3) for X with the eigenvector \φμ) of to get (φμ\Χ — \Ι\ψχ) = 0. On applying {ΦχΛΨχ,} = $ij, ( 1 0.8) n (10.9) i=1 Operate (10.9) with f ( X) and invoke (1.33) to obtain n ( 1 0.1 0) follows that ( 10.11) On combining (10.10) and (10.11) we obtain ( 1 0.1 2 ) 188 10. Method of Eigenvector Expansion 7. An operator acting in an n-dimensional space may admit n linearly inde pendent eigenvectors even when its eigenvalues are not all distinct. That would be the case if the number of independent eigenvectors correspond ing to a multiple root is equal to its multiplicity. There is, however, no general prescription for ascertaining the number of independent eigen vectors corresponding to a multiple eigenvalue except in the special case of normal operators defined in Chap. 1. It can be proved that the number of independent eigenvectors corresponding to an eigenvalue of a normal operator is equal to the multiplicity of the eigenvalue [100]. Hence, a nor mal operator in an n-dimensional space admits n lineraly independent eigenvectors. These eigenvectors may, furthermore, be orthonormalized. More generally, it can be proved that a necessary and sufficient condi tion for an n-dimensional operator to admit n orthonormal eigenvectors is that it be normal. Recall that the hermitian and unitary operators are special cases of a normal operator. Hence the eigenvectors of these operators constitute an orthonormal basis. 8. Let Ι'ί/’λι), · · ·, Ι'ί/’λη) and {φχ^, · · ·, \φ\η) be as in the item 6 above. Let ί be a matrix constituted by the column vectors I'i/’Ai), · · ·, \Ψ\η), and Φ that constituted by the column vectors {φχ^, · · ·, \ Φ\η)· Using the or thonormality relation (10.7), it is straightforward to show that φϊχψ = φ -'χ φ = Dx (Xu..., λη), (10.13) ϋ χ ( λ ί,..., λη) being a diagonal matrix with the eigenvalues Αχ,..., A„ of X as its diagonal elements. This shows that an n-dimensional operator admitting n linearly independent eigenvectors can be diagonalized by a similarity transformation. The transformation in question is generated by the matrix formed by the eigenvectors of the operator as columns. Now, consider the formal solution (10.2) of (10.1). Let X admit n linearly independent eigenvectors |·0 Αι))·"> Ι^λι) corresponding to the eigenvalues A1;..., An. Using (10.9) we represent IV’(O)) as n = (1 0.1 4 ) i = 1 On substituting this in (10.2) we obtain n IV’W) = 'Σ/^χρ(Χίί)(φχί\'φ(0))\·φχί ) i = 1 This is the solution of (10.1) in case X admits n independent eigenvectors. The eigenvectors can not constitute a basis if there is any root not having as many independent eigenvectors as its multiplicity. The vectors required in addition to the eigenvectors to make a complete set are then obtained by invoking the concept of a generalized eigenvector introduced next. (10.15) 10.2 Generalized Eigenvalues and Eigenvectors 189 10.2 Generalized Eigenvalues and Eigenvectors Consider an operator X in an n-dimensional vector space. Let us assume that it has m distinct eigenvalues, denoted by Ai, · · · ,Am and that is the multiplicity of A j. Let us assume that corresponding to an eigenvalue A i there is only one eigenvector, denoted by |V’Ai ( l ) ) ) so that X-A,/]| ^a 1(1 ) ) = 0. (10.16) If ri > 2 then construct \ψχ^k) ) by solving χ - λ ^ λ,Μ ) = hM*- i ) > (10·17) successively for k = 2,..., r;. On operating (10.17) by X — Xil successively for k = 1,... and on using (10.16) it follows that X - A ij ] fc|^A 1 (A:))=0, [ Χ - Κ ή * '{ ψ χ ^ φ Ο. (10.18) Vector \ψ\. (k)) is called a generalized eigenvector of rank k and A, a general ized eigenvalue. The eigenvector of rank 1, |^>a,(1)); is also called the ordinary eigenvector. Note that each of the generalized eigenvectors is arbitrary to the addition of a scalar multiple of the ordinary eigenvector. The importance of the concept of generalized eigenvectors stems from their following properties: 1. If the eigenvalue A j of multiplicity r* has only one ordinary eigenvector, then its r* generalized eigenvectors, |ψχί $ ) ) (k = 1,..., r*), are linearly independent. 2. Generalized eigenvectors corresponding to different eigenvalues are lin early independent. 3. The set of generalized eigenvectors of X is orthonormal to the set of gen eralized eigenvectors of . Let Φ be an η x n matrix formed by the gen eralized eigenvectors 1^ ( 1 )), · · ·, Ιψχ^η) ); · · ·; |^Am(l)), · · ■, I ^ ( r ™ ) ) of X as columns numbered 1 to n. Let the matrix Φ be formed similarly by the generalized eigenvectors \φχτ (k)), · · ·, \φχί (r*)) (i = 1,..., m) of X t corresponding to the eigenvalues A*,...,A^. The orthogonality of the generalized eigenvectors of X and X^ implies that Φ^Φ = I. (10.19) The definition of the generalized eigenvectors and (10.19), imply that & χ φ = φ~λχ φ = j x, (10.20) Jx being the Jordan canonical f orm of X. It is such that (i) its elements along the main diagonal are the eigenvalues of X, (ii) all its elements below the main diagonal are zero, (iii) besides the main diagonal, its only non-zero elements, if any, are in the diagonal above the main diagonal. Each of those non-zero elements is unity. The eigenvalue Αχ appears in fc-l 190 10. Method of Eigenvector Expansion the main diagonal in the rows numbered 1 to ri followed by Λ2 which appears in rows numbered r\ + 1 to r\ + r2 and so on. Unity appears in the diagonal above the main diagonal in rows numbered 1 to r i — 1, ri + 1 to π + 7*2 — 1 and so on. We have thus at hand n independent vectors in an n-dimensional space. We employ it as a basis to evaluate (10.2) by expressing |V>(0)) as m r t W 0 )> = E E ^ W W ° ) ) I ^ W ) · (1 0.2 1 ) i —1 k = 1 S u b s t i t u t e t h i s i n ( 10.2) a nd i nvoke ( 10.16) - ( 10.18) t o show t h a t exp( Xt )\^Xi(k)) = exp(Aji) exp - λ») t \^ ( k ) ) k ~ X 1 — exp(Aji) %\Φ\, {k — I))· (10.22) As a consequence, we get k ~ 1 4 = Σ Σ Σ βχΡ(λ*ί) ^ ( Α) Ι ^ ( ° ) ) | ί Ι ^ ( Α - 0 ) · (10.23) i —1 k — 1 1=0 This is the solution of (10.1) in case there is only one ordinary eigenvector corresponding to an eigenvalue of any multiplicity. The expression (10.23) shows that the evolution of a state involves terms which are products of an exponential in time with a power of time. Note from (10.15) that the time- dependence is an exponential function if the evolution operator admits as many linearly independent eigenvectors as its dimension. The problems that we encounter in Chaps. 12 and 13 involving multiple eigenvalues fall in to the category discussed above. We, therefore, do not dis cuss the general case when an eigenvalue of multiplicity greater than two has two or more independent eigenvectors. In the general case also we can always find Γ; independent generalized eigenvectors corresponding to an eigenvalue Ai of multiplicity r». They possess the property (10.18). Since the maximum value of k is r*, it follows that (10.18) holds for any generalized eigenvector if k = ri- Now, since any vector in the given space is expressible as a linear combination of generalized eigenvectors, it follows that ( X - A 1) r i - - - ( X- Am) rm=0. (10.24) On comparing this with (10.5) we note that an operator satisfies its own eigen value equation. This is the content of the Hamilton-Cayley theorem [100). However, (10.24) need not be the minimum polynomial equation satisfied by X. For, if there are more than one independent eigenvectors corresponding to an eigenvalue Ai of multiplicity two or more then the expansion (10.21) would contain eigenvectors of lower rank corresponding to Ai. Consequently, (10.24) 10.3 Solution of Two-Term Difference-Differential Equation 191 will contain the power of X — Xi less than rt. We have used this property in Sect. 2.2 to express the exponential of a finite-dimensional operator in terms of a polynomial in the operator. We solve (10.1) in the next section for a special form of X without taking recourse to eigenvector expansion. Its eigenvalue equation, along with that of another form, is solved in Sect. 10.4. 10.3 Solution of Two-Term Difference-Differential Equation In this section we solve the equation Cm = a m C rn OVn^m+l TO = 0, 1, ... , TV, assuming that Cn+i = 0, and the equation Cm — OLrnCm -t- p rnCm—\ τη 0,1,..., ( 1 0.2 5 ) ( 1 0.2 6 ) a s s u m i n g C _ i = 0. C o n s i d e r f i r s t ( 1 0.2 5 ). I t s L a p l a c e t r a n s f o r m ( d e f i n e d i n ( 8.8 4 ) ) y i e l d s Cm (0) + ImCm+l ■ (10.27) Cm = z — a. Set m = N in this and use the given condition C/v+i = 0 to obtain -Cjv(0 ). CN = - ^ ~ Z — Qj v Set m = N — 1,7V — 2,successively in (10.27) to get N —m k Cm = 1 z - a r Cm(0) + 53 Cm+k(0) TT J m + l - 1 ^ z Ctm+l k = 1 1=1 Consider next (10.26). Its Laplace transformation yields C 1 '-'m — Z — OL Set to = 0 in t hi s and use C_ i = 0 to show t h a t Cm (0 ) + PmCm — l ■ -C'o(O). Co z - a o Solve (10.30) recursively for m = 1,2,... to obtain m—1 cm = z - a n m— 1 Cm(0) + 53 Ck{0) Π —^— βι+1 Δ ^ AA Z — OH fe= 0 l=k (10.28) (10.29) (10.30) (10.31) (10.32) The inverse Laplace t ransform (defined in (8.85)) of (10.29) and (10.32) determines Cm(t). Now, if am are all distinct, then the poles of (10.29) and 192 10. Method of Eigenvector Expansion (10.32) are simple. The inverse Laplace transform of (10.29) and (10.32) is then straightforward to evaluate. However, it may become an involved exer cise if the poles are not simple. As an example, let am = — (m + a). The solution of (10.25) and (10.26) then reads, respectively, Cm(t) = exp{ —(m + a)t)} Cm(0 ) (1 - exp(—£))fe-rn + Σ k—rn + 1 ' * * 'I k — 1 (k — m)\ Ck( 0 ) (10.33) Cm(t) = exp{ —(m + a)t)} Cm(0 ) m_1 (exp(i) - i ) m~k0 k+i0 k+ 2 - Σ k = 0 ■ βη (m — k)\ Ck{ 0) (10.34) The correctness of these solution may be verified by subst it ut i ng t hem in t hei r respective equation. We use these solutions in Chap. 14 in solving t he problem of strongly coupled atom-cavity system. 10.4 E x a c t l y Sol vabl e Two- and Three- Term Recurs i on Rel at i ons In this section we solve t he eigenvalue equations corresponding to (10.25) and (10.26). It is followed by t he solution of t he eigenvalue problem reducible to t he t hr ee- t er m recursion relation (10.41). 10.4.1 Two-Term Re curs i on Re l at i ons Consider t he problem of determining Crn obeying the eigenvalue equation “t- TTl 0, 1, ... , (10.35) am, 7 m being known functions of m, and A is an eigenvalue. Rewrite this as 0 (a — XI)\C) = / a0 - X 7 o 0 « ί — A 7i 0 0 0 c*2 — A 7 2 0 V = 0. •\ /Co\ c, ^ 2 V : / :/ ( 1 0.3 6 ) By c a r r y i n g i t s e x p a n s i o n i n t e r m s o f t h e f i r s t c o l u mn, we s e e t h a t d e t ( a —A I) in this case is a product of the diagonal elements of a — XI. Hence, det(a — XI) = 0 yields the expression Xn = an, n = 0,1---- (10.37) 10.4 Exactly Solvable Two- and Three-Term Recursion Relations 193 for the eigenvalues. Let Cnm denote the solution of (10.35) corresponding to the eigenvalue λη. Solve (10.35) recursively to obtain Invoking the preceeding arguments, it follows that the eigenvalues in this case also are given by (10.37). Verify also that (10.39) is solved by 10.4.2 Three-Term Recursion Relations We have seen in the last subsection that two-term recursion relations admit analytical solutions in closed form. However, three-term relations are not always exactly solvable. In this subsection we identify exactly solvable cases of frequently encountered quadratic three-term recursion relations of the type [max + m(m - l ) a 2] C m + [(m + 1)71 + m(m + l)^ 2 ]Cm+\ m = 0,1,..., M, (7_i = 0, a's, β's and 7's are fixed constants and A is an eigenvalue. The upper limit M on the allowed values of m is finite if it is given that Cm+u = 0 for k > 1. In case this holds, set m = M +1 in (10.41). It results in a relation between Cm +1, Cm +2 and Cm - This equation will be We solve (10.41) by converting it into a differential equation for the gen erating function M Cnm+n — 0 Til — 1,2,· · · Thi s i s t h e e x a c t s ol ut i on of ( 10.35). Cons i de r n e x t t h e e q u a t i o n α γ η ^ γ η + β ν α ^ γ η — Ι = AC<m , Τ Ϊ Ι = 0, 1, . . . (10.38) (10.39) m Cnm 0 n m m = 0,1, ■ · ·, η — 1. (10.40) + [βο + (m - 1)βχ + (m - 1 )(m — 2)β2 ]0Ύη-\ = ACrn, (10.41) consistent with the given condition Cm +i = Cm+i = 0 if the coefficient of Cm in it vanishes, i.e. if βο + Μβχ + Μ (Μ - 1)β 2 = 0. Write this as a quadratic in M and show that (10.42) Μ = ~\β 2 - β ι ± \/ [β2 ~ β ι ) 2 - 4β2β0 . (10.43) ( 1 0.4 4 ) m= 0 194 10. Method of Eigenvector Expansion so that 1 dm Cm = —r - -----f ( x) . (10.45) mldx™' x=o v ' I t i s t h e n s t r a i g h t f o r wa r d t o show t h a t, by v i r t u e of ( 10.41), ( t oge t he r wi t h ( 10.42) i f M is finite) f ( x) satisfies the second-order differential equation d2 d χ(β 2 χ 2 + a 2x + 7 2 ) ^ 2 + (βι χ 2 + αι2: + 71^ + (β0χ - A)j f ( x) = 0. (10.46) The solution of this equation determines Crn through the relation (10.45). Exact power series solution of a second-order differential equation, re ducible to a hypergeometric or a confluent hypergeometric equation, is known. We recall from Appendix B that an ordinary second-order differential equa tion admits solution in terms of the hypergeometric function if it has at most three singularities, including a singularity at x = 0 0, which are regular and that its solution is expressible in terms of the confluent hypergeometric func tion if two of the three singularities merge. The problem of solving (10.46) is thus reduced to one of determining the nature of its singularities. Recall from the Appendix B that the nature of the singularity at x = 0 0 is determined by transforming to y = l/x. Use (B.2) to show that the change of variable x = l/y transforms (10.46) to d2 y( l 2 y2 + a 2y + β2) - ^ -I- { ( 2 7 2 - 7 1 )y2 + (2a 2 - a ^ y + (2/32 - β ί ) } - ^ + βο - Ay) f ( y) = 0. (10.47) The nature of the singularity of this at y = 0 determines that of the point x = 0 0 of (10.46). Following again the Appendix B, we find that if β2 φ 0, η2 φ 0 then a - α2 ± y/o-l - 4/3272 ,1 Λ,οί x = 0, x = ------------- , x = oo (10.48) 2 p2 are four singular points of (10.46). However, as mentioned above, it is t he case of a t most t hr ee regular singularities which is of i nterest. The number of singularities reduces to t hree if (A) β2 = 0, 7 2 φ 0, a 2 φ 0, or if (Β) η2 = 0, β2 φ 0, ol2 φ 0, or if (C) a\ = Αβ2 η2 φ 0. The number of singular points is two if (D) 7 2 φ 0, β2 = 0, a2 = 0 or if (Ε) β2 φ 0, 7 2 = 0, a2 = 0 or if (F) β2 = 7 2 = 0. Finally, we note that (10.46) reduces to a first-order equation if a2 = β2 = 7 2 = 0. We discuss this below as case (G). Case A: β 2 = 0, 7 2 Φ 0, a 2 φ 0. In this case, the singularities of (10.46) axe at Verify that the first two singularities above are regular whereas the singularity at x = oo is regular only if βο = βι = 0. Under these conditions, the three- term recursion relation (10.41) reduces to a two-term relation [max + m(m - l)a 2\Cm + (m + l)[7 i + m'j2\Cm + 1 = ACm. (10.50) This is a special case of already solved equation (10.35). Case B: 7 2 = 0,( 8 2/ 0, e*2 Φ 0. The singularities of (10.46) in this case are at x = 0, x = - (^-, x = oo. (10.51) P2 Verify t h a t t he last two singularities above are regular whereas t he singularity at x = 0 is regular only if 7 1 = 0. Under these conditions, (10.41) reduces to [max + m(m - l ) a2 }Crn + [βο + (m - 1){βχ + (m - 2)β2 }}Cm-x = ACm (10.52) This is a special case of already solved equation (10.39). Case C: = 4/3272 φ 0. , The singularities of (10.46) in this case are at OLo x = 0, x = —^ 5-, x = 0 0. (10.53) Verify that the first and the last singularity above are regular whereas the second singularity is regular if the coefficient of d f /dx can be factorized as βχχ 2 + αχχ + 7 i = βχ (x + K), (10.54) i.e. if βχ(Κ + ^ ) = α χ, Κ = ψ ^. (10.55) V 2 β2; βχα2 The two equations above determining one unknown K are consistent if — - — = (10.56) βι 2 β2 α2βχ y ’ Th e di f f e r e nt i a l e q u a t i o n (10.46) t h e n r e duc e s t o iA(i+l l ) 2^ +A(*+l|X*+A')l; + ( Α ) ϊ - λ ) ]/( ϊ ) = 0. (10.57) Verify that the transformation x = —α2 ζ/2β2 reduces this equation to the form (B.7) with ρ = - β ι/β 2, q = - 2 Κ β 2/α 2, r = βο/β2, A -4 - 2 X/a 2. A solution of (10.57) is, therefore, given by f(x) = (1 + 2 β2 χ/α 2)α F (a, b; c; —2β2 χ/α 2), (10.58) with c = 2 Κ β ι/α 2, and 0,6 and a determined by solving 10.4 Exactly Solvable Two- and Three-Term Recursion Relations 195 196 10. Method of Eigenvector Expansion a(a - 1) + ~ a + ~ = — (Κβχα - A), (10.59) P2 P2 OL2 a + b+l = 2a+^~, ab = — ( Κ βί α — A). (10.60) P2 Q2 Let us assume t h a t CM+k = 0 (k > 1) so that (10.42) holds. The f ( x) in this case should be a polynomial of degree M. Recall from Appendix B that F(a, b; c; x) is a polynomial of degree n if either a or b is — n. Hence, we must have a = —n, a = M — n. (10.61) On combining this with (10.60) we find that b = 2M — η + — 1, P2 R' + Κβχ ( Μ — ή). (10.62) x = na. 2M - n + — - 1 f t The expression for Cm corresponding to the eigenvalue An is obtained by substituting (10.58) in (10.45). Case D: 7 2 φ 0, e*2 = β ι = 0. The singularities of (10.46) in this case are at x = 0, x = cxd. The singularity at x = 0 is regular whereas that at x = 0 0 is irregular. The equation (10.46) is reducible to the equation (B.13) for the confluent hypergeometric function if, in addition, β ο = β ι = 0. Note that this is obtained as the limit ~^ 0 of the case A above. In this limit, the singularity at x = — 7 2/0 2 of case A merges with already present singularity at 0 0. The recursion relation (10.41) now reads aimCm + (m + 1)(7j + m7 2)C'TO+i = A Crn. (10.63) This is a special case of already solved equation (10.35). Case Ε: β 2 φ 0, = 7 2 = 0. The singularities in this case are at x = 0 and x = 0 0. The singularity at x = 0 0 is regular whereas that at x = 0 is always irregular. Since we are interested only in the case of a regular singularity at x = 0, this case is not of interest. Case F: /32 = 7 2 = 0. The singularities in this case are at x = 0, x = oc. Both the singularities are irregular. However, the singularity at x = 0 becomes regular if 7 1 = 0. The corresponding recursion relation assumes the form [τηαλ + m{m - 1 ) a2 - A\Cm + [βο + (m - 1 )β1 }Οτη-χ = 0. (10.64) This is a special case of already solved equation (10.39). 10.4 Exactly Solvable Two- and Three-Term Recursion Relations 197 Case ( G ) α 2 = β ι = 72 = 0. The equation (10.46) in this case reduces to a first order equation (βιχ 2 + a xx + 7 i ) ^ + {βοΧ ~ λ) f ( x ) = 0. The recursion relation (10.41) then assumes the form m a i C m + (to + 1)71 Cm+i + \β0 + (to - l)/3i]Cm_ i = XC„ Verify that the solution of (10.65) is f {x) ~ (x - x + ) aix - x - ) b, x± = 1 W i X - - Qi ± \J a\~ 4 7 1/3i b = A “t- βοΧ— ( 10.65) ( 1 0.6 6) ( 10.67) ( 1 0.68) β^Χ+- Χ - Υ βι{χ+ - x - )' Let Cm +π = 0 (η > 1). The f ( x) in (10.67) then should be a polynomial of degree M. This implies that a and b should be integers such that a + b = M. We set a = n, b = M — n. Now apply (10.45) to get C„ <-γΣ· {x+)n~k(x-)M~n~rn+k (10.69) (to — k)\k\in — /s)!(M — n — to + &)! We have used this result in Chap. 3 for deriving the expression for the squeezed spin state. 11. Two-Level and Three-Level Hamiltonian Systems In this chapter we present exact analytical approach to studying the dynam ical behaviour of certain classes of two-level and three-level hamiltonian sys tems in a quantized e.m. field. These classes encompass most of commonly encountered systems of interest in quantum optics. This chapter is based largely on [1 0 1 ]. 11.1 Exactly Solvable Two-Level Systems A class of exactly solvable system of a two-level atom in a quantized field is comprised by the hamiltonian Hi = h[ho({C},t) + 5({C},i)5z + {<?({£}, i ) i ^ S _ + h.c.} ] (1 1.1 ) in an appropriate interaction picture. Here, F, are time-independent sums of products of single mode field operators whereas {C} is a set of commuting operators each commuting with Sz, F^ S- and S+F. In other words, {C} is a set of commuting time-independent constants. In the following we suppress displaying explicit dependence on {C}. The Hamiltonian (11.1) describes emission and absorption of the field quanta by the effective field operators F and F'f. The effective detuning S(t) and the atom-field coupling constant g(t) may be time-dependent. The properties of two-level operators are contained in the equations (1.129)-(1.134b). The time-evolution operator generated by Ηχ is U 1 ( t ) = ^ e x p [ - ^ J * H 1 (T)dT\. (11.2) We follow the method of Chap. 2 and write the time-ordered exponential (1 1.2 ) in terms of the products of the exponentials of all those operators which, along with the operators Sz, F^ S- and S+F constituting H\(t), are closed under the operation of commutation. To that end, recall the properties (1.129)-(1.134b) of two-level operators and show that [SZ,F*S-] = - F* S-, [SZ,S+F\ = S+F, (11.3) [S+F,F*S-] = S+S-FF f - F t F 5 _ 5 +. (11.4) 200 11. Two-Level and Three-Level Hamiltonian Systems Now, recall (1.134b) to rewrite S+S- and S - S + respectively as 2SZS+S- and —2SZS - S +. Consequently, (11.4) may be rewritten as [5+F,Ft5_] = 2SZN, (11.5) N = FF^S+S-+F^FS_S+. (11.6) It is straightforward to see that NS- = F^FS_, NS+=FF^S+. (11.7) By repeated use of these relations, it follows that <£(N)S- =0(F*F)S-, Φ(Ν)3+=Φ(ΡΡ^)3+, (11.8) for any ΦΝ expandable as a power series in its argument. The reader should also verify that N commutes with S+F, F^S_ and, of course, with Sz. Hence N commutes with each of the operators in It is, therefore, a time-independent constant of the motion. Hence, the time- ordered exponential (1 1.2 ) may be written as rt Ui = exp 1 jo M T)dr] [exp ( j +(t)S+F} exP ( 2 f z(t )Sz) x exp (/_ ( i ) F t S _ ) ] - i J /i0( T) dr j ( l + f +( t )S+F) x ( c o s h (/z(i)) + 2 smh(/z(t))Sz^j ( l + /l ( i ) F f5_) [ - i f ίι0(τ)άτ jao(i) + x ++(t)S+F + x — (t)F^S^ = exp = exp +az(t)Sz + a+„(t)S+S_FF* (11.9) Here, f ± (t ) and f z (t) are time-dependent functions of time-independent op erator constants of the motion. The second equation above has been ob tained by applying (2.13) and the third by invoking the two-level operator relations. The a's and the x's may be identified in terms of the /'s. How ever, we do not need that relation. We write Sz as (1/2)[S+,<SL], Qo(t) 3 8 ao(t )(S+S - + S ^ S +), and invoke (11.7) to rewrite (11.9) as rt ϋχ = exp i / ho(r)dr J 0 J x+-(t)S+S- + x - +(t)S~S+ + x ++(t )S+F + x [t )F^S- (1 1.10) Xij (t ) being the functions of the time-independent operator constants of the motion. Note that IJ\ for a classically driven two-level system has the same form as (11.10) if the quantum field operator F is replaced by a c-number. 