h n a l e n der Physik. 7. Folge, Band 38, Heft 2, 1981, S. 97-105 J. A. Barth, Leipzig Radiation Properties of Systems of Three and Four Continuously Pumped Atoms By H. STEUDEL Zentralinstitut fur Optik und Spektroskopie der Akademie der Wissenschaften der DDR, Berlin-Adlershof Dedicated to Prof.Dr. Gustav Richter on the Occasion of the 70th Anniversary of hi.? Birthday Abstract. Radiation properties of three and four atoms continuously pumped by an inmherent escitation mechanism are investigated by using Lehmberg’s master equation and the quantum fluctuation-regressiontheorem. For proper configurations we found in comparison with independent atoms simultaneously (i) an enlargement of the radiation rate, (ii) a spectral narrowing and (iii) a reduction of intensity fluctuations. Strahlungseigenschaftenyon Systemen von drei und vier kontinnierlich gepumpten Atomen Inhaltsubersicht. Es werden Strahlungseigenschaften von drei und vier Atomen, die durch einen inkohiirenten Pumpmechanismus kontiiuieriich angeregt werden, untersucht mit Hilfe der Lehmbergschen Mastergleichung und des Quanten-Fluktuations-Regreesionstheorems. Fiir geeignete Konfigurationen konnten im Vergieich zu unsbhengigen Atomen gleichzeitig (i) eine erhohte Strahlungsrate, (ii) eine spektrale Einengung und (iii)eine Verringerong der Intensitatsfluktuationen festgestellt werden. 1. Introduction Current theories of superfluorescence [l- 61 explain a good deal of experimental results [7- 121. Most of these experiments were done in gases at low pressures such that the mean distances between active molecules are of the order of one wavelength or larger. For higher densities near-zone dipole-dipole interaction becomes important, and this is not taken into account in the quoted theories. To establish a theory of superfluorescence for a macroscopic system other than a regular lattice including near-zone interaction is a rather tricky problem. Because of the r-3 dependence of the near-zone field the individual environment of any atom is important and the atoms cannot be “smeared out” t o form a continuum. Another crucial difficulty is connected with the fact that the excitation operators describing a system of atoms, say for simplicity of two-level atoms, fulfil neither Bose-Einstein nor Fermi-Dirac statistics. Therefore collective excitation modes such as excitons or polaritons can be introduced only in the limits of very low or very high excitations. Considerations of this kind gave us the motivation to investigate as complete as possible, and with some numerical effort, the radiation from a few atoms. We calculated radiation rate, radiation pattern, spectrum and photon correlations for two atonis [131 H.STEUDEL 98 and did the same, with exception of photon correlations, for various configurations of three atoms [14]. Corresponding properties of the spontaneous radiation from two and three initially inverted atoms were studied by TH.RICHTER [15, 161. With the restricton t o the high-excitation limit the initial process of superfluorescence was investigated for systems up to 48 atoms both in regular and in irregular geometrical configurations 1171. COFFEYand FRIEDBERG [18] investigated the role of near-zone interaction for two and three atoms in the limit of very short distances. Systems of harmonic oscillators homogeneously distributed over a sphere were investigated by ~ ~ O W I C[19], Z and of three and four harmonic oscillators in triangle and tetrahedron configurations respectively by LEWENSTEIN and RZ@EWSKI[201. Many authors- most recently PEAKASH and CHAXDRA [21]-have investigated the radiation froni N atoms contained in a volume with dimensions small compared with a wavelength without taking into account near-zone interaction. This, in our opinion, is inconsistent. As a remarkable result of our investigations in the present and in the earlier papers we mention the simultaneous occurrence of high radiation rate, small line-width and reduced amplitude fluctuations for proper configurations and the existence of an optimal density of atoms; cf. the discussion in section 5 below. I n section 2 we introduce the N-atom master equation due to LEHMBERG [22]. Results concerning radiation rate and spectrum of three and four atoms are presented in section 3 and those concerning photon correlations in section 4.I n the appendix a general transformation property of the N-atom master equation is derived. 