Dynamical behavior of entanglement in semiconductor microcavities

ARTICLE IN PRESS Physica E 42 (2010) 2091–2096

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Dynamical behavior of entanglement in semiconductor microcavities Sh. Barzanjeh a, H. Eleuch b,c, a b c

Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, Isfahan, Iran Institute for Quantum Studies and Department of Physics and Astronomy, Texas A&M University, USA The Abdus Salam International Centre of Theoretical Physics, ICTP, Trieste, Italy.

a r t i c l e in fo

abstract

Article history: Received 16 November 2009 Received in revised form 17 March 2010 Accepted 18 March 2010 Available online 30 March 2010

We investigate the evolution of entanglement in a semiconductor cavity QED containing a quantum well. By using the Wehrl entropy and introducing generalized concurrence as entanglement measures, we study the influence of the system parameters on the evolution features of the entanglement. & 2010 Elsevier B.V. All rights reserved.

Keywords: Semiconductor cavity QED Exciton Quantum well Wehrl entropy Generalized concurrence

1. Introduction The existence of the entanglement [1] follows naturally from the quantum mechanical formalism. This was first made explicit in the famous paper [2] by Einstein, Podolsky and Rosen (EPR), where they argued that quantum mechanics is an incomplete physical theory. After over eighty years it is still a fascinating subject from both theoretical and experimental point of view. A number of possible practical applications of the quantum inseparable states has been proposed including quantum computation [3] and quantum teleportation [4]. Furthermore, the study of quantum entanglement and its dynamical behavior has been extensively explored in various kinds of quantum optical systems [5–12]. The interaction of the quantum systems with their surrounding environments leads to decoherence effects that destroy quantum entanglement in the real world. The effective decoherence resulting from a quantum system interacting with an environment provides a natural mechanism for the transition from quantum to classical behavior for an open system [13]. The decoherence has been an integral part of several programs addressing the emergence of classicality [14–16]. The physics of decoherence became very popular in last decade, mainly due to advances in the technology. In several experiments the progres-

 Corresponding author at: Institute for Quantum Studies and Department of Physics and Astronomy, Texas A&M University, USA. E-mail addresses: [email protected], [email protected] (H. Eleuch).

1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.03.032

sive emergence of classical properties in a quantum system has been observed [17,18], in agreement with the predictions of the decoherence theory. The second important reason for the popularity of decoherence is its relevance for quantum information processing tasks, where the coherence of a relatively large quantum system has to be maintained over a long time. In this paper we focus on the dynamics of entanglement in a semiconductor cavity QED containing a quantum well coupled to the environment. We explore the evolution behavior of the concurrence and Wehrl entropy [19,20] in order to study the dynamical evolution of entanglement. It has been demonstrated that the Wehrl entropy is a useful measure of various quantumfield properties, including quantum noise [21], decoherence [22], quantum interference [23] and ionization [24]. In contrast to the von Neumann entropy the Wehrl entropy is characterized by the classical entropy properties [25–27]. The paper is organized as follows: In Section 2, we introduce a model for the quantum system. In Section 3, we present the evolution equations. The Wehrl entropy is studied in Section 4. The generalized concurrence behavior is discussed in Section 5. Finally, a conclusion is given in Section 6.

2. Model The considered system is a quantum well confined in a semiconductor microcavity. The semiconductor microcavity is made of a set of Bragg mirrors with specific separation taken to be of the order of the wavelength l. This mirrors could be made with

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metallic film; but often one uses the so-called distribution Brag reflectors (DBR mirrors) that are made of alternative layers of low and high refractive index materials with a layer thickness of a quarter of the wavelength. In the system under consideration, we restricted our discussions to the interaction of electromagnetic field with two bands in the weak pumping regime. The electromagnetic field can make an electron transition from valance to conduction band. This transition simultaneously creates a single hole in the valance band. In these kinds of semiconductors the electron and the hole interact by giving excitonic states. The ground state 1S has the greatest oscillator strength. For this reason we take into account only this state for exciton–photon interaction. One can use an effective Hamiltonian without spin effects for describing the exciton–photon coupling in the cavity as [28–35] H ¼ ‘op ay aþ ‘oe by b þı‘g 0 ðay bby aÞ þ ‘a0 by by bb þı‘ðe0 eıot ay h:cÞ þ Hr ,