11.1 Exactly Solvable Two-Level Systems 201 The time-dependence of the evolution is contained in the Xij{t) which are determined next. To derive Xij{t), write U\ in (11.10) as in (11-2), differentiate it with respect to t and use (1 1.1 ) for H\(t) to get 5(t)Sz +g(t)F'S-+g*(t)S+F U\ = lexp i / ho(r)dT ±+_(i)5+5_ + i++(i )5+ F +x-(t)F^S- + i_+(i)5_5+ (1 1.1 1 ) where ‘dot’ over a quantity denotes derivative with respect to time. On sub stituting for Ui from (1 1.1 0 ), (1 1.1 1 ) reads i_| 5+5_ + x ++S+F + x F* S - + x —I.S-S+ 's(t)Sz + g( t ) F'S- +g*( t ) S+P i + _ 5 + 5 _ + X++S+F + + i - + 5 _ 5 + l. (H-12) On applying the two-level characteristics and the relations (11.7), we express the right hand side of (1 1.1 2 ) as a combination of the same operators that ap pear on its left hand side. On comparing the coefficients of the like operators we arrive at two closed systems of equations 5(t)- ix++ = ~γΧ++ + 9 (t)x-+, m £ - ++Ng(t)x++; (11.13) and IX. m = — — x — + g(t)x+-, i ±+_ = ^ γ - χ +- +Ng*(t )x- (11.14) The s e t s of e q u a t i o n s (11.13) and (11.14) a r e s i mi l ar. The y a r e t o be s ol ved al ong wi t h t h e i n i t i a l c ondi t i on i + + ( 0 ) = x (0) = 0, £+_(0) = i _+(0) = 1. (11.15) Note that, since TV is a constant operator, (11.13) and (11.14) may be treated like c-number equations. Their solution for given time-dependent functions g(t) and S(t) determines x tj for any field operator F in the hamiltonian (11.1). Dynamics of the system is thus determined by solving the c-number like equations (11.13) and (11.14) for £ij. The x'^· are functions of N which con tains, besides the field operators, also the atomic operators. Since Xij appear in (11.10) only in combination with S±, we can get rid of the dependence 202 11. Two-Level and Three-Level Hamiltonian Systems of Xij on the atomic operators by using (11.8). The x tJ may depend also on other constants of motion which may, in turn, contain the atomic operators. Such atomic operators can also be removed by using the relations (11.8) and, possibly, other similar relations. Assuming that the other constants, if any, do not involve field operators in any other combination then what appears in N, the Xij turn out to be functions of the field operators F^F and FF^. The problem of evaluating matrix elements of U\ reduces to that of solving the eigenvalue problem of F^F and FF^. These two eigenvalue problems are, however, not independent. For, if |φ) is an eigenstate of F^F then it follows that F\ip) is an eigenstate of FF^ corresponding to the same eigenvalue. Note also that F^F and FF + are normal operators. Hence, their eigenstates constitute a complete set which may be used as the basis states. The eigenvalue problem in question can be solved effortlessly for single-channel models. For, recall from Sect. 7.5 that F then is a product of single-mode operators and hence F^F a product of number operators. The eigenvalue problem, however, may not be simple for multi-channel models ( see (11.47)). Now, the time-evolution of a state of the system is determined by \Ψ(*)) = & 1 (t)\ip(0 )). (11.16) If the system is described by a density operator p then p(t) = Ul(t)p(0)Ul(t), (11.17) p( 0) being the density operator at the initial time. The dynamics may be investigated in terms of evolution of operators by evaluating A(t) (11.18) A being a system operator. Next, we examine the equations (11.13) and (11.14). Those equations can be solved analytically exactly if (a) ί and g are time-independent, or if (b) δ = 0 and g(t) is real. In the following we derive the expression for U\ (t ) for these exactly solvable cases, and for the case of (c) random time-dependence of 5(t) and g(t). 1 1.1.1 T i m e - I n d e p e n d e n t D e t u n i n g a n d C o u p l i n g Th e s ol ut i on of e q u a t i o n s (11.13) a n d (11.14) i n t h i s cas e i s r e adi l y o b t a i n e d r e a di ng β x +- = cos ( f t ) — Sin(-Tt), £ X-+ = cos (Pt ) + sin (ft ), x ++ = - i g*f ~' sin (ft ), x — = - i g f ^ 1 sin ( f t ), (11.19) Γ ■4\g\2N ( 1 1.20) 11.1 Exactly Solvable Two-Level Systems 203 On substituting (11.19) in (11.10) and on applying (11-8) we obtain tJ\(t) = exp(— ify)£)j cos ( A1 ) — A 1 sin(Ai) S+S- cos( f i t ) + ί ^ Α l s i n ( At) S_S4 g f i 1 8i n ( f t f i r t S - + g*A” 1 sin(f2i ) JF,S+] }, (1 1.2 1 ) A δ2 + 4\g\2 F^F f l = ^ 2 + 4\g\2FF' ( 1 1.22) This determines completely the dynamics generated by time-independent form of the hamiltonian (11.1). It is also of interest to know its eigenstates called its dressed states. Dressed States. Let |A) be the eigenstates of FF^ corresponding to the eigenvalue A: FF^\X) = A|A). (11.23) Since Hi commutes with N, we may reduce the space of states to a sum of subspaces each characterized by an eigenvalue of N. To that end, verify that N\X, 1/2) = A|A, 1/2), NF^\X, - 1/2 ) = AF^A, -1/2 ). (11.24) Hence, the states | A, 1/2) and F^ |A, —1/2) correspond to the same eigenvalue A of N. The space of the eigenstates of Hi is thus split in to manifolds characterized by an eigenvalue of N. Each manifold consists of two states, I A, 1/2) and -1/2 ). We may now express the eigenstates of time-independent form of (11-1) as sin(0 ) \μ) =cos ( 6»)|A, 1/2 ) with exp(iφ) = g/\g\- Let Ηι\μ) = Άμ\μ). Vx ■βχρ(ϊφ)Ρ^\Χ, - 1/2 ), (11.25) (11.26) Redefine the eigenvalues of Hi by absorbing in them the constant contribution from ho to obtain Ηι\μ) = h (cos(6>)^ + |3 |\/Asin(6»))|A, 1/2 ) ^ ( i s l ’/Acos^) - s i n (0 ) ^ ) e x p ^ F + I A,- 1/2 ) ( 1 1.2 7 ) S u b s t i t u t e t h i s i n (11.26) t o ar r i ve a t t h e ei genval ue e q u a t i o n 204 11. Two-Level and Three-Level Hamiltonian Systems μ - i/2 - | g |\/A\ — μ + 5/2 J cos (6>) λ sin(0 ) 1 “ It is now straightforward to see that the eigenvalues are μ± = ±\/X\g\2 + <52/4> and that M± " <V2 t a n (0 ) \g\V^ The states \μ±) are known as the dressed states of the hamiltonian. If ί = 0 then tan(0) = ±1. Hence (11.25) reduces to |λ, 1/2 ) ± ^ exp(i</))i’t |A, - 1/2 ) (11.28) (11.29) (11.30) (11.31) Now we specialize these results to (a) the Jaynes-Cummings model of one-photon transitions, and (c) a two-channel Raman coupled model. J ayne s—Cummings Model. The hamiltonian (11.1) reduces to the Jaynes- Cummings hamiltonian (7.42) of a two-level atom in a single mode field in the interaction picture generated by hu)Sz with ί = — ω, F —,> a. Note also that in this case N = at a + Sz + 1/2. The eigenstates and eigenvalues of are |n) and n = 0,1,.... The dressed states of this model for an arbitrary δ may be derived by using (11.25) and bearing in mind that |A) = |m) where | m) is an eigenstate of aa) with m + 1 as the corresponding eigenvalue. The dressed states for ί = 0, given by (11.31), now read (assuming g to be real) ΙΨ^) = IK !/2) ± \m + 1, -1/2)] . The eigenvalues corresponding to Ιψτη'*) are Mm) = ±\a W m + 1 - The inverse of the relations (11.32) is K 1/2 ) = ^ \Φ^ ]) + I· φ ^ ]) , |m + l,- l/ 2 ) = ~^= 1 ^ 4 ^) “ lV 4 r })] · (11.32) (11.33) (11.34) We may now investigate the dynamics by expressing an initial state in terms of the dressed states. As an example, let the system be initially in the state |m, 1/2). Using (11.34) we get e x p ( - i#i</?i ) | m,l/2 > = [exp(-i|s|*Vm + 1 ) 1^ ) + exp(i|g|iVm + l)|V’4r))] - (11-35) Express | ipm'1) in terms of the bare states | m, ±1/2) using (11.32) to rewrite (11.35) as 11.1 Exactly Solvable Two-Level Systems 205 exp(—i Hi t/h)\m, 1/2) = cos{\g\t\/m + l ))| m, 1/2) —isin(|(/|i\/m + 1 )| m + 1, —1/2 ) = | m,l/2,t ). (11.36) Now, if the atom is initially in its excited state and the field in a state described by /5/(0) then p(0) = J 2 Cmn\m,l/2 ) ( n,l/2\, pf (0) = £ Cmn\m)(n\. (11.37) m,n m,n On applying (11.36) it follows that PW = ^ C mn\m,l/2,t ) {n,l/2,t\ M°) = Σ Cmn \m)(n\. (11.38) m,n m,n It is now straightforward to evaluate operator averages. For example, 1 OO (Sz(t)) = - ^ ( m\p f (0)\m) cos(2gty/mTl)· (11.39) m= 0 The corresponding result when t he atom is initially in it s ground st at e reads - OO (Sz(t)) = - - (m\pf (0)\m) cos{2gty/m). (11.40) m= 0 For a discussion of comparison between t he behaviour of (Sz(t)) depicted by (11.39) and (11.40) in a quantized field with that when the field is a classical dynamic variable, see [80]. Here we compare the evolution in the quantized field with that in an externally prescribed monochromatic field. The hamiltonian of the system in the RWA is then given by (7.24). The atomic operators under it evolve as in (7.29) and (7.30). We let 5 = 0 and find that, if the atom is initially in the excited or in the ground state, then (Sz(t)) = = ± i c o s ( 2 |ff||a|i). (H-41) In the following we compare the characteristics of the evolution described by (11.39) (11.41). 1. If the field is initially in the Fock state |M) then (m|p/(0)|m) = i mM· Hence (Sz (t)) oscillates sinusoidally. This behaviour is the same as that of the atom in an external classical field described by (11.41). 2. The expression (11.39) for the atom initially in the excited state shows that {Sz (t)} exhibits oscillations even when the field is initially in the vacuum state |0). These are called the vacuum field Rabi oscillations. However, the expression (11.40) for the atom initially in the ground state 206 11. Two-Level and Three-Level Hamiltonian Systems shows that it remains in the ground state if the field is initially in the state of the vacuum. The expression (11.41) for the atom in an external field shows that it remains in its initial state in the absence of an applied field. The vacuum field Rabi oscillations are thus a signature of the field quantization. 3. From (11.39) we infer that the time evolution of (Sz (t)) in a field which is not in a Fock state is a result of combination of frequencies proportional to \/m + 1. As examples, we consider the coherent and the thermal states of the field. Recall that (m|p/(0)|m) = e x p ( - | a |2 ) | a |2m/m! (11.42) if the field is in the coherent state |a) and (m\pf(0)\m) = nm/(n + l ) m + 1 (11.43) if the field is in the thermal state with ή as the mean number of photons in it. The sum in (11.39), (11.40) can not be carried analytically exactly for either of these states. Numerical results for the field in the coherent state and the atom initially in the lower state are presented in the Fig. 11.1. It exhibits the phenomenon of collapses and revivals. This phenomenon has been analyzed analytically under the condition \a\ ^ 1 in [102, 103]. The behaviour of {Sz(t)) as a function of time in the thermal field and the atom initially in its lower state is exhibited in Fig. 11.2 by evaluating (11.40) read with (11.43). It is seen that (Sz(t)) rises to become positive and then collapses to oscillate around zero. The nature of oscillations is evidently very much different from that in the coherent state. The analysis of the sum by following the technique employed for the coherent field predicts the collapse but not the revivals [103, 104]. 0 20 40 60 60 100 |g|t Fig. 11.1. (Sz(t)) as a function of \y\t for the field in the coherent state jo·), = 20. 11.1 Exactly Solvable Two-Level Systems 207 0.1 •0 1 Λ V -0 3 -0.5 0 10 20 30 40 50 |9|t Fig. 11.2. (Sz{t)) as a function of \g\t for the field in the thermal state having an average of n = 2 0 photons. We can similarly study the bahaviour of any other observable. A property of particular interest is the squeezing of the field. The coherent field is classical. However, on interaction with the atom according to the Jaynes- Cummings hamiltonian, it exhibits the quantum feature of squeezing. For details, see [105]. Two-Channel Raman-Coupled Model. The hamiltonian in this case is given by (7.60). We let gs = 9 a = 9 - It corresponds to (11.1) with h0 = δ = 0, F = apag + a^aA- Verify that F^F = &lapasal + + al&\ap (11.44) commutes with M and C where Μ = α)ράρ + α+α8 + ά\αΑ, C = a\aA ~ (11.45) Let \mp,mA,ms) be an eigenstate of aj,ap a\aA and with mp,mA,ms as the corresponding eigenvalues. Owing to (11.45), the state space reduces to a sum of spaces labeled by the eigenvalues M and C of M and C. Hence, we may employ the states \M + C — 2ms,ms,ms — C) as a set of basis states. For the sake of illustration, let ms = m^, M —> 2M. An eigenstate \ψ\) of F^F may then be expressed as M \ψ\) = Ύ Cm\2m,M - m,M - m). (11.46) 771=0 Operate this with F^F to obtain [2m(2M - 2m + 1 ) + M - m] Cm +( M — m)\/( 2m + l)(2m + 2)Cm+i +(M — m+ l)\/2m(2m - 1 )C'm _ 1 = XCm. (11-47) 208 11. Two-Level and Three-Level Hamiltonian Systems Define Sm = Cm/(M - m)\y/( 2 m)\. ( 1 1.4 8 ) On s u b s t i t u t i o n i n (11.47) t h i s yi el ds [2 m(2 M - 2 m + l ) - m] S m + (2 m + 1 ) ( 2 m + 2 ) f m+i + ( M - 1 7 1 + l ) 2/m - l = (λ - M)Sm- (11.49) Thi s i s t h e s a me as (10.41) wi t h ot\ = 4 M — 3, a -ι = ~4, β ο = Μ2, β ι = —2M + 1, /32 = 1, 7o = 0, 7 1 = 2, 7 2 = 4. (11.50) Verify that this falls in to the category C of classification of three-term re cursion relations in Sect. 10.3.2. Note that the parameters (11.50) also satisfy (10.42) as they should because Cm+π = 0, (n > 1 ). Invoke (10.62) to show that the eigenvalues are given by The corresponding eigenvectors are also determined by (10.45). For some numerical results and their physical interpretation, see [84]. 11.1.2 On-Resonant Real Time-Dependent Coupling If 5 = 0 and g is real then verify that (11.13) are solved by A realization of particular interest is g(t) = asin(o;i). This accounts for, for example, the effects of spatial mode structure on a moving atom in a cavity. This effect has been studied in [106] for single-mode two-level Jaynes- Cummings model. On inserting (11.52) in (11.10) we can recover not only the known analytic results [106] for the said model but can also study those effects in several other effective two-level models. 11.1.3 Fluctuating Coupling We have thus at hand an apparatus for handling a two-level atom in a quan tized field if the detuning and the coupling are deterministic functions of time. In this section we address the question of studying the dynamics when the said parameters are random functions of time. In the case of fluctuating parameters, we first evaluate the expectation value (A( t )) of an operator A and then average over the fluctuations. Note (for A = n(2n +1), n = 0,1,..., M. (11.51) £ ++ = = —1 (11.52) 11.1 Exactly Solvable Two-Level Systems 209 example from (11.18)) that calculation of an expectation value involves action of U and W together. Hence, the task of evaluating average over fluctuations reduces to that of finding that average over bilinear combinations of {xij}· However, the operator whose expectation value is being evaluated, need not commute with N. Hence, we need to know the average over fluctuations of the quantities of the type Xi j(N\)xki (N2 ), N± φ N 2. It may be determined by constructing equations for the said products. As an example, apply (11.13) and (11.14) to show that x\ = x++(N1 )x*++(N2), x 2 = x++(N1 )x*_ + (N2), x3 = ®_ + (ΛΓι)<+(.Ν2 ), x4 = x- +( N!) x t +(N2), (11.53) obey the following closed set of equations i i i = ~g(t )x 2 +g*(t )x 3, i x2 = - N 2 g*(t)x i +5( t ) x2 +g*(t) x 4, \X3 = Ν ^ ( ϊ ) χ ι - δ(ί)χΆ - g(t)x4, [ ± 4 = N 1 g(t )x2 - N2 g*(t)x3. (11.54) Let us assume that ί is a constant and the coupling is of the form g(t) = iff| exp(—i0(i)), (11.55) |(/| being a constant but the phase <j>(t) a fluctuating variable. In order to find Xi averaged over fluctuations, define xim = exp(imtfi(t))xi, x2m = exp(iπιφ(ί) - \φ(ί))χ2, % 3 m = exp(i τηφ(ί) + \φ(ί))χ3, x4m = 6 χρ(ΐ πι φ(ή)χ4. (11.56) It then follows that Xi=Xio, X2 =X2 1, X3 =X3-1, x4 = X40- (11.57) On invoking (11.54), it is straightforward to show that ^1 m \ s: - «'» ^ ^2m / (11.58) 0 0 0 m — 1 0 0 0 m + 1 0 0 0 m ~\9\ \9\ δ 0 0 - δ Ifflffi -\g W2 Recall from Chap. 6 that analytically exact equation for the average of the variables obeying an equation of the type (11.58) may be derived if φ is either a delta-correlated Gaussian process or a random telegraph noise. We leave it to the reader to derive corresponding equations. The reader may also derive equations for other bilinear combinations of x±. 11.2 N Two-Level Atoms in a Quantized Field The Hamiltonian of N atoms interacting collectively with a quantized field may be written as in (11.1) with the understanding that 5μ obey the SU(2) commutation relations and that S = N/2 is the total spin quantum number. The space of the atomic states is then spanned by N + 1 states |m) (m = —S, —S + 1,..., S. For the sake of illustration, let us assume that F = a. The combined state of the field and the atoms is then spanned by \n,m), n being the eigenvalue of ά^ά. Note that M = a) a + Sz commutes with H. Hence, if initial state of the system is such that Μ\φ(0)) = Μ\ψ(0)), then its state at any time t will be a combination of states | M — m,m), where, the positivity of the photon number M — m and the condition that —5 < m < 5 demand that m = —S, —S + 1,..., M if M < 5, but m = —S, —S + 1,..., 5 if M > S. The eigenstates corresponding to the same eigenvalue M of M are said to constitute a manifold. Thus, the problem of N two-level atoms reduces to diagonalization of at most N + 1 dimensional hamiltonian matrix. The dimension Nj is smaller than N + l if M < S. For example, if M = —5+1 then m = —5, —5 + 1. The problem of N two-level atoms then reduces to solving the eigenvalue problem of only a two-dimensional matrix. Similar considerations apply if the process is multiphoton. For example, if F = ap in (11.1) then the constant opertator is M = a)a + pSz. 210 11. Two-Level and Three-Level Hamiltonian Systems 11.3 Exactly Solvable Three-Level Systems In this section we consider a class of systems consisting of three-level atoms in quantized field describable, in an appropriate rotating frame, by the hamil tonian H2 = h Si(t)A22 + £2(0^33 + j<7*(t)A2iF\ +9 *2(t )A3 2F2 + h.c.}]. (11.59) Here δι (ί ),δ2 (ί) are effective detunings, (71,( 72 are the coupling constants, ^ j = \i)(j\, i,j = 1,2,3 (11.60) are the operators effecting transition between the atomic states |z) and |j), a nd Fi,F\ are the sums of products of single-mode quantized field operators. The orthogonality and completeness relations imply that 3 AijAki = Audjk, = (11.61) i- 1 As a consequence of thi s, t he evolution oper at or t/2 may be expressed as <;i>ij being functions of the field operators. In order to determine these func tions, differentiate (11.62) with respect to time to get 3 Η2 ϋ 2 = ih Σ Φίί(ί)Αί]· (11.63) i,j =1 On substituting (11.59) for H2 and (11.62) for U2 and on comparing the coefficients of Aij, (11.63) leads to a closed set of equations ϊφα = gi{t)F}<j>2i, ιΦ2ί = δ\(ΐ)φ2ί + gi{t)Fi(t>u + g 2 (t)F^(j>3i, \φ3ί = δ2 (ί)φ3ί + 9 Ϊ(ί)Ρ2 φ2ί, i = 1,2,3. (11.64) These are to be solved with the initial condition 4( 0) = 0 for 0ii (0) = φ2 2 (0) = φ3 3 (0) = 1. (11.65) Solving (11.64) analytically exactly is generally a formidable task. An ana lytically solvable case of widespread interest is when the 6's and the g's are time-independent with δ2 = 0. The equations (11.64) then yield d2 - d - - - ^ φ 2 ί + ι δ 1 —φ2ί + Γ 22 φ2ί = 0, i = 1,2,3, (11.66) A22 = N 2A A f + \9 2 M F 2. (11.67) The solution of (11.66) with the initial condition (11.65) is given by = exp ( - i y i ) | cos(fi) - ϊ^ ι Λ; s i n (/,i ) } ii 2 1 Γ δ^ being the Kronecker delta, and Γ 2 = \{ δ Ι + ΑΓ212). (11.69) The rest of the φ^, found on inserting (11.68) in (11.64), read l r / ίδχί\ ( . * . . i i 11.3 Exactly Solvable Three-Level Systems 211 - i — s m( r t ) ^ gl Ργδα + 5 2 ^ 2 ^ 3 } , (11.68) Φη = 1 + \gi\2Pi [exp ( — ^ (^cos(ft) + i-4r s'm(f t )j - 1 Fi, (11.70) 031 = gl gZF2 j ^ [exp ( - ( c o s ( r i ) + 1^ 4; sin ( f t ) } - 1 Fi, (11.71) 212 11. Two-Level and Three-Level Hamiltonian Systems Φ1 2 = - m H -^exp(-i<M/2 )si n(fi ), ( 1 1.7 2 ) Φ32 = - 1 5 2 ^ 2 -^; exp(-i£i£/2 )si n(rf), ( 11.73) Φ33 = 1 + \g2\2 F2 -X- [exp (" - (" cos (Pt) 1 1 2 + ™ i n ( f t ) ) 2 > (11.74) Φ13 gig2Fi r 2 1 12 exp(—i Si t/2) ( cos (Pt ) + · δί + i —- sm 2 Γ i n (Pt )) 1 r t ( 1 1.7 5 ) We h a v e t h u s d e t e r m i n e d c o m p l e t e l y t h e d y n a m i c s o f a t h r e e - l e v e l s y s t e m c o r r e s p o n d i n g t o t h e t i m e - i n d e p e n d e n t f o r m o f t h e H a m i l o n i a n ( 1 1.5 9 ) w i t h δ2 = 0. For the standard models outlined in the Chap. 7, Pi and P2 are single mode operators. The corresponding P is a function of the number operators. In such cases, the number states constitute a convenient choice as a basis. For details of some numerical results, see [102]. Next, we discuss a frequently encountered situation in which a three-level hamiltonian is reducible to an effective two-level one. 11.4 Effective Two-Level Approximation Note from the expressions for the {φ^} derived in the last section that the rate of transition between the levels in the presence of the detuning £1 is determined by ||.