Ann. Phys. (Leipzig) 16, No. 5 – 6, 342 – 349 (2007) / DOI 10.1002/andp.200610238 Berry phase of two atoms in the same quantized light ﬁeld Mai-Lin Liang∗ , Jin-Hua Zhang, and Bing Yuan Physics Department, School of Science, Tianjin University, Tianjin 300072, China Received 9 August 2006, revised 11 February 2007, accepted 13 February 2007 by U. Eckern Published online 8 March 2007 Key words Berry phase, quantized light ﬁeld, Jaynes-Cummings model. PACS 03.65.Vf The Berry phase of two atoms exposed to the same quantized light ﬁeld has been studied. The interaction between each atom and the light ﬁeld obeys the Jaynes-Cummings model. Due to the quantization of the light ﬁeld, the interaction Hamiltonians for the two atoms interacting with the ﬁeld do not commute with each other, which induces new parts in the Berry phase. Meanwhile, there appears a phase which doesn’t exist in the classical regime and is purely induced by the ﬁeld quantization. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction For any wave the phase is very important. The concept of geometric phase was ﬁrst introduced by Pancharatnam [1] in his pioneering work, comparing the phases of two beams of polarized light. The discovery of the quantum counterpart of the geometric phase by Berry in 1984 [2] strongly motivated the research on the phases of wave functions in quantum mechanics. For a quantum system with parame time-dependent ters (R1 (t), R2 (t), . . .) ≡ R(t), the instantaneous eigenstate can be written as n, R(t) with n being the quantum number. When the time-dependent parameters change slowly but periodically, the quantum state evolves gradually and eventually returns to its initial form. The state acquires a quantum adiabatic geomet ∂ rical phase, which is now known as the Berry phase, γ = i n, R n, R · dR, in addition to the usual ∂R dynamic phase. For a spin rotating in a time-dependent magnetic ﬁeld, the Berry phase turns out to be the solid angle in the parameter space. Since the work of Berry [2], there have been many studies on this quantum geometrical phase [3–5]. Theoretically, the original adiabatic and periodic evolution [2] is generalized to, for instance, non-adiabatic evolution [6], non-adiabatic and non-cyclic evolution [7], mixed states [8,9], open systems [10], and even with a quantized ﬁeld driving [11–13]. Meanwhile, various kinds of practical models are studied, which include the spins in a time-dependent magnetic ﬁeld [2,14], the time-dependent harmonic oscillator [15], the spin-orbit interaction [16–18], the composite systems [19–21], etc. Sjöqvist calculated the geometric phases for a pair of entangled spins [19], elucidating how the entanglements sway the Berry phase. Tong et al. discussed the relation between the geometric phase of the system and that of the subsystems [20, 21]. Studies also show the importance of geometric phases in the areas of physics, such as the Bose-Einstein condensates [22], the Jahn-Teller effect [23], the understanding of the anomaly phenomenon in quantum ﬁeld theory [24], and in many interesting topics of quantum optics [25–29]. The geometric phases have been tested in a numerous experiments [30]. A feasible experiment to measure the effects induced by the ﬁeld quantization is proposed in [12], which may be carried out in the near future. Besides the theoretical interest, the geometrical phase has potential applications, among which the geometric quantum computation [31–36] is one of the most importance. The basis of geometric quantum computation is the phase shift of quantum states [35] which is particularly attractive because of the fact ∗ Corresponding author E-mail: mailinliang@yahoo.com.cn, mailinliang@eyou.