ð1Þ where op and oe are the frequencies of the photonic and excitonic modes of the cavity, respectively. The bosonic operators a and b are, respectively, describing the photonic and excitonic annihilation operators and verifying ½a,ay  ¼ 1; ½b,by  ¼ 1. The first two terms of the Hamiltonian describe, respectively, the energies of photon and exciton. The third term corresponds to the photon– 0 exciton coupling with a constant of coupling g . The forth term describes the nonlinear exciton–exciton scattering due to coulomb interaction. Where a0 is the strength of the interaction between excitons [36,37]. The fifth term represents the interaction of external driving laser field with the cavity, with e0 and o being, respectively, the amplitude and frequency of the driving field. Finally, the last term describes the relaxation part of the main exciton and photon modes. We restrict our work to the resonant case where the pumping laser, the cavity and the exciton are in resonance (o ¼ op ¼ oe ). We have neglected also the photon– exciton saturations effects in Eq. (1). It is shown that these effects give rise to small corrections as compared to the nonlinear exciton–exciton scattering [29,38,39]. Furthermore, we assume that the thermal reservoir is at the T¼0, then the master equation can be written as [40–42] @r ¼ ıa½by by bb, r þ g½ðay bby aÞ, r þ e½ðay aÞ, r þ Lr, @t

ð2Þ

in which the time dependent density matrix r ¼ jcðtÞS/cðtÞj is a possible solution of Eq.(4). Also jcðtÞS satisfies the following equation: ı‘

djcðtÞS ¼ Heff jcðtÞS: dt

ð6Þ

The essential effect of the pump field is to increase the excitation quanta number in the cavity which allows us to neglect the term ‘ea in the expression of the effective non-Hermitian Hamiltonian Eq. (5) [43–45]. We can expand jcðtÞS into a superposition of tensor product of pure excitonic and photonic states [43–45,37]: jcðtÞS ¼ j00S þA10 ðtÞj10S þA01 ðtÞj01S þA11 ðtÞj11S þ A20 ðtÞj20S þ A02 ðtÞj02S,

ð7Þ where jijS ¼ jiS#jjS, is the state with i photons and j excitons in the cavity. We then obtain the following differential equations for the amplitudes Aij(t) d A10 ¼ e þgA01 kA10 , dt d g A01 ¼ gA10  A01 , 2 dt

ð8Þ

and pffiffiffi pffiffiffi d A11 ¼ 2gA02  2gA20 ðk þ g=2ÞA11 þ eA01 , dt pffiffiffi pffiffiffi d A20 ¼ 2gA11 2kA20 þ 2eA10 , dt

ð9Þ

pffiffiffi d A02 ¼  2gA11 2ıaA02 gA02 : dt by using following relations A11 ðtÞ ¼ eðk þ g=2Þt A11 ðtÞ, A20 ðtÞ ¼ e2kt A20 ðtÞ,

ð10Þ

A02 ðtÞ ¼ eð2ıa þ gÞt A02 ðtÞ: Eqs. (9) reduce to

where t is a dimensionless time normalized to the round trip time tc in the cavity, and we normalize all constant parameters of the system to 1=tc as: g ¼ g 0 tc , e ¼ e0 tc , a ¼ a0 tc . Lr represents the dissipation term associated with Hr and it describes the dissipation due to the excitonic spontaneous emission rate g=2 and to the cavity dissipation rate k: Lr ¼ kð2aray ay array aÞ þ g=2ð2brby by brrby bÞ:

ð3Þ

pffiffiffi pffiffiffi d  A ¼ 2geðk2ıag=2Þt A02  2geðkg=2Þt A20 þ eeðk þ g=2Þt A01 , dt 11 pffiffiffi pffiffiffi d  A20 ¼ 2geðkg=2Þt A11 þ 2ee2kt A10 , dt