Γ|| defined in (11.69). That rate in the absence of £1 is governed by HA2 II· Now, let |Ji| HA2 II· On ignoring the terms of order ΜΙΙΑΙΙ/ΊΙ Α2 ΙΙ, (1 1 -6 8 ) and (11.70)-(11.75) reduce to Φ12 s Φ21 ~ 0, Φ23 ~ φ32 ~ 0, Φ22 ~ exp(—i<M), (11.76) Φ1 1 a 1 + |5ι| 2^ J - [exp ( i P^ t/δ Λ - 1 12 L V J Pi, (11.77) Φ33 a 1 + \g2\2F2 j ^ - [exp ( i Pf o/δί ) - 1 H, (11.78) Φ13 - [exp ( i P y/S i ) - l] F2, (11.79) 11.4 Effective Two-Level Approximation 213 031 ~ 9l 92^2j ^~ * 12 exp - 1 Fl (11.80) The equations (11.76) show t hat, in t he limit of large detuning, t he levels |1) and |3) are decoupled from t he level |2). Hence, if t he atom is initially prepared in a superposition of only t he st at es |1 ) and |3) then it contin ues executing transitions only between those two states. In other words, the three-level system then acts as an effective two-level one. The corresponding effective two-level hamiltonian can be written knowing the specific forms of the field operators Ργ and F2. However, (11.77)-(11.80) on substitution in (11.62) determine the time-evolution in effective two-level approximation for arbitrary F i,2 without any recourse to the knowledge of the effective hamil tonian. A frequently encountered three-level system is the one in which each of the two pairs of levels is coupled by only one mode. Consider, for example, the levels arranged in a ladder configuration with both the transitions induced by the same field mode described by the annihilation operator a. It is described by (11.59) with F\ = F2 = a. The operator solutions (11.77)-(11.80) in this case read 0 n 9 + 1 1 + \gi\ a 1 a - 2 7i 1 exp 033 ~ 1 + \9 2\2ά α ) ψ [exp 013 « 9ι 92ά ] 2 ψ [ e x P ( n l V ^ i ) “ - 1 - 1 1 031 ~ 9i 92&z^2 [ eXP ( i7 l i/<*l) - 1 (11.81) (11.82) (11.83) (11.84) 71 = (\gi\2 + l52|2)afa - M 2, 72 = ( l 5 i| 2 + l5 2 |2 )afa + 2|ffi| 2 + |ff2|2· (11.85) In writing these expressions, we have made use of (1.21). On combining (11.85) and (11.62) we obtain the operator determining the evolution of the levels |1 ) and |3) as f/eff = exp[-i fl eff/ft] = Σ (1 1.8 6 ) i,j= 1,3 The H,.ff satisfying (11.86) is given by — — ^-(|ffi|2ata|l)(l| + |<72|2άά+|3)(3|^ T-(fl,ifl'2|3)(l|a2 +^|l)(3|at2j. ( 1 1.8 7 ) Note that this is the same as the one written in Sect. 7.5 on qualitative considerations. In order to show that (11.87) is the correct hamiltonian, rewrite it in the form (11.1) and construct the corresponding evolution operator U\ as in (11.10). The correctness of (11.87) is established by showing that U\ so constructed is the same as the Ueg of (11.86). We leave it to the reader to carry the suggested steps. The term in the first brackets in (11.87) is, as already stated in Chap. 7, the Stark shift arising due to virtual transitions to the intermediate level |2). In practice, it is common to ignore the Stark shift either completely or ap proximate it by replacing the field operator aa^ in front of the atomic operator |3) (31 in (11.87) by ά^ά. This replacement amounts to ignoring spontaneous emission from the upper level. However, the and Γ2 in the corresponding Uxit) of (11.21) in these cases turn out to be square roots of an imperfect quadratic form in the number operator a^a. This implies irrational ratio of frequencies in the number states. Correct frequencies (11.85) are, however, linear in the number operator ά^ά. The expressions of and Γ2 in agreement with (11.85) are obtained, as already established above, by treating the Stark shift correctly. For some numerical results in case the Stark shift is treated approximately, see [107]- [109]. The reference [109] compares the results ob tained in the effective two-level approximation with the exact results. The effective Hamiltonians for the three levels in other configurations and in two-mode field may similarly be derived and shown to be the ones given in Chap. 7. 214 11. Two-Level and Three-Level Hamiltonian Systems 12. Dissipative Atomic Systems In the last chapter we outlined an approach to studying lossless two- and three-level atomic systems interacting with lossless quantized field. In this chapter we analyze the equations governing dissipative two and three level atomic systems driven by an external field. 12.1 Two-Level Atom in a Quasimonochromatic Field Recall from Chap. 8 that the dynamical evolution of the density operator of a two-level atom in an external field is governed by the master equation dp di H, ex t ? P • ι ω0 + LTp + Lcp. (1 2.1 ) Here Hext is the hamiltonian of interaction between the atoms and an external field, assumed to be of the form — fi g{t)S+ exp(—iu>it) + g*(t)S_ exp(iwii) (1 2.2) The Liouvillean Lr in (12.1) describes the atomic radiative losses. It is given by (8.95). The Liouvillean Lc governs the losses due to atomic collisions. It is given by (8.106). We transform to the interaction picture by means of pi(t) = exp(iui\Szt)p(t) exp(—iu\Szt). (12.3) The atomic dynamics in this picture is characterized by pi = - i [dSz + g(t )S++g*(t )S-, pij +7 [ ( N + l ) ( 2 5 _ p I5+ - piS + S - - 5 + 5 _ p i ) + N ( 2 S + f n S _ - p i S - S + - S - S + pi ) + {M exp(2 i<y) pi S'- - p j S - S - - S - S - f a ) + h.c.} +7c ( z Sz f aSz - Pi Sz S z - 4 ^ p i ) ]. (12-4) 216 12. Dissipative Atomic Systems Here δ = loq — ωι is the detuning of the atomic levels from the frequency of the driving field, δρ = Ωρ — ωι is the difference in the frequency between the driving field and that of the pump driving the squeezed bath. Invoke (8.37) and the properties of two-level operators to show that the atomic averages obey the equation (with M = \M\ exp(i0)) ^|S(i))=M|S(i ) ) - 7 | 0 | , (12.5) f ( S x{t))\ I S(t)) = (Sy(t)) , (12.6) \(Sz(t))J f - r ph + r c r s - δ 0 \ M{t )=\ r s +δ - r ph- r c -2g( t)\. (12.7) V 0 25 (f) - Γ J Γ = 2 (2N + l) 7, .Tph = — + 7 c, Γα = 2 |M|7 Cos( 0 + £p t), Γ$ = 2 |M|7 sin( 0 + δρ t ). (12.8) The coupling constant g(t) has been taken to be real for convenience. Note from (8.67) that N( N + 1) > \M\2. This implies that (2N + l ) 2 — 4|M| 2 = 4 N ( N + l ) — 4|M| 2 + 1 > 0, i.e., 2 N+l — 2\M\ > 0. As a c onsequence of thi s we have r ph-2| M| 7 >0. (12.9) The equations (12.5) are the so called optical Bloch equations. Their for mal solution evidently is where D( t,t') is the time-ordered integral D{t,t') = ^ e x p QT M(r) dr^ . (12.11) This integral is easily evaluated if M is time-independent (see Sect. 12.1.2). However, its explicit evaluation for general time-dependence of M(t ) is not straightforward. The complexity of the problem, and its possible simplifica tions, can be assessed by rewriting M(t ) in terms of nine elementary matrices Bpq = \p)(q\, p,q= 1,2,3, (12.12) 12.1 Two-Level Atom in a Quasimonochromatic Field 217 the |p) being an elementary column vector which has unity as its only non zero element in its pth row. Consequently, Bpq has unity as its only non-zero element at the position (p,q). Clearly M(t) = 2 g(t) ^.8 3 2 — B2 3 ) + S ^i?2 i — -B12) — A>h (-B11 + -B2 2 ) —ΓΒ 33 + 2 |M| 7 j sin(</> + φρί) ( j l i 2 + ^ 2 1 ^ + cos (φ + φρί) ^.Bn — B2 2 ) (12.13) Analytically tractable cases of the time-ordered integral (12.11) can be iden tified by looking for the cases in which M(t ) reduces to a combination of a set of a fewer number of operators closed under commutation. For example, it can be verified that B 1 2, B21 (1/2 ) ( S n — B 2 2 ) obey the commutation re lations of the SU( 2) operators J±, Jz. We may similarly identify other sets of operators obeying SU( 2). Now, if the values of the parameters are such that M( t ) reduces to such combinations of Bij which are closed under SU( 2) then the problem of evaluating (12.11) is reduced to that of finding an SU( 2) time-ordered integral. That problem has already been addressed in Chap. 2. We discuss in the subsection below the dynamics when M is time-dependent and reducible to SU( 2). In the subsection following it, we discuss the case of time-independent M. 1 2.1.1 T i m e - D e p e n d e n t E v o l u t i o n O p e r a t o r R e d u c i b l e t o S U( 2) The operator M( t ) reduces to an SU( 2) operator if: (a) there is no damping i.e. if 7 = 7 C = 0; or (b) if Γ φ = Γ , \M\ = 0; or (c) if δ = δρ = 0, along with φ = 717Γ. In the following we discuss these cases separately. No Damping. In this case, the equation of evolution (12.5) for the averages is homogeneous. The expression (12.13) for M reads M(t) = 2 g(t) (j332 — B 23) + δ ^i?2 i — -B12) = -i(2g{t)Jx + SJy) (12.14) where, as can be readily verified, the operators Jx = i(i?32 — B 2 3 ), Jy = 1 ( ^ 2 1 — B 1 2 ), Jz = ί(#3ΐ — ^ 1 3 ), (12.15) obey the S U(2) commutation relations. They are the generators of its three- dimensional representation having the property = Jfl, the Jμ being any component of J. The expression (12.11) then reads rt · (12.16) The time-integral can now be performed by using the method of Sect. 2.5.2. The time-ordered integral in (12.16) reduces to the exponential of the integral if δ = 0. In this case, apply (2.14) to show that 218 12. Dissipative Atomic Systems D(t,t') = 1 — isin (29{t,t')) Jx + {cos (29{t,t')) — 1} Jx, ( 1 2.1 7 ) 0 ( M') = J 9 { r ) d T. (12.18) Si nce g(t) is proportional to the envelop function of the driving field, 9(t, t') is proportional to the area enclosed between the field envelop and the time axis between the times t' and t. Substitution of (12.17) in (12.10) along with the use of matrix representation of Jx, yields This determines the dynamics of a resonantly driven non-dissipating two-level atom in a time-dependent field. Now, let the atom be initially in one of the two states | ± 1/2} so that (Sx(0)) = {Sy (0)) = 0, (Sz (0)) = ±1/2. It will be found in the other state | + 1/2) so that (Sx (t)) = (Sy(t)) = 0, (Sz (t)) = +1/2. at the time t which is such that the area 9(t, 0) = (2η + 1)π/2. It will return to its initial state at such time t which generates the area 9(t, 0 ) = ηπ. S t r o n g C o l l i s i o n s a n d T h e r ma l B a t h. Cons i de r t h e cas e J\,h = Γ and |M\ = 0. The condition \M\ = 0 means that the reservoir is thermal. The condition = Γ, by virtue of (1 2.8 ), requires collisions to be so strong that 7 c = Γ/2. This condition is, therefore, referred to as the strong collisions limit. Since, by definition (12.12), B 1 1 + B 2 2 +B33 = I, the expression (12.13) homogeneous part of (12.5) in the limit of strong collisions is thus a product of free and damped evolutions. Resonant Squeezed Bath. Consider the case δ = δρ = 0, φ — ηπ. This implies that /s = 0, /c = 2(—)n\M\^. Note that in this case the equation for (Sx(t)) reading (12.19) for M( t ) in this case reduces to where are as in (12.15). Hence, (12.11) may be rewritten as t1 The time-ordered integral above describes free evolution. The solution of the (Lit)) = - ( r ph - 7b) (Sx(t)) (1 2.2 2) 12.1 Two-Level Atom in a Quasimonochromatic Field 219 gets decoupled from the equations for (Sy (t)) and (Sz (t)): (Sy(t)) = - [rph + r c\ <Sy (t)) - 2g(t )(Sz (t)) (,Sz (t)) = ~ r ( S z (t)) + 2 g(t)(Sy(t)). (12.23) If g is independent of time then these equations can be solved by extending the results of the Appendix C. Else, they may be handled by the method of time-ordered SU( 2) integration outlined in Sect. 2.5.2. 12.1.2 Time-Independent Evolution Operator If the envelop of the driving field is constant and its frequency the same as that of the pump driving the squeezed bath then all the elements of M become time-independent. The master equation then describes the phenomenon of resonance fluorescence in a squeezed bath. The evolution operator M in this case assumes the form + A Γ2 — δ 0 M = I Γ2 + δ - r ph- A - 2 g\, (12.24) 0 2 g - Γ A = 2 |M|7 cos(0 ), Γ2 = 2 |M|7 sin(0 ), (12.25) and r ph is as in (12.8). The formal solution (12.10) now reads IS(t)} = exp ^Mt ) |<i>(0 )) — 7 J exp ^Mr j d r ^ 0 ^ . (12.26) This may be evaluated by applying (C.7). Its application requires eigenvalues of M. It is straightforward to verify that the equation det(M — XI) = 0 determining the eigenvalues A is the cubic /(A) = [(A + r p h ) 2 - Γΐ - Γ | + £2] [A + Γ] + 4g2 [A + Tph - A] = A3 -Κ &2A^ + €lq = 0, (12.27) with α0 = Γ ( r p2h - 4|M| V + δ2) + 4g2 ( r ph - A ), ai = 2/nph Γ + Γ 2^ — 4\Μ\2'γ2 + δ2 + 4 g2, α2 = Γ + 2 r ph. (12.28) Let Xj (i = 1,2,3) be the roots of (12.27). We invoke (C.7) to evaluate exp(Mt) and insert it in (12.26). We get 220 12. Dissipative Atomic Systems (Sx(t)) = [ttl + a 2 (A - Tph) + a 3 {(A - r p h ) 2 + Γ 2 - J 2}] ( 4 ( 0 ) ) + (Γ2 - δ) ( α2 - 2α3ΓρΗ) ( 4 ( 0 ) ) - 2 g a 3 (Γ2 - δ) <5,( 0) ), + 2 7 g ( r 2 - 6 ) ί α3( τ)dr, Jo (4(f)) = (Γ2 + ί ) ( α2 - 2α3-Γρΐι) ( 4( 0) ) + «ί — α2 (/^ + Ah) +α3 {(A + Tph)2 + Γ 2 - δ2 - 452} ] (4( 0) ) - 2 5 [α2 - α3 (Α + Ah + Γ)] (4( 0) ) + 2 7 5 ί [α2{τ) - α3 (τ) (A + Ah + A] d r > Jo (4(f)) = 2ga3 (A + i) (4(0)) + 2 g [a2 ~ a 3 (A + Ah + A] (4( 0) ) + [Ql - Γ α 2 + a 3 ( Γ2 - 452)] ( 4( 0) ) - 7 ί [αι(τ) - Γα 2(τ) + α3 (τ) (Γ2 - 4^2)] dr. (12.29) Jo This gives the atomic averages in terms of {am(f)} which are determined by evaluating (C.8 ) in terms of the roots of (12.27). The problem of studying the radiative properties of a two-level atom in a monochromatic field thus boils down to the one of finding the roots of the cubic in (12.27). Exact analytic expression for the roots of a cubic are known and are reproduced in the Appendix D. However, as elaborated below, the nature of the roots and hence the qualitative features of the evolution can be established even without solving (12.27) explicitly. Recall from the Appendix D that the roots of a cubic are either all real or one of the roots is real and the other two are complex conjugates of each other. We show that the real part of each of the roots of (12.27) is negative. To that end, we invoke the Hurwitz criterion stated in the Appendix D and note that the roots have a negative real part if (a) αο,βι > 0, and (b) αχα2 > ao. Using the definitions (12.28) of the a's and the condition (12.9), verify that both these conditions are satisfied for any non-zero value of 7 and 7 C. As a consequence of this it follows that, as t —> 0 0, the atom evolves irreversibly to a steady state. That state is determined by (C.3). Now, following again the Appendix D, we express the roots in terms of one unknown parameter lying in the range [0,1]. This approach reveals easily the conditions under which a pair of roots become complex. To that end, note that if Γ/2 — 7 C + A > 0 then /( λ) of (12.27) is such that j F ( - r ) = — 4<72 Γ 2 — 7c + A < 0, 12.1 Two-Level Atom in a Quasimonochromatic Field 221 / ( - r ph + 2 |M|7) = i 2 | + 2 | M| 7 - 7c +8g2\M\j {l - cos (φ)) > 0. (12.30) Hence, /(A) has a root, say Ai, in the interval [—Γρ^ + 2\M\j, — JT]. We note that, due to (12.9), —Γρh + 2|M| 7 < 0. Hence Ai < 0. We may write Αχ as Ai = (2|M| 7 - r ph)(l - a) - Γα, 0 < a < 1. (12.31) The other two roots, \2,s, are then the solution of the quadratic (D.5). These roots will be complex if the discriminant of the quadratic is negative, i.e. if (Αχ + (I2 ) 2 — 4 [αχ + Ai(Ai + 0 2 )] < 0. (12.32) This, on inserting the a^s from (12.28), acquires the form D2 - Ω2(δ) < 0 (12.33) where D is a combination of the damping constants and Ω(δ) = y/Ag2 + δ 2 (12.34) is the off-resonance Rabi frequency. This shows that complex roots are ob tained above a threshold value of the Rabi frequency. We have thus at hand a complete description of the atomic dynamics. As discussed in Chap. 8, the atomic averages determine also the statistical properties of the radiation from the atom. We have in (8.76) the expression for the field in the radiation zone. It shows that the positive frequency part of the field at time time t and at the position r with respect to the atom is proportional to S - ( t — |r|/c). Hence, the problem of finding the average of a normal order product of the field operators reduces to that of determining the product of atomic operator in which 5_ are placed at the right of the 5+. Of particular interest are the first-order and second-order coherence functions of the field. These are proportional respectively to GW(r) = (S+(i + r ) l ( t ) ), G<2>(t) = (S+(i)S+(i + r ) 5 _ ( i + r)S_(i)>. (12.35) Note that because of the two-level property S+ = 0, G^ ( 0 ) = 0. This implies that the radiation from a two-level atom is antibunched. The multi-time averages may be evaluated by invoking the regression theorems (Chap. 8 ). To that end, let (5+(i)> = /+„(t) + σ i =±,z <5 +( t ) 5 _ ( t ) > = \ + (5,(t)> = \ + /*„(t) + ^ 0))· (12.36) i= ±,2 The f ( t )'s can be identified by comparing these expressions with (12.29). On using (8.47) and the properties of two-level operators it follows that 222 12. Dissipative Atomic Systems G^(r) = /+0 (r)(5_(i)> + /++(r) (<&(*)} + 0 -\f+z{r)(S-(t)), G(2)(r)= ( 1 + f M r ) ) (S+(t)S-(t)) + £ /« ( ^ { ^ ( ^ ( ^ ( i ) ) i = ±,Z (12.37) Consider a damped atom so that it reaches a steady state. Evaluate the correlation functions in the limit t —>· oo. The corresponding averages are found using (C.3). In the following we highlight some qualitative features of the emission and the absorption spectra obtained by employing the steady state atomic correlation functions. Consider first the emission spectrum. The spectrum is defined in (6.91). We substitute in it the expression (8.75) for the field from the atom in the radiation zone. We assume the free-field to be in the state of the vacuum. We also note that we have evaluated the averages in the interaction picture defined in (12.3). It is in a frame rotating at the driving field frequency ωι. Hence, the expression (6.91) for the spectrum in the interaction picture, with the field given by (8.75), acquires the form This gives the spectrum of the field from the driven atom. It is, therefore, also referred to as the emission spectrum. This involves evaluating expressions of the form Ar and λι being the real and imaginary parts of an eigenvalue λ. Note that there also are time-independent terms in the expressions for the averages and consequently r-independent terms in G(r). They correspond to A r = Ai = 0. 1{ω) due to such terms is the delta function δ(ω — ωι). This component in the spectrum is called the Rayleigh or coherent component. It represents elastic scattering of radiation at its incident frequency. For r-dependent terms in This is a Lorentzian of width A r centered at ω = ωι — A i - A real root con tributes a Lorentzian at the frequency of the driving field whereas a pair of complex conjugate roots contribute Lorentzians at ωι ± Ai in the spectrum. A Lorentzian characterizes an incoherent process. It results due to absorption of (12.38) ■OO d r exp{—ΐ(ω - ωγ)τ} exp { ( - A r + i A i ) r } (12.39) G(r), (12.39) yields ( 12.40) 12.1 Two-Level Atom in a Quasimonochromatic Field 223 a photon from the incident field followed by the process of spontaneous emis sion. Recall that the roots become complex if the field is sufficiently strong. The splitting of the fluorescent spectrum in to three components in a strong field is referred to as ac Stark splitting. We ma y s i mi l a r l y d e t e r mi n e t h e a b s o r p t i o n s p e c t r u m by e va l ua t i ng (9.26). I n t h e i n t e r a c t i o n p i c t u r e i t as s umes t h e f or m Wa(oj) ~ Re j exp{i(w — ω\)r} ^ [s_(r ), S+ ^ dr. (12.41) It consists of a Lorentzian centred at the atomic transition frequency if the field intensity is below its threshold value for the complex roots to occur. Lorentzian peaks displaced from the centre by ±λι appear in a strong field. The qualitative considerations outlined above, however, do not provide information about the relative heights of the peaks. The roots need to be determined explicitly for a quantitative study. As stated before, the ana lytic expression for the roots is generally not simple. It assumes simple form whenever the cubic can be factored in to the product of a quadratic and a linear form in λ. Such special situations are encountered if (cl) the atom is undamped; (c2) the atom is not driven, i.e. g = 0, (c3) \M\ = 0 and the col lisions are strong so that r ph = Γ, (c4) the drive is on-resonance, i.e. (5 = 0, and φ = ηπ. Construction of solution in these cases is a matter of simple algebra left for the reader to carry. Simple results are obtained also when the drive is strong, i.e. Ω( δ) 2 > 7. This case is discussed in the next section for a system of N two-level atoms. 12.1.3 Nonlinear Response in a Bichromatic Field A problem of considerable interest is the nonlinear response of a two-level atom in a bichromatic field consisting of frequencies ωχ,ω2 · This interaction is characterized by (12.42) Hext = h | ^1 exp(-iwif)S+ + e2 exp(-iw2 i)5'+ + h.c. We assume that Hext may be treated perturbatively and that |ei| > |e2 1- We study the response to second order in ei and first order in e2. We are inter ested in the characteristics of the four-wave mixing signal at the frequency Ω = 2ωχ — u)2 - It is characterized by χ ^ ( ω χ,ω χ, —ω2 )· We construct it using the definition (9.18). The summation in that formula runs over ν ^,ν ^,ν ^. In the present situation, each of these frequencies takes values ωχ, ωχ, —ω2 · The distinct combinations of ( ν ^,ν ^,ν ^ ) are { ωχ,ωχ,— ω·2 ), { ωχ,—ω2,ωχ) a n d ( —ω2,ωχ,ωχ). We recall also that L( ±v ) F ~ — i[5+,F]. Consequently, it follows that Χ(3)(ωχ,ωχ, -ω2) ~ Tr[5+p(3)(/2)], (12.43) 224 12. Dissipative Atomic Systems ρ^(Ω) L + [S_ 1 L — \{u)\ — oj2) 1 [S+,· ■[ S-, L - i(w! - ω2) [ ^+ ! 1 ----- [S- , 1 ---— L — 2\ωγ L — ϊωχ L — ιω\ -[S-,Ps 1 L + ϊω2 -[S-,ps: " [5-(-, Psi (12.44) L = Lr + Lc — iwoI'S'z,] and pss is the steady state of L. For the sake of illustration, we asume that N = M = 0. In this case, pss = \ — 1/2) (—1/2|. Verify also that L\ - 1/2 ) (1/2| = [iwo - (7 + 7c)] I - 1/2 ) ( 1/2 |, l | l/2 ) ( - l/2 | = [ - ϊ ω0 - (7 + 7c)] | - 1/2 ) ( 1/2 |, 11/2 ) (1/2 | — I — 1/2 ) ( 1/2 | It is now not difficult to show that χ {3)(ωι,ωι,- ω 2) -2 7 [|1/2 )(1/2 | Ί —1/2 ) ( 1/2 1 (12.45) - 2 ( 7 + 7 C) - ί(ωι - ω 2) —2η — ϊ(ωι — ω2) - 1 - 7 - 7 c - ϊ(2ωι - ω 2 - ω0) . - ί 7 - 7c + Κω 2 ~ ^ο) - 1 7 - 7 c - Κ ω ι ~ ω ο) - 1 (12.46) The signal at 2ω\ — ω2 as a function of ω2 will, therefore, exhibit resonances at ω ο, 2ω\ — wq and a resonance at Wj if 7 C φ 0. The resonance at ω2 = ω\ is thus induced by collisions. It is referred to as the Bloembergen resonance. Note that the resonance at u>i does not correspond to an atomic transition. In Chap. 14 we will come across another important role played by collisions in four wave mixing in a cavity. It is in their ability to bring out otherwise suppressed resonances in non linear response of an atom in a cavity. 12.2 N Two-Level Atoms in a Monochromatic Field The master equation for a system of N two-level atoms in contact with a squeezed reservoir is governed by (8.93). The Hilbert space of N two-level atoms, being a direct product of N two-dimensional spaces, is 2N dimensional. Its density operator has, therefore, 22N elements. The system is evidently an alytically intractable even for small N > 1 unless the atoms-reservoir interac tion is such that it confines the evolution to some lower dimensional invariant 12.2 N Two-Level Atoms in a Monochromatic Field 225 subspace. A case of common interest in which the evolution is confined to a space of drastically reduced dimension is that of the atoms occupying a vol ume so small that all of them see the same field. Such a situation is described by the master equation (8.95) if it is also assumed that Ω^ sa v for all i,j. For, then (8.95) would involve only collective two-level operators confining the evolution to a space of fixed total spin quantum number 5. Now, for an N two-level system, 5 = N/2,N/2 — 1,..., 0 or 1/2 depending on whether N is even or odd. For a given 5, the eigenvalues m of any component of the spin vector is such that m = —S,—S + 1,...,S. A possible state of two-level atoms is the one in which all the atoms are in the ground state. This corre sponds to m = —N/2. This value of m is realized only if 5 — N/2. Hence, we restrict our attention to space of fixed 5 = N/2. Since the dimension of the space of spin quantum number 5 is 25 + 1 it follows that the dimension of the density matrix obeying (8.95) is (N + l ) 2. This is evidently an enormous reduction in the dimension compared with the general equation (8.93). In the following we confine our attention to the collective operator form of (8.95). We let the system be driven by a monochromatic field assuming that the frequency Ωρ of the pump driving the squeezed bath is the same as the frequency ωι of the field driving the atoms. The atomic density operator in the interaction picture defined by (12.3) then reads pi = - i 6SZ + gS+ + g*S_, p\ - \v 5 + 5 -, p\ + 7 [(N + 1) (2S_/)iS+ - /)iS+S_ - S+S-Pi) +7v(2S+pi5_ - piS_S+ - S_S+pi) + | m ( 2 S_piS_ - f r S - S - - S_5_pi) + h.c.