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007) 343 that geometric phases depend only upon the global features and it is robust against certain errors and dephasing [35,36]. One candidate for quantum computation is to use the neutral atoms [37,38]. The atoms can be loaded into the optical lattices [39,40] and one can use external light ﬁelds to manipulate the atoms. So, the study on the geometric phases of atoms under the quantized light ﬁelds [11–13] is meaningful both theoretically and practically. On the other hand, the atom-ﬁeld interaction has been a hot topic in quantum optics [41–46]. The dynamical properties [41–43] and the entanglements [44–46] have been widely investigated. The studies of the geometric phase make us gain new insights into such atom-ﬁeld interaction systems. In [11], a fully quantized version of the phase for the system of an atom interacting with a quantized light ﬁeld has been given that considers the vacuum-induced effects. Under the rotating wave approximation, the Hamiltonian is [11, 47] H = ωsz + ωf a+ a + λ as+ + s− a+ (1.1) where sz = (|e e| − |g g|)/2, s+ = |e g| , s− = |g e| with |g and |e being the ground and excited states of the two level atom respectively, λ is the atom-light coupling coefﬁcient, a+ and a are the creation and annihilation operators for the light ﬁeld. Deﬁning sx = (s+ + s− )/2, sy = (s+ − s− )/(2i), the operators sx , sy and sz are named the pseudo spin operators for the atom. The Hamiltonian (1.1) can be split into two parts, H = H0 + Hint : H0 = (ω − ∆)sz + ωf a+ a , Hint = ∆sz + λ as+ + s− a+ (1.2) where ∆ = ω − ωf is the detuning between the atom and the ﬁeld. One can prove that H0 and Hint commutate with each other. So, in the interaction picture, the interaction Hamiltonian is just Hint . The state vector obeysHint |ψ = i∂ |ψ /∂t. The stationary equation is Hint |ψ = ε |ψ, where ε is the eigenvalue of Hint . The two eigenvales are found to be ε± = ± 12 ∆2 + 4λ2 (n + 1) (1.3) where n is the photon number. The corresponding Berry phases are [11] ∆ α+ = 2πn + π 1 − , ∆2 + 4λ2 (n + 1) ∆ α− = 2πn + π 1 + . ∆2 + 4λ2 (n + 1) (1.4) When n = 0, the Berry phase is nonzero, which is the vacuum-induced phase. In this paper, we deal with the case that two atoms are exposed to the same quantized light ﬁeld. The interaction between each atom and the light ﬁeld obeys the Jaynes-Cummings model [47]. The Hamiltonian for the whole system is (1) (2) + + (1) H (12) = ωs(1) + ωs + ω a a + λ as + a s f z z + − (2) (2) (1) (2) (1) (2) + λ as+ + a+ s− + β s+ s− + s− s+ (1.5) where β is the atom-atom dipole coupling constant. The two terms with λ are the interaction Hamiltonians of atom 1 with the ﬁeld and atom 2 with the ﬁeld respectively. The two atom-ﬁeld interaction Hamiltonians for the two atoms in (1.5) don’t commute with each other and represent a case different from the former studies [11–13], where only one atom exists or two atoms are in different quantized light ﬁelds. It’s interesting to see what new results there will be for the Hamiltonian (1.5). The next section is the calculation of the exact Berry phase and the ﬁnal section is the conclusion. www.ann-phys.org c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 344 Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light ﬁeld 2 The Berry phase The Hamiltonian (1.5) for two atoms in a quantized light ﬁeld can be written as the sum of the following two parts (12) (2) H0 = (ω − ∆) s(1) + ωf a+ a , z + sz (12) (1) (1) (2) Hint = ∆ s(1) + λ as+ + a+ s− (2.1) z + sz (2) (2) (1) (2) (1) (2) + λ as+ + a+ s− + β s+ s− + s− s+ . (12) (12) One can prove that H0 and Hint communicate with each other. So, in interaction picture, the interaction (12) (12) (12) Hamiltonian is Hint . The wave function obeys the evolution equation Hint |ψ = i∂ |ψ /∂t. As Hint (12) (12) is time-independent, the stationary equation is then Hint |ψ = ε |ψ, where ε is the eigenvalue of Hint . By some analysis, it’s found that the eigenfunction can be written as the superposition |ψ = C1 |ee, n + C2 |eg, n + 1 + C3 |ge, n + 1 + C4 |gg, n + 2 . (2.2) The coefﬁcients Cµ , µ = 1, 2, 3, 4 are to be found below. Substituting (2.2) into the eigenvalue equation (12) Hint |ψ = ε |ψ, we have ∆, √ λ n + 1, √ λ n + 1, 0, √ λ n + 1, 0, β, √ λ n + 2, √ C1 λ n + 1, 0 C1 √ β, λ n + 2 C2 C2 √ = ε C3 0, λ n + 2 C3 √ C4 C4 λ n + 2, −∆ (2.3) which can be solved analytically. There are totally four eigenvalues. One eigenvalue is ε1 = −β and the coefﬁcients are C1 = C4 = 0, C2 = −C3 = 1. The corresponding eigenfunction takes the form 1 |ψ1 = √ (|eg − |ge) |n + 1 . 