Quantum decoherence of excitons in a leaky cavity with quasimode

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1 PHYSICAL REVIEW A, VOLUME 63, Quantum decoherence of excitons in a leaky cavity with quasimode Yu-xi Liu, C. P. Sun, and S. X. Yu Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beiing 18, China Received 13 June 2; published 14 February 21 We investigate the quantum decoherence of a mesoscopically superposed states of excitons in a quantum well placed within a leaky cavity. Based on the factorization theory of quantum dissipation, our investigation shows that the coherence of the superposed states of the excitons will decrease in an oscillating form when the cavity field is of the quasimode form. Regarding the effects of the thermal cavity fields on the quantum decoherence of the superposition states of the exciton, we observe that the higher the temperature of the environment, the shorter the decoherence characteristic time. DOI: 1.113/PhysRevA PACS numbers: 42.5.Fx, y I. INTRODUCTION The superposition principle is a basic principle governing the quantum world. The quantum coherence of superposition states in the microscopic world has been manifested in various interference experiments, e.g., the which-way experiments using atoms in Bragg scattering 1 and the Aharonov- Bohm electrons in mesoscopic system 2. Recently, the decoherence of superposed motional states of a trapped single atom, which is induced by coupling the atom or ion to engineered reservoirs, was tested quantitatively in the experiments with cooling atoms and ions 3. However the macroscopic superposed state has not been observed yet. This fundamental dilemma was initially noticed by Schrödinger in his famous cat paradox 4,5. Recent experiments demonstrated that the coherent superposition and its decoherence process can be observed in the laboratory, at least in the mesoscopic domain. In an experiment performed by Wineland s group 6, a superposition of two different coherent states of an ion oscillating in a harmonic potential represents the Schrödinger cat. In another experiment 7, two coherent states of a cavity mode are also superposed coherently by the atoms passing the cavity with large detuning. In fact, the ideal coherent superposition state can only live in the quantum world free from external influences. However, a real quantum system can rarely be isolated from its surrounding environment completely. It is usually coupled to the external world also called heat bath with a large number of degrees of freedom. There are two distinct effects of the bath on the quantum system: quantum dissipation and decoherence. The usual coupling between system and bath allows for an exchange of energy causing dissipation 8,9. Certain interactions between system and bath will lead to the so-called pure decoherence, the decoherence without energy dissipation 1. Due to these two influences of the environment, the quantum-mechanical interference effects are destroyed very rapidly, i.e., a superposition state of the system will evolve into a statistical mixture state. In a solid-state system, maximally entangled states of Bell-type for excitons in two coupled quantum dots, and Greenberge-Horre-Zeilinger type for three coupled quantum dots, have been investigated 11. The decoherence effects on the generation of entangled states of excitons have been investigated for the coupled quantum dot systems 12. The motivation to investigate such kinds of systems is to consider the practical realization of the quantum information processes, such as quantum computing and quantum communications 13. The coherence is the essential requirement for the quantum information, but the decoherence will result in errors in the process of the computation and quantum communication. So the quantum decoherence is the biggest obstacle to its implementation 14. To overcome quantum decoherence, one should know its dynamic details theoretically and experimentally in various physical systems that are the potential carriers of quantum information. We shall study quantum decoherence of excitons in the quantum well immersed in a leaky cavity with the dissipative cavity fields described by quasimodes 15. In Sec. II, we first give a theoretical model and do our best to find analytical solutions of the Heisenberg operators of the cavity fields and the exciton. In Sec. III, by investigating the decoherence of the superposition state of the exciton, we find that, because of the effects of the environment on the system of exciton, the coherence of the superposition states of the exciton is suppressed in a oscillating-decaying way. In Sec. IV, we study the quantum decoherence with the environment of thermal fields at finite temperature. Finally we give our conclusions and some necessary remarks. II. THEORETICAL MODEL AND EXACT SOLUTIONS The quantum well considered here has an ideal cubic lattice with N lattice sites and contains only one molecular layer. We assume that N identical two-level molecules distribute into these lattice sites, with e n and g n denoting the excited state and the ground state of nth molecule. The well is placed at the center of a leaky Fabry-Perot cavity 15 that is described by quasimode. Let operators â (â ) denote creation annihilation operators of the field modes that are labeled by continuous index and the field frequency of each mode is denoted by. All these molecules have equivalent mode positions so that they have the same coupling constant with the same field mode. It is also assumed that the direction of the dipole moment for molecules and wave vectors of the cavity fields are perpendicular to the surface of the quantum well. By using the Dicke model 16 under the rotating wave approxi /21/633/338167/$ The American Physical Society