} ξε i f r. (12.47) This is the master equation for the phenomenon of collective resonance fluo rescence in a squeezed bath. We may solve this equation either by taking its matrix elements in the collective states 1 5, m) or by constructing the equa tions for averages. However, except for small values of N , these equations can be solved only numerically. We have already discussed the case of N = 1 in the last section. The analytical solution of (12.47) for ι/ = Μ = 0 ϊ ο τ Ν = 2 and its numerical solution for N up to 20 are given respectively in [110] and [1 1 1 ]. Solving (12.47) even for its steady state is a non trivial task. For, it does not obey the condition (8.34) of detailed balance. Application of that condition requires us to construct the time-reversed form of the evolution operator. To do that, let us denote by T the operation of time reversal. Let |i) and A denote the state and the operator obtained by performing time- reversal on |i) and A. The operator T is antilinear. We have, by definition, [112] 226 12. Dissipative Atomic Systems |ί> = T\i), A = f ~ l A^f. ( 1 2.4 8 ) I f A in the equations above is a constant c then we find that c = c*. We also note that (12.48) implies Ab = b A. If A = S± then, invoking (12.48), we infer that h = sT, §z = sz. (12.49) (12.50) We use these relations to find the time-reversed form of L given by (12.47). For the sake of illustration, we let ί = Μ = v = 0, N = n. We get Substitute these relations in the detailed balance condition (8.34). It is These equations can not be satisfied simultaneously if g φ 0. However, if g = 0 then the last two equations above are solved by (3.102). In the following we discuss the cases when analytical time-dependent and steady state solution of (12.47) can be derived. The time-dependent solutions can be found (a) if g = Μ — N = 0, (b) in the limit Ω 2 > 7, and (c) in the limit N 2 > 1, whereas exact steady state solution is derivable analytically for (d) m = n = 0, and (e) for some special values of the external field if v = 0, and \M\2 = N ( N + 1). g — Μ = N = 0. The master equation for g = M = 0 reads LtIf = i[gS++g*S., f + 7 [(fi + 1) ( 2 S -/S + - f S +S - - S+S ^ f ) (12.51) Th e f or m of L, f ound by i nvoki ng t h e def i ni t i on ( 8.35), r e a ds L f = i[gS+ +g*S-, f + 7 [(ή + 1) ( 2 S +f S - - f S +S - - S+S- ή (12.52) s t r a i g h t f o r wa r d t o see t h a t t h e r e s u l t i n g expr e s s i on woul d hol d for any / i f g S + + g * S ~, Pss] = 0, S+S-, pss — 0, {τι -\- l )pssS+ nS+pss — 0. (12.53) ^ = 11/ [5+ 5 _, p] + 7 (ft + 1) (2S.p S + - S + S - p - p S + S_ ) = —iv ( 12.54) 12.2 N Two-Level Atoms in a Monochromatic Field 227 It describes collective spontaneous emission in a thermal bath. It can be solved analytically if n = 0. We may solve it by the method of eigenvectors expansion. To that end, take the matrix elements of (12.54) in the eigenstates |S,m) of Sz. Define pm+p,m = ( S—m — p\p\S — m), to arrive at the eigenvalue equation [(7 ΐ^)^τ7ΐ+1 “t“ (^ + i^O^m+p+l] Ρτη+ρ,τη y j A-mArn+pprn+p_i ni_i A ( p )/} m _j_p)77l, (12.55) Am = m(N — m + 1). (12.56) This is to be solved as a recursion relation in m for a fixed p. The method of solving such an equation has been outlined in Sect. 10.3. We note that the eigenvalues are M p ) = ( - 7 + 'w )Ak+i + ( - 7 - w )Ak+P+i, (12.57) k = 0,1,..., N. The real part of the eigenvalues is non positive. They are non degenerate for any ρ φ 0 if υ φ 0. The eigenvalues are two-fold degenerate if p = 0. For, then Afe(0) = —27 (A: + l ) ( N — k) = Ajv-fe-i(O), k = 0,l,...,N. (12.58) The eigenvalue 0 corresponding to k = N is clearly non degenerate. The solu tion of (12.55) in case of p = 0 is to be found by constructing its generalized eigenvectors. It is a tedious but straightforward task. The reader may carry it for N = 2 and compare the solution with the one derived by direct inte gration of the time dependent equation [89], We have outlined that method in Sect. 10.3. The High-Field Limit. Recall from Sect. 8.2 that the dominant contribu tion to a master equation in this limit may be extracted by applying the secular approximation. The task at hand is to solve (12.47) in the limit when the hamiltonian part in it is strong compared with the damping. This task is accomplished conveniently by transforming to a new set of spin operators defined by (assumimg g to be read) 0 1 0 -6/Ω 0 2 g/Ω , (12.59a) 2 g/Ω 0 δ/Ω Ό - δ/Ω 2ς/Ωλ 1 0 0 , ( 12.59b) ν0 2 g/Ω δ/Ω t h e Ω being the Rabi frequency defined in (12.34). The equation (12.47) then reads dp dt ΛΩ \RZ, p] +Lp. (12.60) 228 12. Dissipative Atomic Systems The superoperator L is written, of course, in terms of the new set of operators defined in (12.59a). Now, define The Li (t ) would contain terms some of which are independent of time and the terms which oscillate at the Rabi frequency or its multiples. As discussed in Sect. 8.3, if the hamiltonian part is very much strong compared with damp ing, i.e. if 7 <C Ω then the dominant contribution to the master equation proximation. It is straightforward to see that only the terms in which p occurs in combination with equal number of R + and i i _, and any power of Rz are time-independent under the transformation generated by Rz. Collecting all such terms together we find that, in the secular approximation, (12.60) re duces to For the sake of illustration, we write the equations for averages of operators corresponding to (12.64). If v = 0 then Pi{t) = exp (i QRzt ) /5(f) exp ( —i QRzt ) (12.61) s o t h a t p i ( t ) obeys the equation ^ = i 7 (i)ft(i), (12.62) where Li(t) = exp {i QRzt} Lexp iQRzt ) . ( 12.63) ar i s es f r om t i me - i n d e p e n d e n t t e r ms i n Lj. This constitutes the secular ap- ^ = - i i? [£*, p] - \v A\R + R - + A 2_ R.R + + j ^ R 2z, P +Γχ 2R_pR+ — pR+R_ — R+R - p ‘2RjrpR~ — p R - R + — R - R +p +ΔΖ 2 R zpRz - pRzRz - R zRzp ( 12.64) A = 7 {(n + !) ^ + + η Δ - } , A = 7 {{η + 1)Δ2_ + η Δ^ } . Verify t h a t t he st eady s t at e solution of t hi s equation is (12.65) pss = exp 1η( Γι/Γ2 )Λζ) j Tr exp 1η(Γι/Γ2 )Λ*) . (12.6 6) f t ( R + ) = ( & + ) + 2A ( R + R z ) ~ 2 A ( R z R + } - Δ ζ ( R+) 12.2 N Two-Level Atoms in a Monochromatic Field 229 ^ ( i t ) = - 2 Γ ι ( R+R_} + 2Γ2 (JL.R+) . (12.67) The equation for an operator average is coupled with the average of the products of operators. Each of these products is reducuble to a single operator in the case of a single atom system. We leave it to the reader to solve these equations for N = 1. In general, we note that if Γι = Γι then the terms containing products of operators combine to reduce to commutators. As a result, each of the equation above is closed. Note that we can have i~i = Γ2 provided (5 = 0. We also let ή = 0 so that Γ\ = Γ2 = 7/4 and Δ ζ = η. The solution of (12.67) then reads £ + ( * ) ) = exP ( 2 [ 9 t ~ \ (#4 ^Rz(t)J = exp ( - 7 i) . (1 2.6 8 ) The steady state in this case is pss = I/(N + 1). (12.69) We leave it to the reader to show that C(T) = ~ 2exp(—7 t) + {exp(2igt — 37 t/2) + c.c.} . (12.70) From this we infer that the emission spectrum consists of three Lorentzians, one centred at the driving field frequency and the other two displaced by ±2g from it. The central peak has the width 7 whereas the width of each of the other two peaks is 3 7/2. N 1 . In what follows, it will be found to be useful to work with scaled parameters defined by the relations T =-!»*· 0 = ^ · Δ = ^Ν· I12·71) It will be assumed that N 7 is finite in the limit N 2 > 1. The theory of the behaviour of the atomic system in this limit is based on expressing the averages and the variances as Sa) ™μ0) + ™{,1} + · · · , SnSv) - ( Sp) ( S v) 1 2------------- = ea$ + e2(Tj S H > e = Ν' (12.72) and so on. Now , with N = M = 0 and the parameters scaled as in (12.71), (12.47) yields d ( ^ ) · Λ(*+) A*1*) dr¥= ^ V - - * V + d(s*) w ( s + - s - ) ( s +s - ) d 7 V = <12·73) N 230 12. Dissipative Atomic Systems On using (12.72) and on retaining the terms to zeroth order in e we obtain = i + | 2 m ^ ( l — \v) — i#} τη)ζ\ d — = - 2 m ^ m ^ — — ( m^ — τ η^ Λ . dr + “ 2 V + - J ( 1 2.7 4 ) T o t h e s a m e o r d e r i n e, (S 2)/N2 = 1/4 which, on using the relation S 2 = S+S - + S 2 - Sz gives + m ^ 2 = + 2 4 (12.75) By applying (12.74), it can be checked that the time derivative of this expres sion is indeed zero. As an illustration, we solve (12.74) for its steady state by equating the time derivatives to zero. Show that the inversion in the steady state solves the quartic — 2 + 4m^ 2 - - mi0 ·*2 Its solution for v m i ° ) i = 0 yields m C°) =0 - r yJ 1 m θ\ ( 0) ΪΘ l ± 2Θ 92m ^ 2 < 1, > 1. 0. (12.76) (12.77a) (12.77b) These solutions must be examined for their stability by determining how a small disturbance of the steady state develops in time. To that end, we let - ™(°) m„ = m νμ — ,,ιμ f i nd t h a t (Δ d d r We (12.78) + δτημ in (12.74) and derive the equations obeyed by δτημ. v = 0) 6 mz \ _ / 4m" - ι θ/Ί" δτη+ — δτη^ J \2 { m ^ - m(0) - \θ} 2 m°z 6mz 6m+ — δτπ- Th e n a t u r e of e vol ut i on i s d e t e r mi ne d by t h e ei genval ues of t h e ma t r i x of evol ut i on. I t i s s t r a i g h t f o r wa r d t o see t h a t i f Θ < 1 then the eigenvalues of the matrix of evolution in (12.78) are 2m°,4m°. The steady state is stable if the real part of all the eigenvalues is negative. Hence, (12.77a) is a stable solution if m z < 0. The eigenvalues of the evolution matrix for Θ > 1 are ± W6 2 — 1. The significance of purely imaginary eigenvalues is revealed by comparing the approximate results with exact asymptotic results given in (12.91) and (12.92). We note that the approximate expressions for the av erages above are in agreement with the exact expressions for Θ < 1. In this range, the eigenvalues of the matrix of evolution have a negative real part. However, the approximate results for Θ > 1 are wide off the exact results. Thus fluctuations, ignored in obtaining the approximate results in (12.77b), 12.2 N Two-Level Atoms in a Monochromatic Field 231 seem to play important role in driving the system towards the steady state above 9 =1. T h e e q u a t i o n s ( 1 2.7 4 ) c a n b e s o l v e d e x a c t l y f o r δ = u = 0. Below 9 =1, the time-dependence is purely decaying. Above 9 = 1, the system follows a closed trajectory around the steady state solution determined by the initial condition. But for ignored fluctuations, the system would stay on a trajectory. These fluctuations influence the motion on a time scale much longer than its period. They cause the motion to diffuse on a trajectory as well as between the trajectories. The dynamics of the system on that time scale may be viewed as a result of averaging over the motion on all the trajectories and the initial states. A systematic procedure for carrying such an average is outlined in [113]. Observe that the approximate as well as the exact steady state results show that, for δ = v = 0, the steady state averages are continuous functions of the driving field parameter Θ but their derivatives with respect to that parameter are discontinuous at Θ = 1. This behaviour is reminiscent of a second-order phase transition. The nature of the steady state is entirely different if v, δ Φ 0. In this case (12.76) may be solved numerically. Of course, only its stable real solutions in the range [—1/2,1/2] are physically acceptable. In order to bring out essential features of the solution, we have plotted in Fig. 12.1 mi°·* as a function of Θ for Δ = 0.5, v = —5.0. These are the values of the parameters used in [114]. Fig. 12.1. Solution (Sz)/N of (12.76) as a function of Θ for A = .5, v = —5.0. We have omitted the part of the curve corresponding to positive values of m ^ as that part is unstable (see [114]). The part of the curve between points A and C is also unstable. Let 9a and 9c be the values of 9 corresponding to the points A and C respectively. Notice that, for the values of 9 in the range ( 9 c, Θα), the system admits two stable steady states whereas there is one stable state outside the said range. In order to see which of the two 232 12. Dissipative Atomic Systems states the system exists in, let Θ increase from 0 = 0. The system would follow the lower curve up to the the point A corresponding to θ = Θα- As the field is increased beyond the value corresponding to θ = Θα, the steady state of the system would jump on to B and follow the upper curve. This is reminiscent of the first order phase transition. If the field is decreased while the system is on the upper branch, the system would follow it till the point C corresponding to θ = θα where it will jump to D and follow the lower branch. Thus, transition from the upper to the lower branch takes place at a value of Θ that is different from the value at which it jumps from the lower to the upper branch. This exemplifies the phenomenon of hysteresis. We will see that whereas the exact steady state solution derived below does predict first-order phase transition, it does not predict hysteresis. The hysteresis is thus an artifact of the decorrelation which ignores quantum fluctuations. The S teady S t a t e for Μ = N = 0. The master equation in this case reads dp d i 19 S+ + S^, p — i<5 Sz, P S+S-, P + 7 ( 2S- pS+ - S+S - p - pS+S- j . (12.79) The steady state solution of this equation can be readily derived if <5 = 0 by rewriting it in the easily verifiable form ^ = 2 7 ( S. + iG) β (S+ - iG*) - ( 7 + w) (S+ - iG*) (5_ + iG) p - ( 7 - iv) P (S+ - iG*) + iG) , (12.80) where G = . (12.81) 7 + w It is st rai ght forward t o see t ha t pss - (£_ + i G) _ 1 ^5+ - iG* ) - 1 1 N = — Σ (iG)_m(—iG*)_n5'!n5'" (12.82) m,n—0 is the steady state solution of (12.80). Note that the inverse operators in the first line above exist and that the upper limit in the second line is restricted to N owing to the fact that S ± +rn = 0 for m > 1. The constant D is chosen to have Tr(/3SS) = 1. The solution for v = 0 was derived first in [115]. In the absence of an external drive, i.e. for g = 0, (12.82) reduces to Tr [5™55] = \-N/2){-N/2\. (12.83) 12.2 N Two-Level Atoms in a Monochromatic Field 233 It correctly shows that, in the absence of a drive, the atoms at absolute zero temperature settle to their ground state. Guided by the expression (12.82) for the steady state for <5 = 0, we let the steady state of (12.79) for <5 Φ 0 to be of the form N Σ /~ι om. on (12.84) Substitute this in the steady state version of (12.79) and write the detuning term as [117] -i<S q Qm ηπ &z ) = i<5 Sz, STS^ - 2ιδ q c m qn ^Ζϊ *-5+ q cm on nm on o O j O y = i<5(ra - m) - 2ΐδ Veri fy al s o t h a t 5 + 5 _, SZ*S" = ( m + 1 ) ( 2 & + m)S™Sl - (ra + l ) S™Sl ( 2Sz + ra). The damping part may be treated similarly to arrive at N Σ m,n = 0 | ((m + 1)(7 + i v) + id)cmn + i g(m + 1 )C'm+i„| x { m r 5" + 25'25'ϋι5'" } + h.c. 0 It is readily seen that this equation is satisfied if i<5 τη + 1 H------ —— 1 Cmn 7 + ™ / , i9(m + 1) ^ H--------; : L'm+l n — υ · Thi s is s ol ved by 1 C„ mini -ίδ 7 + i v' (12.85) ( 1 2.8 6) (12.87) (1 2.8 8) (iG)~m(—iG*)~nΓ (m + 1 - \φ) Γ (ra + 1 + ψ ), (12.89) (12.90) The exact steady state solution for v, δ Φ 0 was derived in [114]. The steady state for v = 0, δ φ 0 is derived in [117]. The averages may be evaluated by using (3.71). For the sake of comparison with the results derived above in the limit N 1, we express the averages in terms of parameters normal ized as in (12.71). We exhibit the behaviour of {Sz)/N as a function of Θ in Fig. 12.2 for Δ = .5, v = —5.0 corresponding to N = 10,30,60. Shown along with it is also the curve of Fig. 12.1 depicting the behaviour in the decor relation approximation expected to be valid in the limit N 1. The exact 234 12. Dissipative Atomic Systems Fig. 12.2. Steady state value of (Sz)/N as a function of Θ for A = .5, v = —5.0 and for N = 10 (solid line), N = 30 (short dashes), N = 60 (long dashes). Repro duced also is the plot of Fig.12.1 (dashed-dotted curve) depicting the results of the decorrelation approximation. average exhibits a tendency towards first-order phase transition in agreement with the decorrelation approximation. However, exact result does not predict hysteresis. The expressions for averages assume simpler form in the asymptotic limit N 1 if Ω = δ = 0. It has been shown that in this case, [115] Sx)=0, 'v//N=\, (SZ)/N=- ±V 'μμ - s, /N2 = 0, Θ < 1, (12.91) and Sz)/N = 0, Θ2 sin- 1 (1/0) 1 2 0 2 sin_ 1 (1/0 ) ’ 'yy 4 sin- 1 (1/0) 1 - 1 0 2 ^ 2 2 — T ~~T 4 4 302 02 sin V0 ^ T 0 2sin 1 (1/0 ) "( 1/0) 0 > 1. (12.92) This shows that the zero-variance approximation is valid if 0 < 1 but not if 0 > 1. However, the results found under the zero variance approximation, along with averaging for 0 > 1 mentioned above, are in agreement with exact asymptotic results [113]. We display in Fig. 12.3 the behaviour of (Sz}/N as a function of 0 for N = 10,30,60 and the asymptotic result derived above. 12.2 N Two-Level Atoms in a Monochromatic Field 235 Fig. 12.3. Steady state value of (Sz)/N as a function of Θ. The curves, in order of the lowermost to the uppermost, correspond to N = 10, 30,60 and the asymptotic result N 1. The plots for finite N exhibit the expected tendency of approaching the asymptotic one with an increase in the value of N. S t e a d y S t a t e f or δ = g = u = 0, Μ φ 0. Finding the steady state of (12.47) in this case is facilitated by rewriting it as ^ -β = Ί {ή + 1) ( 2RzpR\ - R\Rzp - p R\Rz) + 7 n ^ 2 R\pRz — R zR\p — pRzR\j . (12.93) where Μ = μν ( 1 + 2 n) exp(iφ), N = \μ\2η + (1 + ή)\ν\2, \μ\2 -\v\2 = 1, (12.94) Rz [|/x| exp(i0)5'_ + |i/|5'+ . (12.95) It can be solved analytically for its steady state if n = 0. Recall from Chap. 3 that the eigenvalues of Rz and hence those of R\ are m = —S, —5 + 1,· · ·, S with S = N/2. This implies that zero is an eigenvalue of Rz and R\ if N is even. The steady state of (12.93) for n = 0 then is Pss = IVoXVoI, &ζ\Ψο} = 0. (12.96) Since Rz is a linear combination of Sx and Sy, j-0o) is their minimum uncer tainty state. We have exhibited the behaviour of \(S,m\ipo}\2 in Fig.3.2 as a function of m. See also the discussion following that figure. However, if N is odd then zero is no longer an eigenvalue. But then R ~ 1 exists. As a consequence pss ~ R ^ R - ^, (12.97) 236 12. Dissipative Atomic Systems is the steady state of (12.93) for n = 0. For further details of its properties and numerical results, see [48]. The steady state of the master equation for <5 φ 0 and for some special values of the driving field has been derived in [48]. Those solutions exhibit several interesting quantum features. 12.3 Two-Level Atoms in a Fluctuating Field So far we have treated the field as a deterministic function of time. Now, we let the field to be fluctuating and assume that it is of the form E(t) = exp(—i0(f) — \uj\t) (E0 + e(f)) + h.c.. (12.98) where the phase 0(f) and e(f) are fluctuating variables, and Eq is a constant. The problem is analytically manageable if the reservoir is thermal, i.e. if M = 0. We assume that to be the case and rewrite the master equation (12.47) in the form pi = - i [(g0 + g(t)) exp(—i0(f))5+ + h.c., pi + Lpi. (12.99) The L in the equation above is time-independent. Define Wm = exp (—im0(f)) exp (i<j)(t)Sz) pi(t) exp (-i<j>(t)Sz) . (12.100) The Wm averaged over the distribution of 0(f), denoted by Wm, provides information about the expectation value of the operators averaged over the distribution of 0(f). For, 0 1 SqzSp_+m) = Tr p1 (t)S%S2SIL+m = Tr exp (+im0(f)) exp ( - i 0 ( f ) 5 2) Wm exp (i0(t )Sz) S ^ S i S p+rn = Tr \WmSp+S qzSp_+m . (12.101) Hence, any expectation value averaged over the phase fluctuations is deter mined by an appropriately chosen W m. For example, the expectation value of an operator that is diagonal in the eigenstates of Sz is determined by Wo and that the expectation value of S±, needed for evaluating the dipole mo ment, is governed by W Tχ. On substituting (12.100) in (12.99) we find that Wm obeys the equation = —i0[mWm - [5„ Wm]\ + [(a, + g(t))S+ + h.c, Wm +tWm, (1 2.1 0 2) 12.4 Driven Three-Level Atom 237 The equation for Wm averaged over the fluctuations of (bit) can be derived if <j>(t) is a delta-correlated Gaussian process. Let 4>{t) = 0, = 2 Ίρδ(ί - t’). Fol l owi ng Ch a p. 5 we f i nd t h a t 2s zw ms z - SzzWm - WmSi (go + g(t))s+ + h.c., w„ (12.103) - r n > ( r n f m -2 [S2,F m]) + LWm. (12.104) This may be solved analytically if g{t) is a constant. See [116] for details and the numerical results. The case of coloured φ(ί) can be handled by treating the equation for W by the method outlined in Chap. 5. For some numerical results, see [70]. 12.4 Driven Three-Level Atom Consider a three-level atom in the ladder configuration interacting with the field reservoir. Its evolution is described by the master equation (8.109). For the sake of simplicity, we let the reservoir to be thermal at absolute zero temperature. The evolution of the atom is then governed by (8.109) with N = M = 0 reading dt p = - 1 + 7 i — pA\\ +7 2 [2j 4i e/9j 4ei pAei + 2 7 x [j4iepAg + Ag\pAe A i i P ~ A e e P (12.105) I t s houl d b e b o r n e i n mi nd t h a t t h e t e r m mul t i p l y i n g c on t r i b u t e s as l ong as Ei — Eg ~ Ee — Ei (see Chap. 8 ). The equations for the operator averages read (^gi ) = —i ( A — -®g)(^gi) — 7 l ( ^ g i ) + 27x(^4ie) (Λθ) = - i (Ee - Ei){Aie) - (7! + 7 2 )(Λθ), (j4ge) = —i (Ee — Es )(Ase) — 72 <-*4ge) ( i gg) = 27 ι (Λί), ( l ee) = - 2 72( i ee). (12.106) These equations are easily solvable. Note that the equations for (Agi) and (Aie) are coupled only when ηχ φ 0. 