ð11Þ

pffiffiffi d  A ¼  2geð2ıak þ g=2Þt A11 : dt 02 The two differential Eqs. (8) describe two coupled harmonic oscillators and their solutions are given by

3. Evolution equations

ıLt=2

A01 ðtÞ ¼ eGt=2 ðaeıLt=2 þ be

In the weak excitation regime ðe=kÞ 51, we can neglect the non-diagonal terms 2aray and 2brby in the master Eq. (3) [43,44]. This approximation is obtained by expanding the equation of motion for the density matrix elements and keeping the dominant terms in e=k. The density matrix can then be factorized as a pure state [43–46,37]. We then obtain, the following compact and practical master equation:

G ¼ k þ g=2,

dr 1 ¼ ðHeff rðHeff rÞy Þ, dt ı‘

L¼

ð4Þ

where we introduce Heff as an effective non-Hermitian Hamiltonian:

g

Heff ¼ ı‘gðay bby aÞ þ ‘aby by bbþı‘eðay aÞı‘kay aı‘ by b: 2

A10 ðtÞ ¼ 

eg , o2

1 Gt=2 0 ıLt=2 eg ða e þ b0 eıLt=2 Þ þ : e 2g 2o2

ð12Þ

with

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4g 2 ðkg=2Þ2 ,

ð13Þ

o2 ¼ g 2 þ kg=2, 0

ð5Þ

Þ

0

a, b, a , b are the constant coefficients. In order to find these coefficients we assume that, at time t ¼0 the vector state jcðtÞS is

ARTICLE IN PRESS Sh. Barzanjeh, H. Eleuch / Physica E 42 (2010) 2091–2096

where

in vacuum state, jcðt ¼ 0ÞS ¼ j00S: Aij ðt ¼ 0Þ ¼ 0:

ð14Þ

Now we substitute Eq. (14) into Eq. (12) and find a ¼ ge



ıLG , ð2ıo2 LÞ

b0 ¼ ½g þðıLGÞb: The coupled differential Equations in (9) can be written in the term: d AðtÞ ¼ MðtÞAðtÞ þ BðtÞ, dt where 0  1 A11 B  C A ¼ @ A20 A, A02 0 0 B pffiffiffi ðkg=2Þt 2ge M¼@ pffiffiffi  2geð2ıak þ g=2Þt 0 ðk þ g=2Þt 1 ee A01 p ffiffiffi B C B ¼ @ 2e2kt eA10 A:

ð16Þ

pffiffiffi sin x x

ð23Þ

The amplitudes A11, A20, A02 can be obtain by evaluating the integral in Eq. (22). One cannot find analytical expression for above integral, thus the numerical calculation can be fruitful. For pure state, the density operator can be written in term of the wavefunction jcðtÞS as rph,exc ¼ jcðtÞS/cðtÞj. The reduced density matrices of photon–exciton system can be written as

rph ¼ trexc ðjcðtÞS/cðtÞjÞ, rexc ¼ trph ðjcðtÞS/cðtÞjÞ:

ð24Þ

by using reduced density matrices we can calculate the photon Wehrl entropy and the generalized concurrence.

4. Wehrl entropy pffiffiffi  2geðkg=2Þt 0 0

pffiffiffi 1 2geðkg=22iaÞt C 0 A, 0

In order to compute the photon Wehrl entropy we use Qfunction. This quasiprobability distribution is defined as 

Qph ðb, b ,tÞ ¼ p1 /bjrph jbS:

ð25Þ



ð17Þ

0 Eq. (16) can be formally integrated, we then obtain Z t 0 0 AðtÞ ¼ eFðtÞ eFðt Þ Bðt 0 Þ dt ,

i.e., Qph ðb, b ,tÞ is proportional to the diagonal elements of the density operator in the coherent state representation. jbS corresponds to the coherent state jbS ¼