2 (2.4) The other three eigenvalues satisfy ε3 − βε2 − aε + c = 0 (2.5) where a = ∆2 + 2λ2 (2n + 3) , c = 2λ2 ∆ + g∆2 . (2.6) The exact solution to (2.5) is ϕ β , + 2R cos 3 3 ϕ π β ε3 = − 2R cos + , 3 3 3 ϕ β π ε4 = − 2R cos − 3 3 3 ε2 = c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2.7) www.ann-phys.org Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007) where R= 1 3 345 3a + β 2 , ϕ = cos−1 9βa + 2β 3 − 27(2λ2 ∆ + g∆2 ) . 2(3a + β 2 )3/2 (2.8) Substituting the eigenvalues (2.7) into (2.3), the coefﬁcients and so the eigenfunction (2.2) are found to be √ √ 2λ n + 1 n + 2 1 2λ ψj = √ |gg, n + 2 , |ee, n + |eg, n + 1 + |ge, n + 1 + εj − ∆ ∆ + εj N (2.9) 4λ2 (n + 1) 4λ2 (n + 2) N =2+ + (∆ − εj )2 (∆ + εj )2 where j = 2, 3, 4. Using (2.9), the Berry phase can be calculated. In the full quantized regime, the geometrical phase is [11–13] + d ψj U (φ) U (φ) ψj dφ (2.10) γj = i dφ c where U (φ) = exp(−iφa+ a)is a phase shift operator and φ changes slowly from 0 to2π. For the state (2.4), the Berry phase is 2(n + 1)π or zero (mod 2π). For the state (2.9), the Berry phase is not difﬁcult to get γj = 2π(n + 1) (2.11) −16λ2 εj n∆ + 4λ2 2(∆ − εj )2 − (∆ + εj )2 . + 2π 2 2 2 2 2 2 (∆ + εj ) (∆ − εj ) + 4λ (n + 1) + (∆ − εj ) (∆ + εj ) + 4λ (n + 2) Though this result is general, in the following we mainly focus our attention on the case that there is no the dipole-dipole interaction orβ = 0 to consider the pure effects of ﬁeld quantization. Classically, if there is no the dipole-dipole interaction orβ = 0, the Berry phase of the two-atom system will be two times that of the one-atom system. From (2.11) and (1.4), one sees that when the effect of the ﬁeld quantization is included, there does not exist such a relation. Numerical results also show the same conclusion. The difference between the Berry phase of the two-atom system and two times that of the one-atom system is plotted in Fig. 1 (For all the ﬁgures in this article, the vertical axis is the Berry phase and the longitudinal axis is the photon number). Figs. 1a and b correspond to the states with ε2 and ε4 respectively. At ﬁnite n, the difference is obvious. In the classical limit n → ∞, the difference tends to zero, which means that now the Berry phase of the two-atom system is two times that of the one-atom system. Such a result can be derived analytically too. Setting n → ∞, we have √ √ ε3 = 0 , ε4 = − a . (2.12) ε2 = a , The corresponding Berry phases are γ3 = 0 and ∆ , γ2 = 2πn + 2π 1 − √ ∆2 + 4λ2 n ∆ γ4 = 2πn + 2π 1 + √ . ∆2 + 4λ2 n (2.13) Thus, in the limit n → ∞, Berry phases in (2.13) are two times that in (1.4) (mod 2π). The Berry phase γ3 as a function of photon number is plotted in Fig. 2 (mod 2π), which becomes zero in the classical limit. www.ann-phys.org c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 346 Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light ﬁeld 5 10 15 20 25 30 -0.25 -0.5 -0.75 -1 -1.25 -1.5 -1.75 -2 a 4 3.5 3 2.5 2 1.5 1 Fig. 1 Differences between the Berry phase of the two-atom system and two times that of the one-atom system. The parameters are ∆ = 0.5 , λ = 1.0 . 