238 12. Dissipative Atomic Systems The problem of resonance fluorescence from a three level atom in external field can also be dealt with by means of the methods developed in preceding chapters. That problem can be handeled analytically in the secular approxi mation. We refer the reader to [118] for details and further references. 13. Dissipative Field Dynamics In this chapter we solve some standard master equations describing evolution of a single and a two-mode field interacting linearly or non linearly with a reservoir. The non linear absorption considered here is a two-photon process. 13.1 Down-Conversion in a Damped Cavity Consider a nonlinear medium placed inside a cavity. Let it be pumped by an external field of frequency 2 ω0 such that it causes generation of two photons of frequency uiq each by the process of down-conversion. The hamiltonian (7.63) for the process of non degenerate two-photon down-conversion assumes the form when the process is degenerate. We assume the cavity to be tuned to u/q. Let the cavity mode in question be coupled linearly also to an external drive. The hamiltonian governing this process is Let the cavity mode interact also with a reservoir which may possibly be squeezed. We will see shortly how such a squeezed reservoir is generated. The interaction with squeezed reservoir is described by (8.82). We assume that the frequency of the pump driving the reservoir is same as the frequency ωο of the cavity mode. In the interaction picture defined by pi(t) = βχρ(ϊω 0ά^ at) p(t) exp(—ϊω0ά^άί), (13.3) the density operator then obeys the master equation [G* exp(2iw0 0 ®2 + G'exp(—2 ίω0 ί)β^2] (13.1) Hex t = h [g* exp(iω0ί)ά + g εχρ(-ίω0 ί)ά+] . (13.2) ^ P i = - i [g*a + ga*, Pi] - ^ [Cat 2 + G*a2, p{\ +k( N + 1) [2apiat — p\a)a — <i)api\ +κΝ [2a^pici — p\<ia) — άίύρ{\ + κ [Μ (2 άριά — pi & 2 — ά2ρι) + h.c.] . (13.4) We assume further that 240 13. Dissipative Field Dynamics See [119] for details of the theory of the process of down-conversion in a thermal bath ( M = 0). We solve (13.4) by converting it in to an equation for p(a, a*) = (a\p\a) by taking its matrix element in the coherent state |a) and by using the relations (4.22)-(4.23). It reads d-pi(a,a*) κ > |G|. (13.5) di 1 + 2 +K 9 d da* 9 d_ da pi (a, a*) n * 9 n 9 Q _ Q a Oa* oa - ( i G' + 2« * ) £ 2 + (iG - N d2 d d 2{N + 1 ) d ^ + d ^ a + d a a pi{a,a*) pi(a,a*) Pi(a, a*) = Li (a,a*)pi (a,a*). We f i r s t e xa mi ne t h e e qua t i ons for t h e aver ages o f a, a*. ( 13.6) 1 3.1.1 A v e r a g e s a n d Va r i a n c e s o f t h e C a v i t y F i e l d Op e r a t o r s Us i ng ( 5.32) we f i nd t h a t d_ d i ( «) ^ _ ( a*) —κ —i G i G* - κ ( a ) <«*> — 1 9 ~9* (13.7) We s ol ve t h e s e e q u a t i ons by a ppl yi ng t h e r e s u l t s of t h e Appe ndi x C ( C.5) a nd f i nd t h a t (G = |G| exp(i0)) <“ (*)> (“ *(*)> = exp(—Kt) c os h ( | G| i ) i e x p ( —ΐφ) sinh(|G|i) —iexp(i0 ) sinh(|G|i) λ cosh(|G|i) J ( a (0 )) (a*( 0 )) + i | ^> i | S). \S) = ng + i Gg* - Kg* + i G*g ( 13.8) ( 13.9) κ2 - \G\2 N o t e t h a t t h e t i me e v o l u t i o n i s g o v e r n e d b y e x p [ — (κ ± |G|)i]. This, by virtue of (13.5), describes damped evolution. As a result, the system decays towards a steady state given by the last term in (13.8). Next we examine the evolution of the variances Δ α 2, Δ α * 2, A\a\2 in a 2, a*2, j« j2. It is straightforward to show that ^ Δ α 2 = - 2 κ Δ α 2 - 2iGZ\|a| 2 + iG - 2 nM*, ^ Δ α * 2 = —2κΔα* 2 + 2 i G M | a| 2 - iG* - 2kM, ^ Z i | a | 2 = —2κ|α| 2 + i(G M a2 - ΰ Δ α * 2) + 2κ( Ν + 1). (13.10) These equations can be combined to see that the equation for G*Δ α 2 +GΔ α * 2 obeys a first-order equation which is solved by ΰ*Δα 2 (ί) + ΰ Δ α * 2 (ί) = e x p ( -2 nt) {G*4a2 {0) + GZla*2 (0)} - ( l - e x p ( - 2 Kt)) [G*M* + GM] . (13.11) The equations for G*Δ α 2 — GAa * 2 and Δ\α\2 are coupled reading d_ ( G*Δ α 2 - G Δα*2' At V ^ M 2 -2k —4i|G| 2 \ ( G*Δ α 2 - ΰ Δ α * 2 i —2 κ J y Δ\ι We sol ve t h i s s e t by r ecal l i ng ( C.5) a nd f i nd t h a t ΰ * Δ α 2 ( ί ) - ΰ Δ α * 2 (ί)' A\a\2 (t) = <W -2 K f\( cosh(2 |G|i) 2i|G| sinh(2|G|i) \ ' ; ^ i Sinli(2 |G|t)/2 |G| cosh(2|G|i) ) 13.1 Down- Conver s i on i n a Dampe d Cavi t y 241 G* Z\a2 (0) — G/\a*2 (0) , Ζ\|α|2 (0) '+Ι ΰ ) ~ IB), (13.13) im 1 ( 2 ik\G\2(2N + 1) + 2 k 2( GM — G*M*) \ 1 1 2(k2 — \G\2) \ |G| 2 — 2 k2( N + 1) + i k( GM — G*M*) J ' 1 ’ We r e s t r i c t o ur a t t e n t i o n now t o a t h e r ma l r e s er voi r. Thi s c or r e s ponds t o M —> 0 and N —► n where n is average number of thermal photons in the reservoir. From (13.11) and (13.13) it follows that, in the steady state, ( Δα 2 (οο) \ _ 1 / - i G « ( 2 r a + l ) \ V 4 | a |2 (oo),/ 2(K2 -\G\2 )\2 n 2 { n + l ) -\G\2)' { ' > Not e f r om ( 13.8) t h a t i f g = 0, i.e. if the cavity mode is undriven then (a(oo)) = (a*(oo)) = 0. Assume that to be the case and compute the steady- state value of the squeezing parameter = ^((exp(—iV>)a + exp(iVO^)2). (13.16) We recall that S = 1/4 in a coherent state and that S < 1/4 implies squeez ing. While evaluating Ξ(ψ), bear it in mind that since p(a, a*) is a Q-function, 242 13. Dissipative Field Dynamics the average (a*man) corresponds to (άηά+7η). Thus ( | a | 2) corresponds to (aat). Verify in particular that if 2ψ + φ = n/2 then s = i M - (1317) This shows that the field is definitely squeezed if n = 0, G φ 0 and that the presence of thermal photons spoils squeezing. Since |G| is restricted by (13.5), it follows that the maximum value of squeezing for n = 0 is 1/8. This amounts to 50% squeezing over the value 1/4 of S' in a coherent state. We leave it to the reader to use the results derived above to obtain two- time correlation functions of a and ά^. It will be found that the correlations (ά(ί)ά(ί + t ) ) and (άΐ(ί)ά+(ί + τ)) are proportional to |G|. Furthermore, use (A.8 ) to show that the two-time correlations reduce to delta-functions in time as κ —► oo. We thus find that the field generated in the process of down-conversion in an ordinary bath is squeezed provided the number of thermal photons is low enough. The output field is also squeezed. The exact relationship between the cavity field and the output field requires further deliberations. We refer the reader to [93], [120]—[123] for a detailed account of the relationship between the cavity and the outcoming fields. The correlations in the output field are the same as in (8.65) with G —► ξ. We wi l l see t h a t t h e knowl edge of t h e aver age s a nd t h e var i ances der i ve d above i s suf f i ci ent t o c o n s t r u c t c ompl e t e s o l u t i o n of t h e ma s t e r e qua t i on. 1 3.1.2 D e n s i t y Ma t r i x I n t h i s s e c t i on we sol ve ( 13.6) for p(a,a*,t). Th e f or ma l s ol ut i on of (13.6) i s p(a,a*,t ) = exp ^Li(a, α*)ί^ p(a,a*, 0). (13.18) Express p(a,a*, 0) as p{a,a*, 0) = j ^ α ορΜ 0 ) δ ^ ( α - α ο). (13.19) On combining this with (13.18) we obtain p(a,a*,t) = J d2 a0 K(a,a*,t -,a0,otQ,0 )p(a0,oio,0 ), (13.20) where the kernel K is defined by K(a,a*,t; αο,^οιΟ) = exP (-^ι(α ι α*)ί) — αο)· (13.21) On using the representation (A.2) of the delta-function, (13.21) reads K( a,a *,t;a0,aQ,0) = ± J d2£exp[—i(£c*o + Γ«ο)] exp (% t ) exp[i(£a + £*«*)]· (13-22) 13.1 Down-Conversion in a Damped Cavity 243 Now, note that Li ( a,a*) is a linear combination of the operators d/d a, ad/da, a*d/da, d2/d a 2, d2/dada* and their complex conjugates and that this set of operators is closed under the operation of commutation. Hence, the exponential in (13.22) can be expressed as a product of the exponentials of the said operators. The scalar coefficients in each of the exponents may be determined by the method outlined in the Chap. 2. There is, however, no need to follow this tortuous path! Instead, we observe that the exponentials of the said operators acting on the exponential function on the right in (13.22) reduce it to the form K (a, a*,t\ a0, a^, 0 ) = - ^ J d2£ exp (Βξ x exp [ϊ ξ(/ια + f 2a *2 + Β*ξ2 - ά\ξ\2) ~ a0 + /o) ~ c.c.]. (13.23) This shows that, if the coefficients of ξ,ξ*, |ξ| 2 satisfy appropriate conditions for the integral in (13.23) to exist then the kernel is a Gaussian in a, a*. Now, recall that a Gaussian is expressible in terms of averages and correlations of the variables as in (5.17). Since the variables in that equation are real, we apply it for the real and imaginary parts, Χχ = ( α + α* )/2 andx2 = ( a —a * )/2 i of a and express the Gaussian kernel as K (a,a* ,t-,a 0,a q,0) 1 exp - — V TMV 8 χ (13.24) V M a — (a) + c.c. —i (a — (a) — c.c.) —Δ(α — α * ) 2 \Δ (a2 — a*2) iZ\ ( a2 — a' 2) Δ { α + α*Υ ( 13.25) V T being the transpose of V and X = ( Δ\α\2 ) 2 - Δ α 2 Δα*2. (13.26) The averages and the variances in the expressions above are given by (13.8), (13.11) and (13.13) with the understanding that the variables a, a* have fixed values a 0, a$ &t t = 0. Hence, (a(0)) = ao in (13.8) whereas Δ α 2 (0 ) = Ζ\α*2 (0) = Ζ\|α|2 (0) = 0 in (13.11) and (13.13). We have thus at hand the expression for the Q-function corresponding to the density operator satisfying (13.4). The kernel for any other quasidistri bution would also have the form (13.24) except that the equations for the averages and the variances are to be derived from the master equation corre sponding to the desired quasiprobability. Alternatively, use the relationship (4.12) between the quantities in question for different quasiprobabilities. The averages of a, a* are same for all the quasiprobabilities but the average of \a\2 is different. For example, recall that ( | a | 2) represents (άά+), (ά+ά) and (flat + aa))/2, respectively for the Q, P and the Wigner functions. Hence, the expression for p(a,a*) stands for any ordering by assigning the value to zl|o;| 2 appropriate to that ordering. 244 13. Dissipative Field Dynamics Density Matrix in Number State Representation. In many applica tions, we need matrix elements of the density operator in the number states. These can be derived by noting that j Qm+n (m\p\n) = -— = exp( H2 )p(a,a*) v t t! da*mdan ex.—ex.* =0 ( 1 3.2 7 ) A s a n i l l u s t r a t i o n, w e c o n s i d e r a t h e r m a l b a t h c h a r a c t e r i z e d b y M = 0, N = n. We also let G = 0 and find that (a) = da ο, Δ α 2 = 0, X = Ζ\|α| 2 = (η + 1)(1 — d2), d = exp (—at). (13.28) Evaluate the kernel by substituting these expressions in (13.24)-(13.26). Com bine the result with (13.20) to show that p(a,a*,t ) /■ = — / d Qo exp πχ a - e x p ( - Kt ) a0 /χ p(a0, a j, 0 ). (13.29) Now we evaluate (m\p\n) by substituting this in (13.27). The operation of differentiation leads to dn da* exp a0d(c X Qn da’ exp - { « * ( 1 - X ) ~ a * d } X a —a *=0 = ( - * ) n e x p ( exp(a0 a*d/x) {a*( l - χ) - a*0 d}r χ J da- { χΥηΒχρ{-\α0\2ά2Ι χ ) Σ { - γ a* = 0 mini .( t o — k)l(n — k)lkl ( i - x ) * x(a*0 d)n- k(a0 d/X)m- k j. Now, rewrite ρ{αΰ,α$,ϋ) as p(a0, Qq, 0 ) = e x p ( - | a 0|2) Σ ~ ^ αοΡαο{ρ\ρ(0 )\<ΐ)· (13.30) (13.31) S u b s t i t u t e t h i s, al ong wi t h (13.30) i n ( 13.27). Ca r r y t h e i n t e g r a t i o n over α ο,α ο as s umi ng m > n. Change the summation over k in (13.30) to n — k. Rewrite the resulting sum over k in terms of the hypergeometric function to obtain (m\p{t)\n) = /to! 1 nl χ + d2 Έ χ ι ( 1 - χ _1)"(χ + d2 )n~mdm~n (m — n + l)l (to — n + Z|p(0)|Z) 11 (x + d2 ) 1 x F ( m — n + l + 1, — n; m — n + 1; x) (13.32) x = d2/(X + d2)( 1- χ). (13.33) If n = 0 then we note that x = 1. The value of the hypergeometric function F(l + 1, —m,n — m + 1,1) is given by (A.38). The expression (13.32) then assumes the form (m\p\m) = exp(—2 nmt) V P' - ( 1 - ex.p(-2 Kt))p~m(p\p\p). (13.34) ' rn\(p — to)! p V' W e h a v e e m p l o y e d t h i s e x p r e s s i o n i n C h a p. 1 4 i n t h e c o n t e x t o f t h e t h e o r y o f a m i c r o m a s e r. 1 3.2 F i e l d I n t e r a c t i n g w i t h a T w o - P h o t o n R e s e r v o i r 2 4 5 1 3.2 F i e l d I n t e r a c t i n g w i t h a T w o - P h o t o n R e s e r v o i r I n t h i s s e c t i o n w e d i s c u s s t h e m e t h o d o f s o l v i n g t h e m a s t e r e q u a t i o n o f a s i n g l e o r a t w o - m o d e f i e l d i n t e r a c t i n g w i t h a t w o - p h o t o n r e s e r v o i r. W e c o n s i d e r f i r s t t h e e v o l u t i o n o f t h e f i e l d o n l y u n d e r t h e p r o c e s s o f t w o - p h o t o n a b s o r p t i o n. W e f o l l o w i t u p b y i n c l u d i n g a m e c h a n i s m t h a t g e n e r a t e s t h e f i e l d i n a t w o - p h o t o n p r o c e s s. 1 3.2.1 T w o - P h o t o n A b s o r p t i o n T h e i n t e r a c t i o n o f a t w o - m o d e f i e l d w i t h t w o - p h o t o n r e s e r v o i r i s d e s c r i b e d b y t h e m a s t e r e q u a t i o n s ( 8.1 1 6 ) w h e r e a s t h a t o f a s i n g l e - m o d e f i e l d i s d e s c r i b e d b y r e p l a c i n g b by a in that equation. By virtue of the relations (2.71) and (2.76) the master equation for both the cases can be expressed in terms of the SU(l, 1) operators K ±,K Z. In the interaction picture generated by the free-field hamiltonian, those equations are of the form — ρ = —ϊη Κ+Κ-, ρ + κ(η + 1) 2Κ_ρΚ+ — ρΚ+Κ- — Κ +Κ- d τ L J L +κή \2Κ+ρΚ- - ρ Κ - Κ + - Κ- Κ+ρ . (13.35) It is convenient to employ the eigenstates {\χ(τη, K) ) } of K z for a fixed eigenvalue label K of Q as a basis. However, we may require the expectation value of such combinations of field operators that do not commute with Q. We, therefore, need to know (χ(ηι, Κ2 )\ρ\χ(η, K{)) even for Κ γ φ K2. We al s o obs er ve t h a t i f we t a k e t h e ma t r i x el e me nt of (13.35) i n t h e s t a t e s | x ( t o, K i )) and |χ(η, K 2)) then the value of to — n in each term is same. Bearing these observations in mind, we solve (13.35) by taking its matrix elements between the states \x(m, ΚΛ )) and \χ(τη + ρ,Κ2)} where K x and K 2 may be different. The corresponding eigenvalue equation, in terms of the function 246 13. Dissipative Field Dynamics F “ ° M m · K M M m + ρ· *»» (13·36) reads (κ = 1 ) p(p) (! - i’lii'm+pf*'2 ) + ( 1 + ί ^ μ ^ Κ χ ) - η {p2m+p + p2m} - λ {ρ) 2(n + 1 )(Κχ + m + 1 )(K2 + m + p + 1 ) F{p)+ 1 2 n( K\ + m + 1 )( K2 + m + p + 1 )F^ } _ 1 = 0, (13.37) μτη(Κ) = \Jm(2K + m — 1). (13.38) On comparing this with the standard form (10.41) we find that this equation does not fall in to any analytically solvable class unless n = 0. If n = 0 then (13.37) reduces to the two-term recursion relation ^m+i,pPm+l,p βητ,ρΡητ,ρ — Pm}p (13.39) for pm,p = {χ{τ η,Κι )\ρ\χ{τ η+ρ,Κ2)) with = 2prn_irp{K2)Pm(.Kl)i βτη,ρ = (1 - ^ ) p 2 m+p{K2) + (1 + i?7)μ2 ηι( Κ1). (13.40) By following Sec. 10.3, we note that the eigenvalues are given by λ£ρ) = - βη,ρ. (13.41) Verify also that the eigenvalues are distinct and that their corresponding eigenvectors are Pm}p = to > n + 1, (n) = ________________ ( C p + 2 n ) r ( Cp + m + n)____________ ( 1.1 4 9 ) m ’ p (ra - m)l y/ml ( m + ρ)ΙΓ ( 2 K r + to) Γ ( 2 K 2 + m + p ) ’ for t o < n, wi t h (Cp + 2 n) as the normalization factor (see (13.46) below), Cp = Κι + K 2 + p - iη{Κχ — K2 — p). (13.43) Note that the adjoint of (13.39) is ^■τη,ρΡτη—Ι,ρ βτη,ρΡτη,ρ ~ λ ^ Ρτη,ρ (13.44) (η) Following Sect. 10.4, the eigenvectors pm’p are given by P Wp = 0, to < η - 1, -(«) = ( ^mV/(m + n)!(m + n + P)! Pm+n,p \ ) m\r(Cp + 2n + m + 1) x y/r ( 2 K x + to + η) Γ{2Κ2 + m + n + p). (13.45) We ascertain the orthogonality of the sets (13.42) and (13.45) by evaluating 13.2 Field Interacting with a Two-Photon Reservoir 247 n n (Cp + 2 n) Γ (Cp + n + fc) r ( C p + 2 k + l ) ( n - k )\ x F (Cp τι -(- k, —τι k; Cp 2k 1; 1) (n - k)\(k - η)\Γ (Cp + n + k + 1) (Cp + 2 n) (Cp + n + fc) (13.46) The density matrix at any time t is then given by (x(m)\p(t)\X(m + p)) = ^ [exp Ρ^,ρΡΖρ 1,η=Ό x { x ( m,K 1)\p ( 0 )\x ( m + p,K 2)). (13.47) As explained above, this enables us to obtain the expectation value of any field operator whether it is an SU(1,1) operator or not. For a different method for evaluating p(t) for η = 0, and for some numerical results, see [95, 124], 13.2.2 Two-Photon Generation and Absorption Consider now the situation wherein the field, in addition to being absorbed in a two-photon reservoir, is generated by another two-photon mechanism. As discussed in [55], this situation depicts closely the experiments of [125]. Assuming that the frequency of the field driving the nonlinear medium to produce two photons is equal to the sum of the frequencies of the photons produced, the hamiltonian describing two-photon generation in the interac tion picture is given by for the two-mode field. These hamiltonians consist only of S U(1,1) operators. The master equation for the field, with (13.35) as the damping part (with n = 0 ), may be written as (κ = 1 ) H ^ ^ G a 2 + ( 13.48) for t h e s i ngl e- mode fi el d a nd by H2 = h[Gab + G*Pa}], ( 13.49) p = 2K ^ p K + — (1 + ί η)ρΚ+Κ - — (1 - ϊη) K+K_p, (13.50) (13.51) 248 13. Dissipative Field Dynamics If there exists a state \ipo) such that Κ~\ψο) — 0, i.e. if iG K- + 1 — 177 \ip0) = 0 then /3SS = I Vo) (Vo I (13.52) (13.53) is the steady state solution of (13.50). Equation (13.52) is the same as (3.120) whose solution is given by (3.121). For single-mode realization, (13.52) defines the pair coherent states [55]. For a discussion of numerical results, see [55]. 13.3 Reservoir in the Lambda Configuration The master equation for a single-mode field interacting with a reservoir in the Raman configuration is given by replacing b by ά in (8.118). It is a function of only the number operators ά^α. It is trivially solvable in the number states representation. We consider the two-mode case dissipating according to (8.118). Let there be a nonlinear medium pumped by an external field such that it gnerates and absorbs the modes in question in the Lambda configuration. The modes in the interaction picture then evolve according to the master equation —β = — ie a^b+b*a, β + κ ( η + 1 ) 2α^%β%^a — β%^aa^b — b^aa^bp^ +κη [26^ά/3ά^6 — βα)Μ^α — . (13.54) We may use the eigenstates {\m,N — m)}, corresponding respectively to the eigenvalues to and N — m οϊ a)a and Wb as the basis states. By following the arguments similar to the ones in the last section, we note that we need to evaluate the matrix elements of β between states of different values of N. We also observe that if we take matrix elements of (13.54) in the states |τη,Νχ — to) and |n,iV2 — n) then each term in (13.54) preserves the value of to — n. We, t he r e f or e, t a k e ma t r i x e l eme nt s of (13.54) i n t h e s t a t e s | m,i Vi — t o) a nd | m + p,N2 — m — p). By following the method of the last section, we find that the analytical solution of (13.54) may be derived by the method of Chap. 10 if ή = 0. Assuming that to be the case, the eigenvalue equation corresponding to (13.54) reads (e = 0, κ = 1) - / W $ = Xn ‘ <13·55) amp = 2\Jm(m +p)(Ni - to + 1 )(N2 - m - p + 1), / ]\f. + No \ Pmp = 2 m l m - p - 1 J +p (N2 - p - 1) + Νχ + N2 +ιη [m (Νχ - N 2 + 2 p ) - p (N2 - p)]. (13.56) By following Sect. 10.3 we note that the eigenvalues are given by = -βηρ- (13.57) The corresponding eigenvectors being Pml = ° ’ m < η - 1, „(.) = & ______________t r ___________ mp n (m — η)\Γ (m + η + p — S — iDp) 13.3 Res ervoi r i n t h e Lambda Conf i gurat i on 249 m\(m + p)\ m > n, (13.58) γ ( Νχ — m )\( N 2 — m — p ) V where An is the normalization constant and Ν χ + Ν 2 ( Νχ - N2 ^ S = ---------1, Dp = η I ----- +p 1. (13.59) It may also be confirmed that the eigenvectors of the equation a^ixpPm+xp - f&pfal = A (13.60) adjoint to (13.55) are = m > n + l, pW = ( - ) ” rrnp (n — m)\r (S — p — m — n + 1 + iDp) ( Νχ - m)\( N2 - to - p)\ x y -----------u , M m < n. (13.61) y to !( to + p )! The scalar product between the eigenvectors (13.58) and (13.61) is V p(k) = V p(n)p(h) / j r m,p r m,p / j r m p r m p m —0 m ~ k An Γ (2k — S + p — 1 — iDp) (n — k)\F (S — p — 1 + iDp —n — k + 2 ) x F (—n + k,n + k — S + p — 1 — iDp\ 2k — S — p — 1 — iDp\ 1) A = Γ (S - p + 1 + iDp - 2n) Γ (2 n - S + p - iDp) Snk' ^13'62^ This determines the normalization constant. As discussed in Chap. 10, these eigenvectors are sufficient to determine the solution of (13.54) for a given p if the corresponding eigenvalues are nondegenerate and that we need to find 250 13. Dissipative Field Dynamics the generalized eigenvectors in case of degeneracy. We note that, for Νχ = N2 = N, = A ^ n_! if P = 0 , i.e. the eigenvalues are two-fold degenerate. The solution of the equation in this case requires a knowledge of the general ized eigenvectors corresponding to two-fold degenerate eigenvalues. However, we do not undertake the tedious but straightforward task of construction of generalized eigenvectors. 14. Dissipative Cavity QED In this chapter we illustrate the general methods developed in previous chap ters by applying them to a dissipative atomic system in a dissipative cavity. Except for atoms modeled as harmonic oscillators, this system can not gen erally be treated analytically exactly. In this chapter we consider a system of two-level atoms. We pay attention to theoretically and experimentally inter esting situation of strong atom-field coupling. We discuss the ways of probing the characteristic features of such a strongly coupled atoms-cavity system by means of an external drive. Finally, we outline the method of treating the situation wherein atoms are pumped in to a cavity one at a time. This con stitutes what is known as a micromaser. 14.1 Two-Level Atoms in a Single-Mode Cavity We consider a system of N identical two-level atoms of transition frequency ωο coupled to a cavity mode of frequency ω. The corresponding hamiltonian is H = h ωα^α + u>qSz + H a.