2. The N-atom Master Equation We consider N identical two-level atoms at fixed space points x,, ,LA = 1, 2, .., N . The twolevels are connected by a dipole transition with the common polarization vector e. a t , a, are raising and lowering operators. The occupation number operators are denoted by ml, = a,a,f, n, = aza,. The temporal development of the expectation value of any atomic operator Q is described by the master equation [22, 23, 131 . (6)= i CQa,([afa,, &I> - 2 y~,(a,ta,C? + Qa,i'-ap- 2afapQ) a,, a,, 5 C (n,Q + Qn, - 2a$Qap> - x C (mpQn, a -. I 2'(nap& + Qm, - 2a,Qa$> - c1 + naQmJ (1) P G,,,, = coo and 2y,, = 2y are transition frequency and decay rate for onesingleatom. GAP and ynpwith 3, ,u describe the mutual electromagnetic interaction of t,he atoms with + the near-zone interaction included. After taking out a factor y , Qa, = Yqap, Y A P = YPa,, we are left with the pure geometrical enbities q2+, pnP defined by -qAp + ip,, 3 = (koor)-l [-1 - (ek)2] + [i(kor)-2- (kor)-3][ l - 3(e;)2]exp (ikor), (3) = Irl, i = r / r , ko = o o l c . The term containing 7 describes a non-radiative relaxat)ion, the one with the factor I' accounts for the incoherent pumping process with W being the pumping rate, and the term Containing x = TF1 - (2T1)-l with T,,T,being the commonly used relaxation times describes phase interrupting collisions. These three terms were not included in the ~nast~er equation as it was derived by other authors [21,22]. They are, however, uniquely 1' Radiation of Three and Four Continuously Pumped &oms 99 determined by the requirements i) to give the phenomenological one-atom terms commonly used and supported by quantum-mechanical models and ii) t o act independently at the individual &toms.For our numerical computation we put j j = x = 0. The transformation property derived in the appendix applies to the more general version. The method we utilize here, as we did in the earlier papers, works in three steps: (i) The dynamics of N atoms is described by the master equation (1). (ii) Multi-time correlations of atomic operators are reduced t o equal-time expectation values by use of the quantum-fluctuation regression theorem, see LAX[24]. (iii) Field correlations are calculated from atomic correlation functions. The 4N-dimensional operator space of our N-atom system may be spanned by products of one-atom operators a t , up, mp, n,withone factor for each atom. Each base operator Q,,.r = 1 . 4N,built in this way maps exactly one state with distinct excitation number Mi t o one state with excitation number M , = Mi A M (excitation number M = number of excited atoms). Fortunately the 4N equations resulting from eq. (1)couple only expectation values of operators with the same A M . Their number is . + 2N! (4) (N - A M ) ! (N + AM)!' I n order t o calculate the radiation rate, spectrum and intensity correlations we are only interested in the subspaces AM = 1,-1, 0. I n general v ( N , d M ) = Y ( N ,- A M ) . We find ~(3,0) = 20, ~ ( 3 . 