2 YU-XI LIU, C. P. SUN, AND S. X. YU PHYSICAL REVIEW A mation, we write down the Hamiltonian for the quantum well and the cavity fields HS z â â g â S â S, 1 â t i â ig bˆ. Equation 5b can be integrated formally as 5b where is the transition frequency of an isolated molecule and the molecular collective operators are defined as N S Z n1 s z n, N S n1 s n, where s z (n) 1 2 (e n e n g n g n ), s (n)e n g n, and s (n)g n e n are quasispin operators of the nth molecule. The coupling constant g between the molecules and the cavity fields takes a simple form that is proportional to a Lorentzian g 2, 2 where depends on the atomic dipole 17 and is the decay rate of a quasimode of the cavity. The central frequency of the cavity is, which is the same with the transition frequency of the molecule because the molecules interact resonantly with the quasimode field. In this paper we restrict our investigation to only one quasimode cavity. In the case of low excitation, with the intermolecular interactions neglected, the collective behavior of many molecules can be described by a bosonic exciton 18. Therefore the total system can be modeled as a standard harmonicoscillator bath or environment system. For a typical system without any practical considerations, one of the authors C.P.S. and his collaborators systematically studied its quantum dissipation and decoherence in association with the quantum measurement problem by considering factorization 9,19 22 and partial factorization 23 of the wave function of the total system. This paper will generalize the partial factorizaton method to discuss quantum decoherence of excitons in a leaky cavity. Because the excitation of the molecules is somewhat low, we will make the so-called bosonic approximation 18,24,25 by defining bˆ S/N and bˆ S /N with bˆ,bˆ 1. The Hamiltonian 1 is then changed into 2 3 â tâ expitig t bˆ texpi ttdt. After substituting Eq. 6 into Eq. 5a to eliminate the field operators and denoting bˆ (t)bˆ (t)e it to remove the high-frequency behavior from Eq. 5a, we obtain the equation of motion for the slowly varied exciton operator Bˆ t t Bˆ tktt dtt, where the general memory kernal function is and Ktt ti g 2 expi tt g â expi t. Through the Laplace transformation, we could easily obtain Bˆ (t) and therefore bˆ (t). A simple calculation shows bˆ tbˆ te it utbˆ u tâ e it, 8a where the time-dependent coefficients are utl 1 ũp, ũppk p 1, 6 7 8b 8c where K ( p) is the Laplace transformation of the general memory kernal function 1 K(tt) and g u tl pi ũp, 8d Hbˆ bˆ â â g â bˆ bˆ â, 4 where L 1 denotes the inverse Laplace transformation. Substituting Eqs. 8 into Eq. 6, we have which describes an interaction between excitons and cavity field with g( )Ng. The Heisenberg equations of motion for the operators of the field modes â k (â k ) and the excitons bˆ (bˆ ) can be easily obtained as with â te i t â v tbˆ v, tâ 9 bˆ t ibˆ i g â, 5a v tig expit t utexpi t dt, 1a

3 QUANTUM DECOHERENCE OF EXCITONS IN A LEAKY... PHYSICAL REVIEW A v, tg expit t u texpi t dt. 1b In order to obtain the time-dependent coefficients above, at first we change the sum into the integration L/c d where L is the length of the cavity and c is the speed of the light in the vacuum 27, i.e., Ktt 2 2 NL expi tt d c Then we assume that is much larger than all other quantities of the dimension of frequency and is a small quantity. Therefore we may adopt to the standard approximation of extending the lower limit of the integral Eq. 11 to. By integrating Eq. 11 we have KttMe tt 12 with MN 2 L/c. In the following calculation, we only need time-dependent coefficients u(t) and v (t). With the help of the Laplace transformation of the function K(tt), we obtain u(t) by use of Eqs. 8b,8c as follows: ut cos 2 t sin 2 t e /2t, 13 where 4M 2 and v (t) as follows: v tig 1i/ 2 exp i 2 t 2 texpi t i 2 ig 2 1i/ 2 expi 2 t 2 texpi t. i /2/2 14 III. DECOHERENCE OF MESOSCOPIC SUPERPOSITION STATES OF THE EXCITON Now we prepare a superposition state of the exciton. We choose a superposition of distinct coherent states as the Schrödinger s cat. That is, the exciton initially is in the state C 1 D 2, where 1 or 2 is a coherent state for the exciton, and the cavity fields are in the vacuum states zero temperature. Why we choose coherent states instead of number states can be ustified by the predictability sieve 26. Thus the whole initial state for the exciton and the cavity fields is the product of the initial state of the exciton and the cavity fields C 1 D 2 ). 15 In order to discuss the coherence properties for the system of excitons, we have to calculate the time evolution of the wave function tututc 1 D 2 ), 16 where U(t)e i(ht/). For any coherent state of the exciton we have expb *b, therefore we obtain tutc exp 1 b 1 *b D exp 2 b 2 *b Taking into account the fact that U (t)ou(t)o(t) and U(t) for any operator O, we interpolate operator U (t)u(t) into Eq. 18 and use the property of the evolution operator U(t) and easily obtain the wave function of the whole system at any time t: tc 1 u*t 1 u *t D 2 u*t 2 u *t. 19 We could calculate the reduced density matrix of the exciton system at any time t by Tr R ((t)(t)), and obtain the decoherence factor by calculating one of the nondiagonal elements, such as Ft 1 *u t 2 u*t exp * 2 u t 2. 2 Since bˆ (t),bˆ (t)1 and from Eq. 8a together with its complex conugate, we have u t 2 1ut 2 1cos 2 t sin 2 t 2 e t. 21 The decoherence factor in Eq. 2 becomes