- f, (14.1) Ha- f = gh &*§- + S+a (14.2) We a s s ume t h a t t h e c a vi t y mode i n que s t i on a n d t h e a t o ms a r e coupl ed t o a t h e r ma l b a t h ha vi ng an aver age of n photons. The master equation for the density operator p of the combined system of the atoms and the cavity mode in the interaction representation generated by ω(ά^ά + Sz) then reads (with δ = ωο — ω) ^ [H0, p] + Atp + λ Άρ, = Lp, (14.3) (14.4) Af = κ(η + 1) [2άρα^ — ρά^α — α*αρ\ +κή [2ά^ρα — ραα* — αά)β\ , (14.5) 252 14. Dissipative Cavity QED i a = 7 (n + 1 ) 2S- pS+ - pS+S_ - S'+S'.p + 7 « 25'+/5S'_ — p S - S + — SLS'+pJ . (14-6) The equation (14.3) may be solved by converting it in to a c-number equation by taking its matrix elements between the the states {|τη,Μ) } where |m) is a number state of the of the field and |Μ) (M = —N/2, —N/2 + 1, · · ·, N/2) is an eigenstate of Sz. Its steady state solution can be found by applying the principle of detailed balance (see Sect. 12.2). Else, verify by direct substitution that if n — 0 then the steady state is given by pss= \0,-N/2)(0,-N/2\. (14.7) The steady state for η φ 0 reads Pss = exp[-/3(afa + 5Z)], exp(-/3) = ■ (14.8) n + 1 The time-dependent solution of (14.3) even for a single two-level atom is a formidable task except when n = 0 and the eigenvalue of ά^ά + Sz in the initial state is ±1/2 [127]. However, some situations of practical interest in quantum optics that can be tackled much more conveniently by approximation methods include (i) the · limit of strong atom-field coupling compared with dissipation, and (ii) weak atomic coupling with an external drive. The former enables us to invoke the secular approximation whereas the latter can be handled perturbatively. 14.2 Strong Atom-Field Coupling The situations in which the reversible atoin-cavity coupling is very much stronger than irreversible dissipation are of immense interest. Recall from Sect. 8.2 that such situations can be treated conveniently by the method of the secular approximation. We outline this method (i) for a single two-level atom when the atom and the field are in an arbitrary initial state, and (ii) for N two-level atoms in particular initial state. 14.2.1 Single Two-Level Atom Recall that the secular approximation consist in retaining only those terms in the dissipative operator which are time-independent in the interaction picture generated by the strong reversible part. The reversible part in the present case is the hamiltonian Ho- We express the density operator of the system as a linear combination of the eigenstates of Ho- For the sake of illustration we let <5 = 0. Corresponding eigenstates |ipm^) and the eigenvalues μ^ = ±gy/m + 1 of Ho are given respectively by (11.32) and (11.33). The density operator may then be expressed as 14.2 Strong Atom-Field Coupling 253 p = |0,-l/2)(0,-l/2| + ]T ^^-^’-VSKV^I+h.c.} m=0 a=± + Σ Σ σ£Λ&°χΛ m,n —0 a,/3 —± (14.9) We, therefore, derive the equation for \ψ ^ ) ( ψ ί ^\ by letting ρ —> \· φ^ ) ('φ^\ in (14.3). The secular approximation consists in retaining only those terms in [ i a + i f ] | v 4 a) )('φη>'> | which have same time dependence as jipm ’^{'ψη ^Ι under the action of the evolution generated by Hq. This means that if \'φρί'>){'φ^δ'> \ is a term in the expression for [ΑΆ + Af]\%pm'>) {‘φτι'> | then it may be retained in the secular approximation if — μ ^ = μ $ — ■ The action of the damping operator on a dressed state may be found by first writing the dressed state in terms of the bare states |ra, ±1/2). Use the knowledge of the action of the operators on the bare states and rewrite the resulting bare states in terms of the dressed states. As an illustration, we determine the action of a on |V>m^)· Using (11.32), we find that, for ra = 1, 2,..., “IV’m '*) = [Κ V2) ±\m + l, -1/2)] = ~^= [y/m\m — 1,1/2) ± \/m + l|ra, —1/2)] v 2 = ^ l ^ l ) - ^ l ^ l ) r m ] = \ [Vm + 1 ± Vrn] , ά|4±)> = ± ^ l °,- l/2 ) · In the same manner show that 1 2 Bearing these considerations in mind, it straightforward to see that the master equation (14.3) for n = 0 = 7 = 0, ω = ωο leads to the following equations in the secular approximation ^ I V ^ X ^ I = - [i(/4n"} - μ (ηβ)) + κ(τη + η + 1 )] (14-14) (14.10) (14-11) (14.12) (14.13) m φ n and any a, β = ± or a φ β and any ra, n, d:|0,- l/2 ) ( ^ }| = \ΐ μ ^ - κ ( η + 1/2 ) 1 |0,- 1/2 )< ^ > |, di _d di |0, —1/2)(0, —1/2| = 0. (14.15) (14.16) The equations for the diagonal components read, ra = 1,2,..., d 254 14. Dissipative Cavity QED di di - ( m + l/2 ) h/i +)X</4+ ) l \i>L }XV4 2 k ^ι^Λχ^Λι+^“)2ι^+Λχ^ιΐ — (πι + 1/2)\'Φ{ηι ))(Ψί - ) l ( 1 4.1 7 ) (14.18) Wh e r e a s t h e e q u a t i o n (14.14) for of f - di agonal e l e me nt s i s s i mpl e, t h e e q u a t i ons ( 14.17) a n d (14.18) c o n s t i t u t e c oupl e d s e t of r e cur s i on r e l a t i o n s for t h e d i a g o n a l el eme nt s. Th e i r s um a nd di f f er ence l e a d t o t h e e qua t i ons ( wi t h r = 2 κΐ) d d r a nd F+(t) — — ^ra + F+ + ^ra + ί ’+_1, ' ( m + + V/^ ”(m + it*-® (14.19) (14.20) for -)i (14.21) The equations (14.19) and (14.20) are solved easily by following the Sect. 10.3. The resulting expressions for F^(t ), which may also be verified by direct substitution in the respective equations, read Fm{t) = exp ( — (2 m + ί)κί) Γ (ra + 3/2) (exp(2/ti) — 1))* Σ n = 0 (ra — η)\Γ (n + 3/2) -F+( 0), (14.22) and Fm(t) = exp ( —(2ra + l)nt) m »Σ n = 0 + 1 ra! (e2Kt - 1 )* + 1 (m — n)\n! ~F-{ 0 ). (14.23) The solution of the equations above, on substitution in (14.9) provides the description of the system in the secular approximation. For some detailed numerical results see [103]. For a different approach to the strong-coupling limit, see [128]. 14.3 Response to an External Field 255 N Two-Level Atoms. In order to apply the secular approximation in this case, we need to know the eigenvalues and the eigenstates of Ho for a general value of N. As stated in Sect. 1 1.2, the eigenvalue problem of H0 can be solved analytically only if the eigenvalue M of a)a + Sz is small. We let N be arbitrary and M = —N/2, —N/2 + 1, —N/2 + 2. We also let ω = ujq. Th e s t a t e | 0, —N/2) is the only eigenstate of H if M = —N/2. It corre sponds to the eigenvalue —ΝΤιω/2. For M = —N/2 + 1 there are two eigenstates of H: 1 ^ ) = [| 0, —N/2 + 1) ± |1, —N/2)\. (14.24) The corresponding eigenvalues of H are Η(—Νω/2 + μ^), μ^ = ω ± 3\/ΪΫ. (14.25) Thus, if the initial excitation is —N/2 + 1 then the iV-atom system behaves mathematically like a two-level one. Next, for M = —N/2 + 2, H admits three eigenstates: 3 N \Ψί) = ^ 2 aij\j - i, - y + 3 - j ), (14.26) j =i w h e r e a^j are the elements of the matrix A defined by /y/( N - l )/2 ( 2 N - l ) 1/V2 \JN/2(2N - 1) \ - ^/Ν/2 Ν - 1 0 (TV — 1)/2N — 1 (14.27) V ^ { N - \)/2 {2 N - 1 ) - 1/λ/2 y j N/2 (2 N - 1 ) / The eigenstates (14.26) correspond to the eigenvalues Η(—Νω/2 + μ») (i = 1,2,3) where μ ι,3 = 2u>o ± gV4.N — 2, = 2ωο· (14.28) The inverse of the relation (14.26) is A T 3 U-l, - y + 3 - j ) = Y^ai j l ^i ). (14.29) If the initial excitation of the system is M < —N/2 + 2 then it is confined to the subspace formed by the eigenstates constructed above. We use these results in the next section for determining the atomic response to an external field. 14.3 Response to an External Field A useful means of probing the characteristic features of atom-cavity interac tion is by monitoring its response to a weak external drive. It is determined 256 14. Dissipative Cavity QED by the method outlined in Chap. 9. Recall that it consists in evaluating an appropriate susceptibility of the system. We use that approach to examine the response of a system of two-level atoms in a cavity to (i) a monochromatic field, and (ii) a bichromatic field. 14.3.1 Linear Response to a Monochromatic Field We let the atoms to be driven by a monochromatic field of frequency v. Recall from Chap. 9 that the rate of absorption of the energy by the system from the field is given by (9.26): Wa(v) ~ Re T¥<Js_ ( l + iv) 1 [S+, A»] j λ/iVReTr S- (L + iv) 1 |0, - N/2 + 1) (0, —N/2\ (14.30) wher e we have us e d t h e f a ct t h a t pss is given by (14.7)). The expression (14.30) can, of course, be evaluated exactly by using the results of the last section. However, simpler expression is obtained in the secular approxima tion. To apply it, we invoke (14.24) to express 10, —N/2 + 1) in terms of the' dressed states. Then, following the procedure outlined in the last section, it is straightforward to see that, in the secular approximation, ( I + iv) 1 ^ X 0,- y l = A t l ^ X O - N/2\ A ± = - i (ω0 - ν ± g\/N) - K +^ - N The equation (14.31) yields ( L + i v) (14.31) (14.32) - 1 N N |0,- - + 1 )(0,- - | ί /V = - ^ [ Α - ι\ψ+)+Αζι\ψ0)} <0, — — | = ^ [{A- 1 + A^jlO, - N/2 + 1) + {A;1 - A l ^ l, - N/2) ] (0, - N/2|. (14.33) On substituting this in (14.30) it follows that the rate of absorption of energy from the applied field is proportional to x 21 - 1 κ + 7 N W a ( v ) (ω0 - ν - gy/N) + (ω0 -ι> + g VN) + ^ ~ · 27/V) (14.34) 14.3 Response to an External Field 257 The rate of absorption thus exhibits resonances at ωο ± g V N as a function of the applied frequency v. Compare it with the case of an atom in free space. In that case, the absorption exhibits resonance at the atomic transition frequency ωο· The splitting of the absorption resonance at ωο in a damped cavity to two resonances at ωο ± g\/N is referred to as the vacuum field Rabi splitting. The reader may verify by numerical evaluation that the exact expression for for -χ/ϊν|ί?| κ,Νη is in good agreement with the one derived above in the secular approximation. 14.3.2 Nonlinear Response to a Bichromatic Field Next, we examine the nonlinear response of a two-level atom coupled strongly with a single cavity mode to a bichromatic field consisting of frequencies ωχ,ω2. This coupling is described by the hamiltonian given by (12.42). We are interested in the response at the frequency Ω = 2ωχ — ω2. It is characterized by the susceptibility given by (12.43) read with (12.44). The superoperator L in this case is given by (14.3) plus the Lc where Lc, defined in (8.106), incorporates effects of collisions that the atom in question undergoes with other atoms. We assume ή = 0 so that pss is given by (14.7). Recall that in Chap. 1 2 we investigated this process in free space. We evaluate χ(3 )(α;ι,α;ι, —ω2) numerically and compute the signal 5 ( β ) = \χ^ ( ωχ,ωχ,- ω2)\ (14.35) as a function of ω2. We assume that ωχ = ωο. The resonances in the suscep tibility are expected to occur at the imaginary parts of the eigenvalues of L. In the absence of damping, L -» La-f· The eigenvalues of i a- f are purely imaginary and are given by i{pm — μ«} where {μ™} are the eigenvalues of Ha- f. We observe from the expression for the susceptibility that if n = 0 then it involves trasnsitions between the states up to the second excited manifold of the dressed states. The corresponding eigenvalue spectrum is depicted in in Fig. 14.1. The figure shows also the allowed transitions between the levels. The eigenvalues are expected to be only slightly perturbed if the damping is small. The resonances in the susceptibility are then expected to occur at (a) ω2 = ω0 ± g, (b) ω2 = 2 ωχ - ω o ± g, (c) ω2 = 2 ωχ — ωο ± g (\f 2 ± l ). Like we found in the case of a free atom, the collisional damping may lead to the appearance of new resonances. In Fig. 14.2, we plot normalized signal Sm = S ( n )/S ma.x( n) as a function of (ω2 - ω0 )/9 for κ = 0.03g and 7 = O.Olg. The solid line curve is in the absence of atomic collisions (-yc = 0) whereas the dashed curve is for 7 C = 0.04. Note that in the absence of collisions, the 258 0>o 14. Dissipative Cavity QED |Ψ,Μ> |Ψ,Η> -|Ψο(*’> -|Ψ.Η> A ' ' ..................j Ll... 1 *—10,-1/2 > Fig. 14.1. Schematic diagram of the spectrum of the Jayens-Cummings hamilto nian of a single two-level atom up to the second excited manifold. (fflirtOcJ/g Fig. 14.2. Normalized four-wave signal as a function of = (u>2 — ωο)/g for κ = 0.03g, 7 = 0.01<? j c = 0 (solid line curve) and 7C = 0.04<? ( dashed curve). signal exhibits resonances only at ωο ± g (resonance (a) above). Since ωι is assumed to be the equal to ωο, the resonance (b) is same as (a). However, the resonance (c) is different but is suppressed by the one at ωο ± g· The dashed curve for the case when the atom undergoes collisions exhibits the resonances (c). It also exhibits the collision induced Bloembergen resonances at ω2 — ωο = ±2g. These do not correspond to any dipole allowed transition. The resonance (a) is due to transitions to the first manifold of the spec trum of the interacting atom-field system. This part of the spectrum is the same as the one for a harmonic oscillator model in a classical field. The part of the spectrum corresponding to the second and higher manifolds is charac teristic of a two-level model and the quantized field. This part is brought out by the resonance (c). These resonances are thus a signature of field quantiza tion and hence may be referred to as quantum resonances. The observations here establish important role played by collisions in bringing out the quantum resonances. For further details, see [129]. That reference also shows that the 14.4 The Micromaser 259 results of the secular approximation in this instance are not only in quanti tative but also in qualitative disagreement with exact numerical results. The secular approximation need not hold for evaluation of a nonlinear suscep tibility. The second of the reference [129] discusses also the case of mixing by a collective system of N atoms. For a description of quantum effects in four-wave mixing by a trapped atom, see [130]. 14.4 The Micromaser A setup in cavity QED that offers immense possibilities of realizing a variety of quantum effects is the micromaser [131]-[135], A micromaser refers to a setup in which atoms prepared in a particular state pass through the cavity one at a time. In these experiments, the field is too weak for measurements. The experimentally measurable property in these experiments is the atomic population. The statistical characteristics of the cavity field are, therefore, required to be determined in terms of the statistics of the atomic population in a given state. The relationship between the statistical properties of the cavity field and that of the atomic population is, however, still an open question even for the simple model of a two-level atom in a single-mode cavity. In the following we outline briefly the theory of a micromaser. 14.4.1 Density Operator of the Field Let an atom, prepared in a desired state (henceforth referred to as an active state) described by the density operator pa pass one at a time through a cavity. Let pf (ti) be the density operator of the field at the time the ith atom enters the cavity. Let tint be the time taken by an atom to traverse the cavity. If Lint is the Liouvillean describing the evolution of the combined system of the atom and the cavity field then the state of the cavity field at the time ti + iint when the ith atom exits the cavity is evidently given by pi(ti+tint) = Tra exp papi(ti) = F(tint)pf(ti), Where F(t int) is the so called production Liouvillean: F(tint) = Tra exp Pa (14.36) (14.37) Let the field evolve under the action of the Liouvillean Lf from the time U + iint the ith atom exits the cavity till the time U+i of the entry of the (i + l ) t h atom. The state of the cavity field at that time is given by 260 14. Dissipative Cavity QED (14.38) where D( t ) = exp t ) ( 1 4.3 9 ) i s t h e damping Liouvillean. The second line in (14.38) is obtained by assum ing, in accordance with the usually encountered experimental conditions, that the time interval r = ti+\ — ti between the arrival of two successive atoms in the cavity is very much longer than the time tint that an atom takes to tra verse the cavity. Note that the trace of the density operator does not change during an evolution. Hence, The equation (14.38) determines the dynamics of the field in a micromaser for fixed values of r and i int. Its formal solution reads This may be evaluated, for example, by the method of expansion of p(0) in the magnitude of all its other eigenvalues is less than one. Note that the terms in the summation in (14.43) for which \μν\ < 1 vanish in the limit n —» oo. Hence, as n -¥ oo, Pf{tn) —> pss where pss is the eigenstate of D(T)F(tint) corresponding to eigenvalue unity i.e. Tr [Dp] = Tr[p], Tr[Fp] = Tr[p] (14.40) for any p. pf{tn) = D( r) F( t iat) pf(0). (14.41) terms of the eigenvectors of D(r)F(t i nt). Let /?„ be an eigenvector of DF corresponding to the eigenvalue μ„, i.e. DFpv pvpv. (14.42) Let ρν be the eigenvector of the adjoint of DP corresponding to the eigenvalue μ*. Let us represent an operator / by |/). The expression (14.41) may then be written as exp [ηΐη(μι,)] \pv) (p„|pf(0 ) ), (14.43) the (A\B) denotes scalar product of A and B. We note that 1. In the problems of interest, one of the eigenvalues of DP is unity whereas D(r)F(i(tin^ p ss — pss· (14.44) 14.4 The Micromaser 261 2. The time scale of approach to the steady state is determined by the eigenvalue nearest to unity in magnitude. Thus, if μι is the eigenvalue nearest to unity in magnitude then the steady state will be reached when a number n ~ 1/ 1η(1/|μι|) of atoms have passed through the cavity. If R is the rate of pumping then this corresponds to the time scale t ~ The relation (14.38) determines the state of the field when the time in terval r between the arrival of two successive atoms in to the cavity and the time of interaction ijnt are constants. These times, however, need not be con stants. Consider first the time r. The spread in r arises due to the fact that the process of preparing atoms in the desired state at the entry of the cavity is generally a random process. For, in the experiments, the atoms are pumped in to the cavity from a source at regular intervals of time, say, T. At the cav ity, the atoms are excited to the active state by shining appropriate lasers. The process of excitation is random. It is described adequately by assuming that the atoms are excited to the active state with probability p and that there is no correlation between the excitation of any two atoms. If K atoms arrive at the cavity in time ί χ = K T then the probability P( N,K) that N of them are excited to the active state is given, under the aforementioned assumptions, by the Binomial distribution Now, let Ρ{(Ν,ί κ) denote the state of the field at the time ί χ of the arrival the active state. At the cavity, it may get excited to the active state with probability p or not with probability 1 — p. If it is not excited then it passes through the cavity leaving the state of the field unchanged. Else it interacts with the cavity field transforming its state, by virtue of (14.36), by the action of F on it. In either case, the field decays for time T till the arrival of the next atom numbered K + 1. The density operator ρ{(Ν,ί κ+ι ) of the field seen by atom number K + 1 is thus expressible as This recurrence relation may be used to determine the field density operator the cavity or the field density operator pr(N) at the time of entry of the N ih active atom without any reference to how long it takes to prepare that many active atoms. The density operator pf(tK) at the time ΐ χ is given by Pf (N,t K) summed over all N. On carrying summation over N in (14.46) it follows that 1 / R\n(\/\μι\). (14.45) of the K th atom under the condition that N atoms before it got excited to P i ( N,t K+i ) = exp ( Lf T) (1 — p)pf (N,t K) +pFpf (N - M k ) ]. (14.46) Ρί(ίκ) at the time ί χ with no regard for the number of active atoms traversing 262 14. Dissipative Cavity QED Pf(tK+i) = exp ( Lf T) (1 - p ) + p F p f ( t K ). ( 1 4.4 7 ) T h e d e n s i t y o p e r a t o r β{(Ν), on the other hand, is given by pf(N, ί χ ) summed over all K: Le t us e xa mi ne t h e s t e a d y s t a t e s o l ut i on of t h e ma ps (14.47) a nd ( 14.48). Th e s t e a d y s t a t e of t h e ma p (14.47) i s t h e s t a t e r e a c he d i n t h e a s y mp t o t i c l i mi t t x oo whereas that of the map (14.48) is the state reached in the limit N —y oo. Those states correspond to the solutions βί ( ί χ+1 ) = βί(ί χ), pt (N + 1) = β{(Ν) respectively of (14.47) and (14.48). The steady state pss of (14.47) is the solution of the operator equation It is easy to see that the steady state of the map (14.48) also obeys (14.49). Hence the steady state density operator of the cavity field is the same whether the dynamics is described in terms of time or in terms of the number of active atoms traversing the cavity. An experimentally important case of random pumping is the Poisson pumping. This corresponds to the limit p —> 0, T —>■ 0 of binomial pumping such that p/T —> R where the constant R is the average rate of pumping of the active atoms. In this limit, the atoms enter the cavity continuously. In other words, in the Poisson limit, the discrete time t x becomes a continuous variable t. Hence we can write β{ ( Ν,ί χ+ι) — β{( Ν,ί χ) = Τ(ά/άί )β^(Ν,ί). The equation (14.47) then reduces to Finally, the density operator of the field after the passage of a fixed number N of active atoms, irrespective of the time for the case of Poissonian pumping, is found from (14.48) to be given by The properties of the cavity field including the effect of the pumping statistics may thus be studied for any model of atom-field interaction. We must also account for the fact that all the atoms may not move with the same velocity. The spread in the atomic velocities results in a spread in the time of their interaction i;nt. This may be included by averaging the density operator found above over the distribution W (tjnt) of iint· In realistic situ ations the atomic velocities follow the Maxwellian distribution. The average over W (tint) may be carried knowing the functional dependence of operators on i int. It has been computed in [132] for the two-level atomic micromaser with a single mode cavity discussed next. p exp ( Lf T ) F pex p (14.48) 1 - exp ( i t· τ ) ( ( 1 - p ) +pP) Pss = 0. (14.49) (14.50) βί(Ν + 1) = ( l - U/R ) 1 Ρβί (Ν). (14.51) 14.4 The Micromaser 263 Consider two-level atoms pumped in to a cavity. The atoms interact reso nantly with a single cavity mode by executing one-photon transitions between its levels |<7) and |e). This interaction is described by (14.1) with ω = ωq. The micromaser theory is based on determining the field superoperators defined in (14.37) and (14.39). Assume the Q-factor of the cavity to be very high so that the field damping is negligibly small during the time of transit of the atom. The evolution of the combined system of the atom and the field is then governed only by (14.1) so that 14.4.2 Two-Level Atomic Micromaser exp PaPf exp i l l PaPf exp \ί ΐί\ηΐ . (14.52) The expression for the production operator may be found by using the results of Chap. 11. Show that if the atoms enter the cavity in their excited state then Fpt = cos (gt intx) pf cos (gtiat\) sin (gtintX) sin (gt int\) + a f ^----- Pi----- ^ ----- - a, (14.53) Λ Λ Λ = gy/a^a + 1 (14.54) We examine the characteristics