1= ) 15, ~ ( 4 ~= 0 )70, ~(4.1) = 56, (5) = 210 ~ ( 5 , 0= ) 252, ~(5.1) and conclude from these numbers that with an appropriate computational effort and without further approxiinations or imposition of symmetries the method characterized above is applicable for atomic numbers N 5 4. For details we refer t o earlier papers [13, 141. The numerical computations were carried out with a BESM-6 computer. Y(N,d M ) = 3. Radiation Rate and Spectrum We calculated the total radiation rate for three different geometrical configurations of four atoms: (a) a square, polarization vector e orthogonal t o the configuration plane (b) a n equidistant linear chain, e orthogonal t o the configuration axis and (c) an irregular plane arrangement, e in the configuration plane. The ratio of pumping rate and spontaneous decay rate is chosen as I'/y = 4. I n Fig. 1 the dependence on the smallest distance E is depicted when the configurations altered similar to themselves. While for the regular arrangements (a, b ) the radiation rate increases monotonically with decreasing distance I , for the irregular arrangement (c) the radiat,ion rate passes a maximum a t I = 0.1151. Such an effect was already found for three atoms [I41but was less significant there. Due t o the near-zone interaction hhe spectrum of the radiation from three or four atoms is highly structured for atomic distances 1 < Al2n. At another place [I41 the splitting into three components of the spectrum radiated by three equidistant atoins was depicted. Fig. 2 of the present paper demonstrates that the 8-level scheme (1 3 + 3 1) of three two-level atoins is able t o produce a much more complicated radiation spectrum. No less than six components are seen. At the other hand for intermediat,e distances E M 1/2n and for proper pumping rates a line-narrowing takes place. For three equidistant atoms, 1 = 0.181, e orthogonal t o the configuration line, Fig. 3 shows the depen- + + H.STEUDEL 100 0.1 0.2 0.4 0.3 0.5 MINIMUM DISTANCE 1 (WAVELENGTH) Fig. 1. Total radiation rate of four atoms in dependence on the smallest distance I for a) a square, b) a linear equidistant chain and c) an irregular configuration. Pumping rate divided by spontaneous decay rate = Fly = 4 - 20 -10 0 Am /2y 10 20 Fig. 2. Spectrum of the radiation emitted from three atoms for the configuration and the detection direction indicated in the upper right. rly = 1/8 Radiation of Three and Four Continuously Pumped Atoms 101 dence of radiation rate and spectral width (full width at half maximum) on the pumping rate. The direction of observation is chosen orthogonal both to e and to the configuration axis. It is seen that for Ply 2 2 the linewidth is smaller while at the =me time the radiation rate is larger than the corresponding values for independent atoms. Radiation rate and spectral width for three equidistant atoms were also studied in the earlier paper [14], but there the dependence on the atomic distance 1 was depicted with the pumping rate fixed. -cN X v) Z 0 I- 0 -a I W I- a Z 0 I- a P 4 u I I- 0 5 z -I +' a Z \ I : I- -I a a IV W L v) PUMPING RATE/ 2~ Fig. 3. Rate and spectral width of the radiation from three equidistant atoms, the latter measured orthogonal to the configuration line, in dependence on the pumping rate for atomic distance 2 = 0.18A (full line) and for three independent atoms (dmhed line). Polarization vector e orthogonal to the configuration axis H.STEUDEL 102 For N = 4 we calculated only a few spectral data. For 4 equidistant atoms, 1 = 0.17A, = 4, e orthogonal to the configuration line, observation orthogonal to both e and the axis, we found a spectral narrowing by 10% and an enlargement of radiation rate by 9.5% compared with independent atoms. For 1 = 0.18A, r / y = 3 the corresponding values were -7.1% and +7.3%. 4. Intensity Correlations Now we are studying second order field correlations defined by From ($8) the normalized intensity correlations function g(2) in the far-field is defined as follows rl 7 r,+Oo. The operator I ( r , t ) = Hn)(r,t ) E(+)(r,t ) (8) measures the intensity of the radiation. With :: indicating the normal operator product the relative variance in the far-field is defined as v = ( : ( I - <I))a:)/<Zy = ( : P : > / < Z ) Z - 1 = g q i . , i ;0) - 1. (9) 0.4 I... INDEPENDENT ATOMS 1 ,. .........-“f.:: I ).= . ---.-. ....................... AT MINIMUM_. 