4 YU-XI LIU, C. P. SUN, AND S. X. YU PHYSICAL REVIEW A FIG. 1. Decoherence factor F(t) as the function of the time t with a given set of parameters:.5 mev, 2.1, and M2 mev. Ftexp * 2 1cos 2 t sin 2 t 2 e t. 22 Therefore, with the evolution of the time, the coherence of two coherent states of the excitons is suppressed. It is evident when the time t, because of the environment effect, the energy of the exciton will be dissipated and states of the exciton will turn into vacuum states. Now we consider a behavior of the short time, that is t,(/2)t1, then the decoherence factor is Ftexp * 2 t. 23 The characteristic time t d of the decoherence of the exciton superposition states is ( * 2 ) 1. The coherent properties of the exciton states is determined by their initial superposition states and the decay rate of the quasimode. The smaller the superposition of 1 and 2, the shorter the decoherence time t d. A special case of heavy interested 28 31, is when the system is prepared initially in odd or even coherent states of the exciton, i.e., 1 2. Then the decoherence factor is Ftexp 2 2 1cos 2 t sin 2 t 2 e t. In Fig. 1, we give a sketch of the decoherence factor F(t) as the function of the time t with a given set of parameters M,, and. We find that the coherence of the exciton system will decrease in the oscillating decay form. It is similar to the case observed in the experiment done in Ref. 3. In terms of the definition of the coherent state of the exciton, we know that 2 is the mean number of the excitons. Hence by adusting the initial number of the excitons we could completely determine the decoherence time. The smaller the mean number of the exciton is, the longer the decoherence time. IV. TEMPERATURE EFFECT ON COHERENCE In this section, we will discuss the temperature effects on the coherent properties of the exciton superposition states. For every single mode of the cavity field of the frequency in a thermal state, its density matrix is given by ˆ 1 n 1 n n n 1 n n n, 24 where n exp( /k B T)1 1 is the average thermal photon number and k B is the Boltzman constant and T is the temperature of the thermal fields. Under the P representation Eq. 24 is rewritten as ˆ d2 n e 2 /n. 25 It is clear that the P representation of the thermal fields is given by a Gaussian distribution. If we assume that every mode of the cavity field is initially in a thermal state, then the density operator of the whole system is ˆ ˆ e ˆ bath with density operators of exciton ˆ ec 1 D 2 ) 1 C* 2 D* By using the same method as in the previous section, we obtain the time evolution of the system as tut ) u*t u*t e i t v *t s v*t s, s. 28 By virtue of the normalized condition bˆ (t),â (t), we obtain the decoherence factor by calculating the reduced density matrix of the exciton system as follows:

5 QUANTUM DECOHERENCE OF EXCITONS IN A LEAKY... PHYSICAL REVIEW A Ftexp * 21ut 2 with d 2 1 n exp 2 n ** utv * *. Taking into account the relation 1 d 2 exp 2 ** 1 exp 1 2 for Re, we finally obtain the decoherence factor as with Ftexp * 2 1ut 2 exp ut 2 t,t a t,t v t 2 n. 32b We may transform of Eq. 32b into L/c d as in Ref. 27: t,t L c v t 2 n d. 33 We find that the integration in Eq. 33 is so strongly peaked near, so we may remove the slowly varying factor n. Although we may calculate an exact integration of Eq. 35 for the details, please see Appendix, we are more interested in the limit of the short time, we finish the integral of Eq. 35 and obtain T,tn Dt 34 with n (e /k B T 1) 1 and D being given in Eq. A6. So in the case of the thermal fields, by keeping only the firstorder terms of t and (/2)t, the decoherence factor of superposition states of the exciton system becomes Ftexp * 2 t exp n D t. 35 Under the condition of the high temperature, the decoherence factor is simplified: Ftexp * 2 t exp D Tk 2 B t. 36 So the characteristic time t d of the decoherence of the exciton system reads t d * 2 D Tk 1 2 B. 37 It is shown that the higher the temperature of the environment, the shorter the time of decoherence of the exciton system. If the exciton system is initially in the odd or even coherent states, then the characteristic time is t d 1 2 (2 DTk B /) 1. So the decoherence time becomes shorter when the cavity fields are initially in the thermal radiation. V. CONCLUSIONS The decoherence of the mesoscopic superposition states for the exciton in a quasimode cavity is investigated, where an exact solution is readily found. We find that the coherence of the superposition states of exciton is reduced by the interaction between the cavity fields and the excitons. In the long run, we find that the energy of the exciton will dissipated because of the environment effect and the states of the exciton will turn into vacuum states. We find that the smaller the superposition of two coherent states of the exciton, the shorter the decoherence time for a short-time process. The temperature effect on coherence of the system of exciton also is studied by virtue of the P representation. We find that the higher the temperature of the environment is, the shorter the decoherence time is. ACKNOWLEDGMENTS One of the authors Y.x.L. would like to express his sincere thanks to Dr. X. X. Yi for his many useful discussions. This work is supported partially by the NSF of China and 9-5 Pandeng proect. APPENDIX: SOLUTION OF T,t In this Appendix, we will give an integration of (T,t) in detail. From Eqs. 14 and 35, we have T,t n L2 2 i 2 1e /2ti/2t e i( /2)t/2t e i( /2)t/2t 4 2 c 2 2 /2i/2 H.c. /2i/2 d A

6 YU-XI LIU, C. P. SUN, AND S. X. YU PHYSICAL REVIEW A n L e t 2e /2 cos /2t 4 2 c 2 2 /2 2 2 /4 d n L e t 2e /2 cos /2t 4 2 c 2 2 /2 2 2 /4 d. We set x, and extend the lower limit of the integral Eq. A1 to as we do in the Sec. II, then finish following integral formulations: dx x 2 2 x/2i/2x/2i/2 /2i3/2i1/2/2 i2 2/2i/2i3/2i1/2/2 A2 exp ix/2t/2tdx x 2 2 x/2i/2x/2i/2 expi/23/2t /2i3/2i1/2/2 cosx/2t] x 2 2 x/2 2 2 /4 dx i2 expi2/2t 2/2i/2i3/2i1/2/2 et 2 /23/4 2 cos/2t2/2sin/2t]2e (/2)t 2 /23/4 2 2 /29/4 2 2 /2 2 /4 A3 A4 dx x 2 2 x/2 2 2 /4 3/2i3/2 /2i/2 2 /29/4 2 A5 from Eqs. A1 A5, we can obtain obtain an exact expression of (T,t), but we are only interested in the behavior of the short time, because after evolution of the long time, all states of the system will return to vacuum reduced by the environment. So in the condition of t1 and (/2)t1, we only keep the first-order small quantity, then we have T,tn L2 2 2 /2 2 2 /22 2 2c 2 /2 2 /4 2 2 /29/4 2 tn Dt, A6 which is needed in Eq S. Durr, T. Nonn, and G. Rempe, Nature London 395, E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, Nature London 391, C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Nature London 43, W. H. Zurek, Phys. Today 441, S. Haroche, Phys. Today 51, C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, Science 272, M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche. Phys. Rev. Lett. 77, A. O. Caldeira and A. J. Leggett, Ann. Phys. N.Y. 149, ; A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, and W. Zwerger, Rev. Mod. Phys. 59, L. H. Yu and C. P. Sun, Phys. Rev. A 49, ; C.P. Sun and L. H. Yu, ibid. 51, W. H. Zurek, Philos. Trans. R. Soc. London, Ser. A 356, L. Quiroga and N. F. Johnson, Phys. Rev. Lett. 83, F. J. Rodriguez, L. Quiroga, and N. F. Johnson, Phys. Status Solidi A 178, C. H. Bennett, Phys. Today 471, ; A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, ; P. Shor, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science 1994 IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. l C. P. Sun, H. Zhan, and X. F. Liu, Phys. Rev. A 58,

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