of the micromaser first by ignoring the field damping at all times. Undamped Micromaser. A simplified version of the micromaser is ob tained by assuming that the field does not decay significantly even during the time interval between the arrival of two atoms in to the cavity [136]. In this case, Lfp = — ίω[ά^ά, p}. Use this form of Lf and (14.53) for F in (14.38), take the matrix element in the number state | m) of the field to obtain (with Pmn = (m\pf\n)) {m\pf(k + l)|n) ξ pmn(k + 1 ) βτηβηβτη—\η — \(}ζ), (14.55) a m = cos (gtint\/m ), β τη = sin (gtint^/m). (14.56) The equation (14.55) gives the density operator of the field at the time of entry of (k + l )th atom in terms of that at the time of entry of fcth atom in to the cavity. Let us examine the steady state solution of this equation. Consider first the diagonal elements. Writing pm = pmm we see that pm(k + 1) = a 2 m+1pm(k) + /3^pm_i(fc). (14.57) Its steady state solves βτη+ΐΡπΊ = βτηΡτη—Ι (14.58) 264 14. Dissipative Cavity QED This shows that β\ρο = P’i pi = · · · = β'η+ιΡη = · · · = 0. Hence, a non-trivial steady-state is possible only if In this case Φ 0 whereas pmm = 0 for m < Νχ. Note that if (14.60) holds then (14.59) is satisfied for all those positive integers JVj which satisfy In order to determine Ci, note from (14.57) that under the condition (14.60), the equations for pm(k-1-1) for the values m > Νχ + l are decoupled from pn(k) n < Νχ. It may in fact be verified that the equations reduce to uncoupled blocks defined by Ν _ ι + 1 < m,n < Ni (with N 0 = —1 ). Since the equations in each block are decoupled from each other, it implies that the trace of p in a block remains unchanged during the evolution. Hence pm{ 0) being the initial photon number distribution function. Thus, if the cavity is initially in the vacuum state then pmm(0 ) = Smo and Q = Sn i.e. the cavity field is the number state \Νχ ). If the initial field has components in other blocks then the steady state would be a mixture of the number states. For the steady state of an undamped micromaser when the atoms enter the cavity in a superposition state, see [137]. A micromaser operating with three- level atoms in a two-mode cavity has been analyzed in [138]. A micromaser in which atomic levels are coupled by two channels is studied in [139]. Damped Micromaser. Let us now account for the field damping during the time interval between the arrival of successive atoms. Let the field damping be described by Af defined in (14.5) so that Lf = Af. Let n = 0. We substitute that form of Lf in (14.39) to evaluate the field damping superoperator D( t ). Its matrix elements in the photon number representation are given by (13.34). It is then straightforward to show that, owing to the map (14.38), and (14.57), the photon number distribution pm obeys /3jvj+i = 0 for some positive integer Νχ, which is such that, for an integer p, (14.59) gtint y/Νχ + Ι = ρπ. (14.60) N i = q 2i {N i + l·) — 1, qx = 1. Hence, u n d e r t h e trapping condition (14.60) (14.61) (14.62) (14.63) m — N i - i + l k—m oo = Σ D mk \a l +i Pk ( t i ) + 0kPk-\( t i ) ] , (14.64) k=m 14.4 The Micromaser 265 D mk = n ,dm( 1 - d)k'm, d = exp(—2 k t ). (14.65) (κ — m)!m! The equation (14.64) shows that if there is a trap at Nk then pNk+n{ti+1 ) (n > 1) are no longer coupled with pm(U) [m < Nk). The equation for Pm{k + 1 ) in a block is coupled with pm(k) in higher blocks but not with the ones in lower blocks. Hence, if pm(to) = 0 for m in blocks higher than the Mth then pm(U) = 0 for all U for m in a block higher than the Mth. An interesting and useful consequence of the trapping condition is that pm in the steady state is confined to the lowest block. In order to see that, perform the summation over m from zero to N on the steady state form of (14.64). Owing to (A.14), Dmy. = 0 if k < m. Hence, we may extend the lower limit of summation over k in (14.64) to zero. The summation over m then involves only Dmk- Break the summation over k in two parts, one running from zero to N and another from N + 1 to oo to obtain N N N oo N Y jPm = Y j Fk Y j Dmk+ Σ FkYsDmk, (14.66) m —0 k —0 m —0 k = N + 1 m = 0 the Fk being the m-independent terms under the summation in (14.64). Since the value of k in the first sum above is less than the upper limit N of the summation over m in it, and since Dmk = 0 for m > k, the upper limit of summation over m there may be replaced by k. The resulting binomial sum over m then yields unity. Change the sumation index over k —> k + 1 in the term having β^. to get OC N —β 2 Ν+\ΡΝ + Σ Fk Σ Dmk = 0· (14.67) fc=7V-j-l m —0 We let N = Νχ, which is the lowest value of N for which /3λγ+ι = 0, to arrive at OC («fc+iAvifc + βΙ +1 Σ>Ν^+ι) Pk = 0, (14.68) k=Nt + 1 where N Drik = Σ Dmk (14.69) m = 0 are positive definite if d φ 1. The term in the parentheses in (14.68) is, therefore, positive definite if d φ 1. Since pk are positive semidefinite, it follows that in order to satisfy (14.68) they should vanish if k > Νχ + 1. In case of d = 1, i.e. if the cavity damping is ignored, then D\\k = 0 for k > Νχ. The equation (14.68) then can be satisfied for non-zero p^. Thus, the problem of finding the steady state reduces to solving equations only in the first block. This is evidently a significant simplification. 266 14. Dissipative Cavity QED The problem of determining the time-scale of approach to equilibrium, however, requires an elaborate analysis of the eigenvalues. We refer to [140] for the details of such an analysis. Here we quote the substance of the results. Those results state that if e ξ kt <C 1 then the population in the kth block leaks to the lower block on the time scale ~ eN*- N*-i. The population in the lowest block approaches the steady state on the time scale of the cavity damping time κ- 1. For a numerical study of the time-scale, see [141]. Approximation Method. Since solving even the zero temperature micro maser photon number distribution equation (14.64) is generally a formidable task, approximation methods are employed to investigate its characteris tics. These methods consist in reducing the stroboscopic map (14.38) to a continuous-time evolution by noting that [pf(ij+i) — pf(tj)]/(tj+i — U) « (d/dti)p{(ti) if U.|_i — ti = τ ti. On replacing ti by t, (14.38) in this approximation reduces to ( r = R -1) ^pt(t) = R\n D( t )F(i int) pf(t). (14.70) This equation is further approximated by treating the logarithm of operators as c-number logarithms to obtain ^ P f ( i ) = Lt + Rl n( F( t int)) pt(t). (14.71) Furthermore, ln(F(iint)) is expressed in powers of 1 — F(iint) retaining terms to a desired order. In the case of an atom in a single-mode field, further approximation is introduced to treat the photon number m as a continuous variable μ. This approximation applied to the Jaynes-Cummings model described to second order in 1 — F(imt) in (14.71) then turns out to be a Fokker-Planck equation. We refer the reader to [140, 142] for comparison of the exact and approximate results. 14.4.3 Atomic Statistics As mentioned before, the characteristics of a micromaser are examined exper imentally by studying the statistics of the atomic population and correlation between the population of atoms exiting the cavity at different times. The analytical results for the variance in the observed population of atoms exiting the cavity and correlation between atomic population at different times were reported first in [143] by assuming the pumping mechanism to be Poissonian. It included also the effects of finite detector efficiency. See also [144, 145] for analysis of atomic correlations. The study of the atomic statistics has been carried in [146] by constructing the joint distribution for observing atoms exiting the cavity at different times in one state or the other. It is applicable to a binomial pumping which includes Poissonian pumping as a special case. See [147, 148] for experimental results. Appendices A. Some Mathematical Formulae The delta function is defined by dA: exp(i kx). 5(05(C) = ^ J exp{i(z£ + ζ*ξ*)}ά2ζ where the integration is over the complex plane. f O O γ / dkexp(±i kx) = πδ(χ) ± i P — Jo x wher e P indicates Cauchy principal value defined by pj =£s[/J+/ l i m = Ρ — =ρ ΐπί(χ). €->o x ± le x δ(χ - y)F(x) = δ(χ - y)F(y), ± ί ( Χ - ν) = - ± ί ( Χ - ν) i l i m 7 βχρ(—7 |τ|) = δ(τ). Z 7 —^00 dx - F( x ). x 1 l i m a —^0 OiyJ7Γ exp {—x jar) = δ(χ). I f t h e n ^ ( £ > 0 = ^2 J ^(2,2*)βχρ{ΐ(.2:ξ + 2 * ξ * ) ^ 2.ζ, 1 F(z,z*) = J Ρ(ξ,ξ*)βχρ{-ΐ(ζξ + ζ*ξ*)}ά2ξ. ( A.l ) (A.2) (A.3) (A.4) (A.5) (A.6 ) (A.7) (A. 8 ) (A.9) (A.1 0 ) (A.ll) • The Gamma function Γ(ζ) is defined by p O O Γ(ζ) = I exp(—x)xz_1dx, Re(z) > 0. Jo 268 Appendices Γ(ζ + 1) = ζΓ(ζ). I f m a positive integer or zero then 1 Γ(πι + 1 ) = ml, Γ{ζ) Γ( 1 - ζ ) = Γ( - τ η) 0. 8 Ϊη(πζ) ο 2 ζ - 1 / ί \ Γ(2ζ) = _ Γ ( 4 Γ ^ + _ ). • T h e b e t a f u n c t i o n Β( α,β) is defined by Β(α,β)= f xa_1(l — x)/3_1 dx, Re(a) > 0, Re(/3) > 0. Jo • If x is a real variable and Re(a) > 0 t hen dxexp( —ax + βχ) = ../ — exp(/3 /4a), a dx x e x p ( —a x ) 1 π / 1 Γ ( m + - y/ a 2m+i \ 2 (A.14) (A.15) (A.16) (A.17) (A.18) (A.19) (A.2 0 ) (A.21) (A.12) (A.13) If A is an N χ N symmetric matrix whose eigenvalues are positive then det(4) > 0 and / °o /-OO . d x i... / dxjv exp (- X T A X + C TX\ - o o J —oo π N exp - C f A ^ C 4 (A.22) det(^) ' ' - ’ where X is a column constituted by the elements X\,..., x n, C is a column of N constants and CT denotes the transpose of C. Integration over complex a plane may be carried by substituting a = x+iy, ( —0 0 < x, y < 0 0 ) so that A. Some Mathematical Formulae 269 / /♦OO /»OC d2a^ L dxl (A.23) dx / dy. J — o c J — O C I t may a l t e r n a t i v e l y be per f or me d i n t h e p o l a r r e p r e s e n t a t i o n a = y/r exp(i0) ( 0 < r < 0 0, O < 0 < 2 π) so that /» 2 π άθ. Using this representation, it may be verified that, if Re(a) > 0 then t o! im+lSmn· — J d2a ama*n exp(—a | a | 2) IJ d2 aexp(—| a | 2 + β*α)/(α*) = /(/?*). ^ J d2 a e x p ( - | a| 2 + p*a)amf (a*) = — J d2a exp(—a|o:J2 + 6 1 a + b^a*) = - exp . If a2 — 4|c| 2 > 0 then — J d2a exp (—a | a | 2 + ba + b*a* + ca2 + c*a*2) I 1 Γ b2 c* + b*2c + a|6 |2" “ γ a2 — 4|c| 2 8ΧΡ ί a2 — 4|c| 2 _ ' The Hermite polynomial Hn(x) is defined by dn Hn(x) = (—)n exp(x2) — exp( -x2), ( n/2 ) Hn(x)= V ( - ) m ■ n · —(2x)n_2m. y ^ y 1 m\(n — 2 m)\ ’ 7 7 1= 0 I t s i n t e g r a l r e p r e s e n t a t i o n a nd t h e g e n e r a t i n g f unc t i ons ar e 2n roc (A.24) L (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (A.31) • The spherical harmonics are such that Υι.μ (Θ,Φ) = (-) μΥΙ_ μ (Θ,Φ), (A.34) y„o(M) = - i =, (A.35) \/4π /»7Γ pQ.'K / sin(0)d0 / d φΥι Μ(θ,φ)Υκο(θ,φ) = S K l $ m q > (A.36) Jo Jo 270 Appendices Σ Σ L = 0 M - - L = δ(φ — φ')δ(ο os(0) — cos(0')). (A.37) If F(a, b; c; z) is the Gauss hypergeometric function then <a- > B. Hypergeometric Equation In this Appendix we address the question of reducibility of a second-order differential equation ^ + P(X)^ +Q(X)]/(X) = ° (B-1} to an equation for the hypergeometric or the confluent hypergeometric func tion. For details see, for example, [149, 150]. The nature of solution of (B.l) is governed by its singularities. We recall that if P(x) and Q(x) are analytic at a point xq then xq is called an ordinary point of (B.l). The xo is a singular point of (B.l) if it is an isolated singularity of either P(x) or Q(x) or both. The singular point Xo is called a regular singularity if (x — x0 )P(x) and (x — x 0)2Q(x) are analytic at x = xo- In other words, x = xo is a regular singularity if P(x) has a pole of order no more than one and Q(x) has a pole of order no more than two at Xo- The nature of singularity at infinity is examined by transforming to y = 1/x. The eq.(B.l) in terms of y = 1/x reads d ^ ( 2 _ P ( l/y )\ d_ Q( l/y) A v 2 \y v 2 ) dv y4 f(y) = o· ( B -2) The singularity of this equation at y = 0 corresponds to that of (B.l) at X = oo. Let xo be a regular singular point with lim (x - xo)P(x) = Po, lim (x - x 0 )2 Q( x) = qo■ (B.3) X—¥X q X—¥Xq B. Hypergeometric Equation 271 7 2 - ( 1 - p o h + qo = 0 (B.4) is called the indicial equation and its roots the exponents or indices cor responding to Xq. The equati on ( B.l ) is called a Riemann equation if it has three regular singularities including the singularity at infinity. The sum of the exponents corresponding to those singularities is unity. By an appropriate transforma tion of x —> z, its three singular points can be mapped on to z = 0,1, oo. This, followed by a transformation of f ( x) —> w(z) transforms (B.l) to the hypergeometric equation d2 d ί z( 1 - z ) — r + {c - (a + b + 1 )z}~ ab w(z) = 0. (B.5) dz 1 dz If c is not an integer then, within the unit circle \z\ < 1, linearly independent solutions of (B.5) are F(a,b; c;z) and z l ~cF(a — c + 1, b — c + 1; 2 — c; z) where F(a,b\c\z) is the Gauss hypergeometric function T h e e q u a t i o n F(a, b; c; z) F(c) Γ(πι + a) r ( m + b) r{a) r( b) ^ Γ{τη + c) r{m + 1) zm. (B.6 ) Note that if a = —to or b = —to where to is a positive integer then F(a, 6; c; z) is a polynomial of degree to. The equation (B.5) has z = 0,1, oo as its regular singular points the sum of whose indices is 1. In addition to this, one of the indices of each of the singularities at z = 0,1 is zero. The transformations that reduce a Riemann equation to a hypergeometric equation are known [150]. However, we outline below the procedure for transforming (B.l) to (B.5) only for the particular forms of P(x), Q(x) encountered in Chap. 10. A form of (B.l) that concerns us is x(l - x)2- ^ + p { l - x){x+ q)-^ + (rx - X) f(x)= 0, (B.7) p, q, r, A being constants. Verify that f(x) = (1 - x) aw(x) (B.8 ) transforms (B.7) to an equation for w(x) which is of the form (B.5) if a(a — 1 ) — ap + r = apq + λ, (B-9) with a, b determined by solving a + b + l = 2 a — p, ab = a(a — 1) — pa + r, (B.10) and c = pq. (B.ll) 272 Appendices A solution of (B.7) thus reads f(x) = (1 - x)aF(a,b-,c;x), (B.12) with a and a,b, c determined by (B.9)-(B.ll). Confluent Hypergeometric Function. Consider the hypergeometric equa tion (B.5). Rewrite it by changing the independent variable z to x = bz. The singular points of the transformed equation are then at x = 0, b, oo. Let 6 —> oo so that the singularity at x = b merges with that at x = oo. The transformed equation x ~!h + (c - x ) - ^ - a\f ( x ) = ° ’ (B·13) is called the confluent hypergeometric equation. Note that this equation is thus obtained by the confluence of two singularities at infinity. Also, x = 0 is a regular singularity of (B.12) whereas its singularity at x = oo is irregular. If c is not zero or a negative integer then a solution of this equation is the confluent hypergeometric function Φ{α. c.z) = I M y ... r { m + a)----- Γ(α) t-ί r(m + c)r(m + 1) v 7 m=0 v 7 v 7 Note that Φ(—to; c; x ) is a polynomial of degree to. C. Solution of Two- and Three-Dimensional Linear Equations In this Appendix we derive exact solution of a linear equation ±\iP(t))=Mm)) + \S(t)) (C.l) when M is a ί-independent 2 x 2 o r a 3 x 3 matrix. Its formal solution is given by | ψ(ί)) = exp (Mt) 1^(0) ) + f dre xp(—Mt )\S( t ) ) Jo (C.2) If t he real par t of all t he eigenvalues of M is negative then, in the limit ί —^ oo, exp (Mt ) —>· 0. If, in addition, | S) is independent of time then (C.2) yields = - M - 1 |S). (C-3) Consider first the case of two-dimensional M given by <a4) D. Roots of a Polynomial 273 Let Ai and λ2 be the eigenvalues of M. Assume that λι φ A2 and invoke (1 0.1 2 ) to obtain exp (Mt ) = - ——— ( Μ — λ2) exp(Aii) — ( M — XY) exp(A2 i) Ai — λ2 λι-λ 2 V/MO M t ) )' ^ βιι (t) = ( a- A2 )exp(Aji) - (a - Ai)exp(A2<), β22(ή = (d - X2) exp(Xit) - (d — Ai) exp(A2 i), βΐ 2 (t) = b(exp(Xit) - exp(A2 t)), β ΐ\(t ) = c(exp(A,i) - exp(A2 i)). (C.6 ) Combination of (C.2), (C.5) and (C.6 ) yields the solution of (C.l). Its limit Ai —» A2 gives the solution for Ai = λ2 · Next, let M be a 3 x 3, ί -independent matrix. Let A1;A2,A3 be its eigen values assumed to be distinct. Invoke (10.12) to obtain 0 exp ( m t j = ^ exP(Am<) M - Xn m — 1 ηφτη 171 T 3 = ^ am(t ) Mm~\ (C.7) = 1 t he a m(t) being unknown functions. The second line in the equation above is owing to the fact that the maximum power of M in the first line is two. The am(t)’s, obtained by comparing equal powers of M in the two equations in (C.7), are found to be given by al(t)\ 1 ί Χ2Χί(Χ3 — X2 ) AiA3(Ax — λ3) AjA2 (A2 — Ai) a 2 (t) = 7, Ai - Al A?- A| Al - A? ,a 3(t) J ^ \ X3 ~ X2 A] — Αβ λ2 — Ai 'exp(Ai t )\ exp( A2 t) I , (C.8 ) v exp(A3 t ) ) D = (Ai — A2)(A2 — A3)(A3 — Ai). (C.9) The exp (Mt) is determined by inserting (C.8 ) in (C.7). The resulting expres sion, on combination with (C.2), determines the solution of (C.l). D. Roots of a Polynomial In this Appendix we list some general properties of the roots of a polynomial and the exact expression for the roots of a cubic. For details, see [151]. 274 Appendices 1. Consider the polynomial equation /n(A) = o„A" + o„_iA" 1 + · · · + οιλ1 + ao = 0 (D.l) where ao,... ,an are real constants. This equation admits n roots Ai,..., A„. The roots may be real or complex. However, if A j is a root then, since {ak} are real, it follows that /^(Aj) = /„ ( A*) = 0. Hence A* is also a root. 2. If /( a ) and /( 6 ) are of opposite sign then /„ ( A) = 0 has a root in the interval (a, 6 ). 3. If one of the roots of a polynomial /n(A) = 0 is known to be, say, Aj then on dividing that polynomial by A — Ai we obtain the quotient which is a plynomial </>n_i(A) of degree ra — 1. The other roots of /n(A) = 0 are the roots of (A) = 0. 4. In the problems concerned with dynamical stability, it is crucial to know whether or not the roots are positive and the conditions under which the real part of the roots is negative. This task is helped by the following: • Descarte’s rule of signs: If r+ is the number of positive real roots of (D.l) and V is the number of changes of sign in the sequence of the coefficients an, a„_i, an_2,..., ao then r+ = V - 2h, (D.2) the h being a nonnegative integer. In other words, the maximum value of r + is V and, if less, always by an even number. • Hurwitz criterion: The real part of every root of (D.l) is negative if, and only if, ao > 0 and n determinants D\ — a\, D 2 — ai ao 0 D3 = «3 a 2 ai a^ (I4. «3 Dn = a 1 «3 ^ 2 n — 1 ^ 2 n —2 ' * * are all positive. 5. Consider the cubic equation /3 (A) = A3 + α2 λ2 + αχλ + αο == 0 (D.3) (D.4) If Ai is its root then, following t he i t em number 3 above, we find t h a t its ot her two root s are t he solutions of t he quadr at i c A2 + (Ai + α2)λ + αχ + Αι(Αι + a2) = 0. (D.5) I n o r d e r t o l i s t t h e r o o t s of ( D.4), we def i ne P 1 2 al — g a2l q = ao r i a 2 + ^ a: 2 » A— i + VS. B— f - s/A ( D.6 ) ( D.7 ) D. 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Index absorption spectrum 182 ac Stark splitting 223 algebra - harmonic oscillator 41 - 5(7(1,1) 43 - 5(7(2) 42 - SU(m) 44 - SU(m,n) 45 antibunching 132 antinormal ordering 49, 83 Bargmann representation 64 Bell’s inequality 35 Bloch-Siegert shift 142 Bloembergen resonances 224 Born approximation 157 bunching 134 cat paradox 31 cavity QED 151 Chapman-Kolmogorov equation 103 characteristic function 100 coherent multiphoton process 148 coherent states 58 - generalized 57 - Glauber 58,62 - of e.m. field 58, 133 - of harmonic oscillator 62 - of spins 70 - of 5(7(2) 70 - pair 78, 248 - Perelomov 57 coherent states, completeness relation 57 - for harmonic oscillator 62 - for 5(7(1,1) 78 - for 5(7(2) 72 coherent states, minimum uncertainty 59 - of harmonic oscillator 64 - of spins 70, 72 - of 5(7(1,1) 77 coherent states, uncorrelated equal variance minimum uncertainty 59 - of harmonic oscillator 62 - of spins 70 - o f 5(7(1,1) 77 collapses and revivals 206 collisional damping 172 complementarity 12,27 cumulants 101 density operat or 21 Descarte’s rule 277 detailed balance 106, 108,160, 225 differentiation, parametric 8 - of exponential operator 37 - of operat or product 8 disentangling an exponential 48 - harmonic oscillator algebra 49 - 5(7(1,1) algebra 51 - 5(7(2) algebra 50 down conversion 151, 239 dressed st at es 204 e.m. field - chaotic, classical 127 - chaotic, quantum 133 - coherence time 130 - coherent 127, 133 - correlation functions, classical 123 - correlation functions, quantum 130 - quant i zat i on 121 effective two-level approximation 212 effective two-level atom 147 eigenvalue 7, 186 - generalized 189 eigenvector 7,186 - generalized 189 entangled st at e 20, 25 EPR Paradox 32 equal variance minimum uncertainty state 13 284 Index Fokker-Planck equation 107 four-wave mixing 223, 257 - collision induced resonances 224, 257 - quantum resonances 258 Gaussian process 102 geometric phase 16 - in adiabatic evolution 18 - of a harmonic oscillator 18 - of a two-level system 18 Hamilton-Cayley theorem 190 Heisenberg equation 14 hidden variables theory 34 - local 34 Hilbert space 1 homodyned detection 134 Hurwitz criterion 277 Husimi function 87 incompatibility 12 interaction picture 15 interference 26,27 Jaynes-Cummings model 143,144, 204 Jordan canonical form 189 Lie algebra 40 Lie group 56 Markov approximation 158 Markov process 103 master equation 104, 105, 158 measurement problem 30 micromaser 259 - trapping condition 264 minimum uncertainty states 12, 59 - of harmonic oscillator 61,67 - of spins 70 - o f 5(7(1,1) 77 mixed state 22 moments 100 multi time joint probability 15 multi-channel models 149 noise - additive 110 - coloured 109 - delta correlated 109 - Gaussian white 109 - multiplicative 1 1 0,1 1 2 - white 109 non-classical st at es 95 - of e.m. field 95 - of spin-1/2s 97 normal ordering 49, 83 Ornstein-Uhlenbeck process 110-112 P-function 86 - for spins 91 parametric processes 150 phase - dynamic 16 - geometric (see geometric phase) 16 photon 122 Poisson process 115 probability amplitude 11 probability density 11,99 - conditional 103 - joi nt 99 pure st at e 22 Q-function 87 - for spins 92 quant um eraser 29 quasiprobability distribution 83 - for spins 89 Rabi frequency 142, 221 random telegraph noise 116 regression theorem 105, 163 representations - by eigenvectors 55 - equivalent 56 - labeled by group parameters 56 - of harmonic oscillator algebra 60 - of 5(7(1,1) algebra 76 - of 5(7(2) algebra 68 resonance approximation 144 resonance fluorescence 171, 219 - collective 225 rot at i ng wave approximation 143, 181 Rydberg atom 152 s-ordering 83 Schmidt decomposition 25 Schrodinger equation 