0.2 0.4 0.6 I D ISTA Nc E/ WAVE LENGTH Fig. 4. Relative variance V of the radiation intensity measured in the direction of the maximum (full line) and in the direction of the minimum (dashed line) of the radiation pattern for three equidistant atoms on a straight line. r / y = 3, e orthogonal to the configuration line. The radiation pattern is considered only in the plane orthogona1to e. For comparison the value V = 1/3for three indepndent atoms is marked (dashed line) 103 Radiation of Three and Four Continuously Pumped Atoms V > 0 means “bunching”, V < 0 “antibunching” of photons. In Fig. 4 the relative variance is depicted for three equidistant atoms in dependence on the atomic distance I , Fly = 3, e orthogonal to the configuration axis. As directions of observation we chose both that of the maximum and that of the minimum of the radiation pattern in the plane orthogonal to e. The variance in the direction of the maximum is always smaller than the value V = 113 for the radiation from three independent atoms. The variance in the direction of the minimum on the other hand is always greater than 113 provided that 1 > 0.21. For 4 atoms arranged in a linear equidistant chain- r / y = 3, e orthogonal to the configuration line, direction of observation orthogonal to both these directions-we found V = 0.36 for 1 = 0.181 and V = 0.39 for 1 = 0.201. The corresponding value for four independent atoms is V = 0.5. In Fig. 5 the temporal correlation function q@)(i,i ;A t ) is seen for three equidistant atoms with various atomic distances and for various detection angles. Depending on the parameters such a curve may lie above or below the one corresponding to three independent atoms. Contrary to the cases N = 1or 2 [13] we found no antibunching for N 2 3. However there may occur a delayed anticorrelation, see curve (d) where values q(2)< 1 are taken for finite A t . \ \ \ 0.9 I Q5 1 DELAY T IM E X 2 y Fig. 6. N o m l b d intensity cornlation function g@)(?, ; At) in the far-zone for three equidistant atoms for a) I 3 co,i.e. independent atoms, b) I = O.U, Q = Oo, c) 1 = 0.18rZ, Q = 0’ and d) 2 = O.l&I, Q = 90’. e orthogonal to the c o n f i a t i o n line, I’/y -- 3. The direction of detection is always chosen orthogonal to e and forms the angle @ with the aonfiguration line. It coincidee with the minimum (b and c) or with the maximum (d) of the radiation pattern H. STEUDEL 104 5. Discussion (A) That the radiation rate for an irregular arrangement of atoms passes a maximum when the smallest atomic distance is 1 m O.#, see curve (d) a t Fig. 1,is in agreement with results obtained froni our investigations on the initial process of superfluorescence for 24 to 48 atoms [17]. There we concluded that for a statistical distribution of atoms there is an optimal density of atoms corresponding t o mean atomic distances between 1/10 and 1/20. Such a value for the optimal density seems t o be in rough quantitative agreement with experimental resulta, compare Fig. 1in the paper by OKADAet al. [ll], though for a more exact comparison inhomogeneous broadening would be of importance which was not accounted for in our calculation. (B) Let us look a t a special linear arrangement of three atoms with distances 1 = 0.181 and polarization vector e orthogonal to the configuration axis. The pumping rate may be three times the spontaneous decay rate. When we compare the radiation emitted by these atoms and detected a t a direction orthogonal both t o the axis and t o e with radiation from three independent incoherently pumped atoms we find (i) an enlargement of the total radiation rate by GAYo, see Fig. 3, (ii) a line-narrowing by 6.8%, see Fig. 3, and (iii) a decrease of the relative variance from 1/3 t o 0.12, see Fig. 4, i.e. nniplitude stabilizat ion. With reference to these three properties we are tempted t o speak of a ‘(three-atom laser”. We are aware however, that one essential laser property, namely the