0.5 b 5 10 15 20 25 30 0.5 0.4 0.3 0.2 0.1 5 10 15 20 25 30 Fig. 2 Berry phase of the state with eigenvalue ε3 . The parameters are ∆ = 0.5 , λ = 1.0 . That the phase γ3 is zero in the classical case means that there is no such a phase classically, or this phase is generated purely by the ﬁeld quantization. Now let’s have a look at the changes of Berry phase with the dipole-dipole coupling constant. In [13], all the Berry phases for the eigenstates go to zero in the limit β → ∞. For the present system, it is different. In the limit β → ∞, it is found that ε2 = β and ε3 = ε4 = 0. For ε2 = β, the state (2.9) becomes ψj → √1 [|eg, n + 1 + |ge, n + 1] = √1 [|eg + |ge] |n + 1 2 2 c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2.14) www.ann-phys.org Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007) 347 and the Berry phase is zero. For ε3 = ε4 = 0, the eigenfunction (2.9) becomes √ √ 2λ n + 1 2λ n + 2 ψj = √1 |ee, n + |eg, n + 1 + |ge, n + 1 + |gg, n + 2 , −∆ ∆ N (2.15) 4λ2 (n + 1) 4λ2 (n + 2) N =2+ + . ∆2 ∆2 The Berry phase is γj = 2πn + 2π ∆2 + 4λ2 (n + 2) ∆2 + 2λ2 (2n + 3) (2.16) which is a ﬁnite value and can’t be written as 2πm(m integer) or zero. According to the deﬁnition, whether the Berry phase of a state is zero or not is determined by the form of the eigenfunction. For the toy model in [13], the states reduce to |ee, n, |eg, n + 1, |ge, n + 1, |gg, n + 2, and all the Berry phases for the eigenstates are zero. Actually, for the Ising-type interaction in [13], not all the Berry phases are zero either in the limit β → ∞ if the two atoms are both exposed to the same quantized ﬁeld. Replacing the atom-atom interaction in (1.5) by βσ1z σ2z , the equation (2.3) becomes √ √ C1 ∆ + β, λ n + 1, λ n + 1, 0 C1 √ √ λ n + 1, −β, 0, λ C C n + 2 2 2 (2.17) √ √ = ε . C3 λ n + 1, 0, −β, λ n + 2 C3 √ √ 0, λ n + 2, λ n + 2, −∆ + β C4 C4 One of the eigenvalues is ε1 = −β and the eigenfunction is the same as (2.4). The corresponding Berry phase is zero. The other three eigenvalues satisfy ε3 − βε2 − β 2 + ∆2 + 2λ2 (2n + 3) ε + β 3 − β∆2 + 4nβλ2 + 2λ2 ∆ = 0 . (2.18) The corresponding eigenfunctions can be written as √ √ 2λ n + 2 2λ n + 1 ψj = √1 |ee, n + |eg, n + 1 + |ge, n + 1 + |gg, n + 2 , ∆ + εj − β N εj − β − ∆ 4λ2 (n + 1) 4λ2 (n + 2) N =2+ + . (∆ + β − εj )2 (∆ + εj − β)2 (2.19) 2 In the limit β → ∞, Eq. (2.18) reduces to (ε + β) (ε − β) = 0, solutions of which are ε2 = ε3 = β and ε4 = −β. For ε4 = −β, the state (2.19) tends to (2.14) and the Berry phase is 2π(n + 1) or zero. But for ε2 = ε3 = β, the state (2.19) becomes √ √ 2λ n + 1 n + 2 1 2λ ψ j = √ |ee, n + |eg, n + 1 + |ge, n + 1 + |gg, n + 2 , −∆ ∆ N (2.20) 4λ2 (n + 2) 4λ2 (n + 1) + . N =2+ ∆2 ∆2 The Berry phase is the same as (2.16) and is nonzero. One may notice that, in the strong coupling limit β → ∞, the parametersλ, ∆disappear from the energy eigenvalues, but still appear in the eigenfunctions. This phenomenon may be understood as follows. In the total energy, different forms of energy are summed up. If one part is large enough, other parts will be negligible. But for the eigenstates, different interactions stand for different actions on the states. One interaction can’t replace all other ones, even if this interaction is very strong. www.ann-phys.org c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 348 Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light ﬁeld 3 Conclusions New results for the Berry phase of two atoms interacting with the same quantized light ﬁeld are presented. The atom-ﬁeld interaction Hamiltonians for the two atoms don’t commute with each other, which induces new phenomena for the geometric phases: (1) The geometric phase for the two-atom system is not two times that of the one-atom system even if there is no atom-atom interaction. But, in the large quantum limit, the geometric phase for the two-atom system is two times that of the one-atom system, which is just the classical case. (2) When the atom-atom coupling is inﬁnite, the geometric phase for the eigenstate |ψ2 is not zero, which is different from the results in [13], where the geometric phase for each eigenstate is zero. In [13], the atom-ﬁeld interaction Hamiltonians for the two atoms commute with each other. 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