13 Schwarz inequality 2 - generalized 3 secular approximation 162, 227, 253, 256 semiclassical approximation 139 similarity t ransformation 39 - harmonic oscillator 41 - 517(1,1) 43 Index 285 - SU(2) 42 - SU(m) 44 - SU(m,n) 45 Sneddon’s formula 37 spectroscopic squeezing 75 spectrum 136 - absorption 223,256 - emission 222 spin operators - collective 69 - lowering 23 - raising 23 squeezed reservoir 166 squeezed states - of harmonic oscillator 67 - of spins 73, 74 squeezed vacuum 166 squeezing operator 65 Stark shift 214 stationary process 100 stochastic differential equation 109 sub-Poissonian distribution 132 superoperator 10 - adjoint 10 superposition, principle of 26 susceptibility 179 - optical 180 symmetric ordering 83 - for spins 94 thermal reservoir 164 three level atom 145 time-ordered exponential integration - harmonic oscillator algebra 52 - S U( l, 1) algebra 53 - S U (2) algebra 53 trace 6 transition probability 103 two-channel Raman-coupled model 150,207 two-level atom 144 two-photon process 146 two-photon reservoir - in ladder configuration 245 - in Lambda configuration 248 uncertainty relation 12 uncorrelated equal variance minimum uncertainty state 13, 62 vacuum field Rabi oscillations 205 vacuum field Rabi splitting 257 vacuum fluctuations 122 wave mixing 149, 181 wave-particle duality 26 welcher weg 28 which path 28 Wiener process 110, 111 Wiener-Khintchine theorem 136 Wigner function 87 - for spins 92 Zeno effect 16 Springer Series in OPTICAL SCIENCES Ne w e d i t i o n s o f v o l u me s pr i o r t o v o l u me 60 1 S o l i d - S t a t e L a s e r E n g i n e e r i n g B y W. K o e c h n e r, 5 t h r e v i s e d a n d u p d a t e d e d. 1 9 9 9, 4 7 2 f i g s., 55 t a b s., X I I, 74 6 p a g e s 1 4 L a s e r C r y s t a l s T h e i r P h y s i c s a n d P r o p e r t i e s B y A. A. K a m i n s k i i, 2 n d e d. 1 9 9 0, 89 f i g s., 5 6 t a b s., X V I, 4 5 6 p a g e s 15 X - R a y S p e c t r o s c o p y A n I n t r o d u c t i o n B y Β. K. A g a r w a l, 2 n d e d. 1 9 9 1,2 3 9 f i g s., XV, 4 1 9 p a g e s 36 T r a n s m i s s i o n E l e c t r o n M i c r o s c o p y P h y s i c s o f I m a g e F o r m a t i o n a n d M i c r o a n a l y s i s B y L. R e i m e r, 4 t h e d. 1 9 9 7,2 7 3 f i g s. X V I, 5 84 p a g e s 45 S c a n n i n g E l e c t r o n M i c r o s c o p y P h y s i c s o f I m a g e F o r m a t i o n a n d M i c r o a n a l y s i s B y L. R e i m e r, 2 n d c o m p l e t e l y r e v i s e d a n d u p d a t e d e d. 1 9 9 8, 2 6 0 f i g s., X I V, 5 2 7 p a g e s P ub l i s he d t i t l e s s i n c e v o l u me 60 60 H o l o g r a p h i c I n t e r f e r o m e t r y i n E x p e r i m e n t a l M e c h a n i c s B y Y u. I. O s t r o v s k y, V. P. S h c h e p i n o v, V. V. Y a k o v l e v, 1 9 9 1,1 6 7 f i g s., I X, 248 p a g e s 61 M i l l i m e t r e a n d S u b m i l l i m e t r e W a v e l e n g t h L a s e r s A H a n d b o o k o f c w M e a s u r e m e n t s B y N. G. D o u g l a s, 1 9 8 9,1 5 f i g s., I X, 278 p a g e s 62 P h o t o a c o u s t i c a n d P h o t o t h e r m a l P h e n o m e n a I I P r o c e e d i n g s o f t h e 6 t h I n t e r n a t i o n a l T o p i c a l M e e t i n g, B a l t i m o r e, M a r y l a n d, J u l y 31 - A u g u s t 3,1 9 8 9 B y J. C. M u r p h y, J. W. M a c l a c h l a n S p i c e r, L. C. A a m o d t, B. S. H. R o y c e ( E d s.), 1 9 9 0,3 8 9 f i g s., 23 t a b s., X X I, 545 p a g e s 63 E l e c t r o n E n e r g y L o s s S p e c t r o m e t e r s T h e T e c h n o l o g y o f H i g h P e r f o r m a n c e B y H. I b a c h, 1 9 9 1,1 0 3 f i g s., V I I I, 1 78 p a g e s 6 4 H a n d b o o k o f N o n l i n e a r O p t i c a l C r y s t a l s B y V. G. D m i t r i e v, G. G. G u r z a d y a n, D. N. N i k o g o s y a n, 3 r d r e v i s e d e d. 1 9 9 9, 39 f i g s., X V I I I, 41 3 p a g e s 65 H i g h - P o w e r D y e L a s e r s B y F. J. D u a r t e ( E d.), 1 99 1, 93 f i g s., X I I I, 252 p a g e s 6 6 S i l v e r - H a l i d e R e c o r d i n g M a t e r i a l s f o r H o l o g r a p h y a n d T h e i r P r o c e s s i n g B y Η. I. B j e l k h a g e n, 2 n d e d. 1 995, 6 4 f i g s., X X, 4 4 0 p a g e s 67 X- Ray Mi c r os c opy II I P r o c e e d i n g s o f t h e T h i r d I n t e r n a t i o n a l C o n f e r e n c e, L o n d o n, S e p t e m b e r 3 - 7,1 9 9 0 B y A. G. M i c h e t t e, G. R. M o r r i s o n, C. J. B u c k l e y ( E d s.), 1 9 9 2,3 5 9 f i g s., X V I, 49 1 p a g e s 68 H o l o g r a p h i c I n t e r f e r o m e t r y P r i n c i p l e s a n d M e t h o d s B y P. K. R a s t o g i ( E d.), 1 9 9 4,1 7 8 f i g s., 3 i n c o l o r, X I I I, 328 p a g e s 6 9 P h o t o a c o u s t i c a n d P h o t o t h e r m a l P h e n o m e n a I I I P r o c e e d i n g s o f t h e 7 t h I n t e r n a t i o n a l T o p i c a l M e e t i n g, D o o r w e r t h, T h e N e t h e r l a n d s, A u g u s t 2 6 - 3 0,1 9 9 1 B y D. B i c a n i c ( E d.), 1 9 9 2,5 0 1 f i g s., X X V I I I, 7 31 p a g e s Springer Series in OPTICAL SCIENCES 70 El ec t ron Hol ogr aphy B y A. T o n o m u r a, 2 n d, e n l a r g e d e d. 1 9 9 9,1 2 7 f i g s., X I I, 1 62 p a g e s 71 Ene r g y - Fi l t e r i ng Tr ansmi s s i on El ect ron Mi c r os c opy B y L. R e i m e r ( E d.), 1 9 9 5,1 9 9 f i g s., X I V, 4 2 4 p a g e s 7 2 Nonl i near Opt i c al Ef f ec t s and Mat eri al s B y R G i x n t e r ( E d.), 2 0 0 0,1 7 4 f i g s., 43 t a b s., X I V, 5 4 0 p a g e s 73 Evanes cent Waves F r o m N e w t o n i a n O p t i c s t o A t o m i c O p t i c s B y F. d e F o r n e l, 2 00 1, 2 7 7 f i g s., X V I I I, 268 p a g e s 74 I nt e rnat i onal Tr ends i n Opt i c s and Phot oni cs I C O I V B y T. A s a k u r a ( E d.), 1 9 9 9,1 9 0 f i g s., 1 4 t a b s., X X, 4 2 6 p a g e s 75 Adv a nc e d Opt i c a l I mag i ng The or y B y M. G u, 2 0 0 0, 93 f i g s., X I I, 2 1 4 p a g e s 76 Hol ogr aphi c Da t a St orage B y H.J. C o u f a l, D. P s a l t i s, G.T. S i n c e r b o x ( E d s.), 2 000 228 f i g s., 6 4 i n c o l o r, 1 2 t a b s., X X V I, 486 p a g e s 77 Sol i d- St at e Lasers f or Mat er i al s Pr ocessi ng F u n d a m e n t a l R e l a t i o n s a n d T e c h n i c a l R e a l i z a t i o n s B y R. I f f l a n d e r, 2 0 0 1,2 3 0 f i g s., 73 t a b s., X V I I I, 350 p a g e s 78 Hol ogr aphy T h e F i r s t 50 Y e a r s B y J.- M. F o u r n i e r ( E d.), 2 0 0 1,2 6 6 f i g s., X I I, 4 6 0 p a g e s 79 Mat hemat i c al Met hods o f Quant um Opt i c s B y R.R. P u r i, 2 0 0 1,1 3 f i g s., XI V, 285 p a g e s 80 Opt i c al Pr oper t i e s o f Phot oni c Cryst al s B y K. S a k o d a, 2 0 0 1, 85 f i g s., 23 t a b s., X, 1 92 p a g e s 81 Phot oni c Ana l o g - t o - Di g i t a l Convers i on B y B. S h o o p, 2 0 0 1,2 5 2 f i g s., 11 t a b s., X I I I, 328 p a g e s Ravinder R. Puri Mathematical Methods of Quantum Optics With 13 Figures Springer Contents 1. Basic Quantum Mechanics.................................................................. 1 1.1 Postulates of Quantum Mechanics................................................ 1 1.1.1 Postulate 1............................................................................. 1 1.1.2 Postulate 2............................................................................. 11 1.1.3 Postulate 3............................................................................. 11 1.1.4 Postulate 4............................................................................. 11 1.1.5 Postulate 5............................................................................. 13 1.2 Geometric P h a s e................................................................................ 16 1.2.1 Geometric Phase of a Harmonic Oscillator........ 18 1.2.2 Geometric Phase of a Two-Level System............ 18 1.2.3 Geometric Phase in Adiabatic Evolution .......... 18 1.3 Time-Dependent Approximation Method................................... 19 1.4 Quantum Mechanics of a Composite System............................. 20 1.5 Quantum Mechanics of a Subsystem and Density Operator . . 21 1.6 Systems of One and Two Spin-1/2 s.............................................. 23 1.7 Wave-Particle Du a li t y..................................................................... 26 1.8 Measurement Postulate and Paradoxes of Quantum Theory . . 29 1.8.1 The Measurement Problem............................................... 30 1.8.2 Schrodinger’s Cat Paradox ................................................ 31 1.8.3 EPR Paradox......................................................................... 32 1.9 Local Hidden Variables Theory...................................................... 34 2. Algebra of the Exponential O p e r a t o r.......................................... 37 2.1 Parametric Differentiation of the Exponential........................... 37 2.2 Exponential of a Finite-Dimensional Operator........................... 38 2.3 Lie Algebraic Similarity Transformations................................... 39 2.3.1 Harmonic Oscillator Algebra............................................. 41 2.3.2 The SU{2) Algebra...................................................................... 42 2.3.3 The S17(1,1) A l g e b r a................................................................. 43 2.3.4 The SU(m) Algebra................................................................... 45 2.3.5 The SU(m,n) Algebra.............................................................. 45 2.4 Disentangling an Exponential............................................................... 48 2.4.1 The Harmonic Oscillator Algebra......................................... 49 2.4.2 The SU(2) Algebra...................................................................... 50 X Contents 2.4.3 SU( 1,1) Algebra ..................................................................... 51 2.5 Time-Ordered Exponential Integral.............................................. 52 2.5.1 Harmonic Oscillator Algebra................................................ 52 2.5.2 SU( 2 ) Algebra.......................................................................... 53 2.5.3 The S'C/(1,1) Algebra............................................................. 53 3. R epre s ent at i ons of Some Lie A l g e b r a s.......................................... 55 3.1 Representation by Eigenvectors and Group Parameters...................................................................... 55 3.1.1 Bases Constituted by Eigenvectors..................................... 55 3.1.2 Bases Labeled by Group Parameters................................. 56 3.2 Representations of Harmonic Oscillator Algebra...................... 60 3.2.1 Orthonormal Bases................................................................. 60 3.2.2 Minimum Uncertainty Coherent States............................. 61 3.3 Representations of SU( 2 )................................................................. 6 8 3.3.1 Orthonormal Re pr e s ent at i on................................................ 6 8 3.3.2 Minimum Uncertainty Coherent S t a t e s............................. 70 3.4 Representations of 5 ( 7 ( 1,1 )............................................................. 76 3.4.1 Orthonormal B a s e s................................................................. 76 3.4.2 Minimum Uncertainty Coherent S t a t e s............................. 77 4. Q u a s i p r o b a b i l i t i e s a n d No n - c l a s s i c a l S t a t e s............................... 81 4.1 Phase Space Distribution Func t i ons............................................... 81 4.2 Phase Space Representation of S p i n s............................................. 8 8 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Co mp o n e n t s.......................................................................... 93 4.4 Classical and Non-classical S t a t e s.................................................. 95 4.4.1 Non-classical States of Electromagnetic F i e l d................ 95 4.4.2 Non-classical States of Spi n- l/2s ....................................... 97 5. T h e o r y o f S t o c h a s t i c P r o c e s s e s ......................................................... 99 5.1 Pr obabi l i ty Di s t r i b u t i o n s................................................................. 99 5.2 Markov Processes ................................................................................102 5.3 Detailed B a l a n c e..................................................................................105 5.4 Liouville and Fokker-Planck Eq u a t i o n s.........................................106 5.4.1 Liouville Equ a t i o n.....................................................................107 5.4.2 The Fokker-Planck Equat i on ................................................107 5.5 Stochastic Differential E q u a t i o n s....................................................109 5.6 Linear Equations with Additive No i s e.....................................110 5.7 Linear Equat i ons with Multiplicative N o i s e..........................112 5.7.1 Univariate Linear Multiplicative Stochastic Differen ti al Equat i ons.............................................................................113 5.7.2 Multivariate Linear Multiplicative Stochastic Differ ential Equat i ons........................................................................114 5.8 The Poisson P r o c e s s............................................................................115 Contents XI 6. T he El ectromagnetic F i e l d....................................................................119 6.1 Free Classical Fie l d...............................................................................119 6.2 Field Quantization.................................................................................121 6.3 Statistical Properties of Classical F i e l d...........................................123 6.3.1 First-Order Correlation Function.........................................125 6.3.2 Second-Order Correlation Function.....................................126 6.3.3 Higher-Order Correlations......................................................126 6.3.4 Stable and Chaotic F i e l d s......................................................127 6.4 Statistical Properties of Quantized Field.........................................130 6.4.1 First-Order Correlation ..........................................................131 6.4.2 Second-Order Correlation ......................................................132 6.4.3 Quantized Coherent and Thermal Fields ..........................132 6.5 Homodyned Detection .........................................................................134 6.6 Spectrum..................................................................................................135 7. Atom—Field I n t e r act i o n H a m i l t o n i a n s............................................137 7.1 Dipole Interaction.................................................................................137 7.2 Rotating Wave and Resonance Approximations..............................140 7.3 Two-Level Atom.....................................................................................144 7.4 Three-Level A t om.................................................................................145 7.5 Effective Two-Level Atom ..................................................................146 7.6 Multi-channel Models...........................................................................149 7.7 Parametric Processes ...........................................................................150 7.8 Cavity Q E D............................................................................................151 7.9 Moving Atom..........................................................................................153 8. Qu a n tu m Theory of Da m p i n g..............................................................155 8.1 The Master Equation...........................................................................155 8.2 Solving a Master Equation...................................................................160 8.3 Multi-Time Average of System Operators........................................162 8.4 Bath of Harmonic Oscillators ............................................................163 8.4.1 Thermal Reservoir....................................................................164 8.4.2 Squeezed Reservoir..................................................................166 8.4.3 Reservoir of the Electromagnetic F i e l d..............................167 8.5 Master Equation for a Harmonic Oscillator...................................168 8.6 Master Equation for Two-Level A t o m s...........................................170 8.6.1 Two-Level Atom in a Monochromatic F i e l d.....................171 8.6.2 Collisional Damping................................................................172 8.7 Master Equation for a Three-Level Atom.......................................173 8.8 Master Equation for Field Interacting with a Reservoir of Atoms..................................................................174 5.9 Stochastic Differential Equation Driven by Random Telegraph N o is e.......................................................116 XII Contents 9. Linear an d Nonlinear Response of a System in an E x t e r n a l F i e l d................................................................................177 9.1 Steady State of a System in an External Field..............................177 9.2 Optical Susceptibility..........................................................................179 9.3 Rate of Absorption of Energy............................................................181 9.4 Response in a Fluctuating F i e l d.......................................................183 10. Solution of Linear Equations: M et h o d o f Eigenvector Expansion ..................................................185 10.1 Eigenvalues and Eigenvectors ............................................................186 10.2 Generalized Eigenvalues and Eigenvectors ....................................189 10.3 Solution of Two-Term Difference-Differential Equation .............191 10.4 Exactly Solvable Two- and Three-Term Recursion Relations..............................................................................192 10.4.1 Two-Term Recursion Relations.............................................192 10.4.2 Three-Term Recursion Relations ........................................193 11. Two-Level a n d Three-Level Hamiltonian S y s t e m s..........................................199 11.1 Exactly Solvable Two-Level Systems...............................................199 11.1.1 Time-Independent Detuning and Coupling.......................202 11.1.2 On-Resonant Real Time-Dependent Coupling.................208 11.1.3 Fluctuating Coupling..............................................................208 11.2 N Two-Level Atoms in a Quantized Field......................................210 11.3 Exactly Solvable Three-Level Systems............................................210 11.4 Effective Two-Level Approximation.................................................212 12. Dissipative Atomic Systems.................................................................215 12.1 Two-Level Atom in a Quasimonochromatic F i e l d.........................215 12.1.1 Time-Dependent Evolution Operator Reducible to SU{2)..................................................................217 12.1.2 Time-Independent Evolution Operator..............................219 12.1.3 Nonlinear Response in a Bichromatic F i e l d.....................223 12.2 N Two-Level Atoms in a Monochromatic Field............................224 12.3 Two-Level Atoms in a Fluctuating Field........................................236 12.4 Driven Three-Level Atom....................................................................237 13. Dissipative Field D y n a m i c s.................................................................239 13.1 Down-Conversion in a Damped C a v i t y..........................................239 13.1.1 Averages and Variances of the Cavity Field Operators . 240 13.1.2 Density Matrix..........................................................................242 13.2 Field Interacting with a Two-Photon Reservoir...........................245 13.2.1 Two-Photon Absorption.........................................................245 13.2.2 Two-Photon Generation and Absorption...........................247 13.3 Reservoir in the Lambda Configuration..........................................248 14. Dissipative Cavity QED .......................................................................251 14.1 Two-Level Atoms in a Single-Mode Cavity....................................251 14.2 Strong Atom-Field Coupling.............................................................252 14.2.1 Single Two-Level Atom...........................................................252 14.3 Response to an External Field .........................................................255 14.3.1 Linear Response to a Monochromatic F i e l d.....................256 14.3.2 Nonlinear Response to a Bichromatic F i e l d.....................257 14.4 The Micromaser....................................................................................259 14.4.1 Density Operator of the Field...............................................259 14.4.2 Two-Level Atomic Micromaser............................................263 14.4.3 Atomic Statistics......................................................................266 Appendices............................................................................................................267 A. Some Mathematical Formulae...........................................................267 B. Hypergeometric Equation....................................................................270 C. Solution of Two- and Three-Dimensional Linear Equations......................................272 D. Roots of a Polynomial ........................................................................273 References..............................................................................................................277 I n d e x.........................................................................................................................283 Contents XIII

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