threshold, is absent. We believe that something of these remarkable properties we found for highly idealized systems of a few atoms will survive in a more realistic macroscopic system. However, because of the difficulties pointed out in the introduction we cannot draw quantitative conclusions besides the above rough estimate of an optimal density. The author wishes to thank Dr. TH. R~CHTER for detailed discussions. Appendix A Transformation Property of the N-Atom Master Equation The level scheme of a system consisting of N two-level atoms has a spindle-shaped structure with one top being the ground state and the other being the fully inverted state. The structure of the uppermost “store” with transitions starting from the fully inverted state reflects the structure of the lowest “store” with transitions ending a t the ground state. We widely made use of this symmetry when we were dealing with the initial process of superfluorescence [17]. Now we raise the question whether a corresponding up-down symmetry could be established for the complete master equation (1). Once again we iise the operator base {Qr}introduced in section 2 and define a transformation w1iir.h we will call conjugation $?, w: a? + a?, aA+ aA, InA+ s n A , nn+ mA. Gf can be written as a product of the inversion f , (A 1) a 2 ++ aA, rn1 f + nn, (A 2) which exactly exchanges up and down, and the operation of hermitian conjugation%. Let a,,Z,, p,, vr denote the numbers of factors a, a+,rn, n respectively in the base operator Q,. Obviously my Er pr v, = N. Then it is easy to write down explicitly 9: + + + Radiation of Three and Four Continuously Pumped Atoms 105 the diagonal part of eq. (1)with respect to our fixed operator base, After a thorough discussion of the behavior of all terms under the conjugation V we find The diagonal part transforms in a transparent way while the off-diagonal part simply transforms to its hermitian conjugate. References [l] J. C. IVL~CGILLIVRAY and M. S. FELD,Phys. Rev. A 14,1169 (1976). and F. HAAEE,Phys. Lett. A 68, 29 (1978). [2] R. GLAUBER [3] F.U,H. ICING, G. S ~ O D E RJ., Hans and R. GLAWE, Phys. Rev. A 90, 2047 (1980). [4] D. POLDER, M. F. H. S C ~ M A Nand S Q. H. F. VREIXEN, Phys. Rev. A 19,1192 (1979). Phys. Lett. A 72, 306 (1979). [6] M. F. H. S m m m m s and P. POLDEB, ~ R. K. BULLOUGH, Opt. Commun. 81,219 (1979). [6] J. A. H m t and [7] N. Smmmowmz, J. P. HERMAN, J. C. MACGILLIVRAY and M. S. FELD,Phys. Rev. Lett. 80, 309 (1973). [8] M. GROSS,C. FABRE, P. P I I J J Z and T S. m o o = , Phys. Rev. Lett. 86, 1035 (1976). [9] A. FLUSBERG, T. MOSBERG and S. R. HARTBUNN, P h p . Lett. A 68, 573 (1976). [lo] C. M. BOWDEN,D. W. HOWGATE and H. R. ROBL(ed.), Cooperative effects in Matter and Radiation, Plenum, New York 1977. [ll]J. OKADA, K. IKEDA and M. MATSUOKA, Opt. Commun. 27, 321 (1978). [12] Q. H. F.VREHEN,H. M. J. HIKSPOORS and H. M. GIBBS,in: Coherence and Quantum Optics IV,ed. L. & imm, and E. Worn, Plenum, New York 1978. [13] H. STEUDEL and TH.RICHTER,Ann. P h p i k Leipz. 36,122 (1978). [14] H.STEUDEL,J. Phys. Lond. B 12, 3309 (1979). [15] TH.RICHTER,Ann. P h p i k Leipz. 86, 266 (1979). [16] TH. RICHTER,to be published. [17] H. STEUDEL, Ann. P h p i k Leipz. 87, 67 (1980). Phys. Rev. A 17,1033 (1978). [l8] B.COFFEY and F. FRIEDBEBG, [19] W. '~AKOWICZ, Phys. Rev. A 17, 343 (1978). and K. RZ.@EWSKI,J. Phys. Lond. A 18, 743 (1980). 1201 M. LEWENSTEIN [2l] R. PRAXASH and N. CEANDRA,Phys. Rev. A 21,1297 (1980). [22] R. H. LEHMBERG, Phys. Rev. A 2, 883 (1970). Springer Tracts Mod.Phys. 70, 1(1974). 1231 G. S. AGARWAL, [24] M. LAX,Phya. Rev. 129, 2342 (1963);172, 360 (1968). Bei der Redaktion eingegangen am 11.September 1980. Anschr. d. Verf.: Dr. H. STEUDEL Zentmlinstitut fiir Opt& und Spektroskopie der Akademie der Wissenschaften der DDR DDR-1199 Berlin-Adlershof Rudower Chaussee 6 8 Ann. Physik. 7. Folge, Bd. 38

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