ð18Þ

0

1 X

Cn jnS ¼ expð12jbj2 Þ

n¼0

1 X bn pffiffiffiffiffi jnS: n ¼ 0 n!

ð26Þ

Using Eqs. (7) and (22), the quasi-probability distribution Eq. (25) turn out to be

where t

0

Mðt 0 Þ dt :

ð19Þ

0



Qph ðb, b ,tÞ ¼

In order to solve Eq. (18), we need to write 0

pffiffiffi B cos x B B B b pffiffiffi YðtÞ  eFðtÞ ¼ B B pffiffiffi sin x B x B @ c pffiffiffi pffiffiffi sin x x

ðg=2kÞt



be

1þ

pffiffiffi x

pffiffiffi sin x

2 ðg=2kÞt

b e

pffiffiffi ðcos x1Þ x

pffiffiffi ðg=2kÞt bce ðcos x1Þ x

pffiffiffi ceðg=2k þ 2iaÞt pffiffiffi sin x x pffiffiffi ðg=2k þ 2iaÞt bce ðcos x1Þ

pffiffiffi eðg=2kÞt 1 2g , kg=2

c¼

pffiffiffi eðg=2k þ 2iaÞt 1 : 2g kg=22ia

1 2iy 2 jbj A20 Þj2 e 2! þ jA01 þjbjeiy A11 j2 þ jA02 j2 , 2

ejbj ½jð1þ jbjeiy A10 þ

ð27Þ

1

C C C C C, C x C C pffiffiffi 2 ðg=2k þ 2iaÞt c e ðcos x1Þ A 1þ x

ð20Þ

where b ¼ jbjeıy . In analogy to the classical entropy, the Wehrl entropy can be written as [19,20] Z   SW ðtÞ ¼  Qph ðb, b ,tÞlnðQph ðb, b ,tÞÞd2 b ð28Þ

x ¼ b2 eðg=2kÞt þ c2 eðg=2k þ 2iaÞt ,

b¼

1

p



with

ð21Þ

Thus, the expression eFðtÞ can be determined by replacing, b!b, c!c. The resulting expression for A is Z t AðtÞ ¼ YðtÞ wðt0 Þ dt0 , ð22Þ 0

pffiffiffi

ð15Þ

a0 ¼ ½gðıL þ GÞa,

Z

pffiffiffi

wðtÞ  eeðk þ gÞt A01 cos x þ 2bA10 pffiffiffi

pffiffiffi pffiffiffi   sin x pffiffiffi cos x1 bA01 pffiffiffi þ 2A10 eðkg=2Þt þb2 x x pffiffiffi pffiffiffi  sin x pffiffiffi cos x1 : cA01 pffiffiffi þ 2bcA10 x x

ıL þ G , b ¼ ge ð2ıo2 LÞ

FðtÞ 

2093

The SW (t) exploits the unique property of the Q-function, which is always positive in contrast to other quantum quasiprobabilities. To explore the influence of decoherence on the dynamical behavior of the Wehrl entropy, we have plotted the time evolution of the photon Wehrl entropy SW (t) as a function of time t for three different set of parameters in Fig. 1. Fig. 1a shows the behavior of the Wehrl entropy in the absence of excitonic spontaneous emission rate g ¼ 0. It is shown that the entanglement is decreases with the time.

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large values of g the Wehrl entropy approach rapidly zero with time. This figure explicitly shows the influence of g on the dynamical behaviors of the Wehrl entropy.

0.0012

SW (t)−Ln()−1

0.0010 0.0008

5. Generalized concurrence

0.0006

5.1. Entanglement measurement

0.0004 0.0002 0.0000 0

2

4

6

8

10

t

i

where r is the reduced density matrix, li is the i th eigenvalue of r. For the mixed states, the entanglement of formation takes the form [47] X ð30Þ EðrÞ ¼ min pi Eðci Þ:

0.0008

SW (t)−Ln()−1

Several measures to quantify entanglement have been proposed in recent years. The partial entropy of the density matrix can provide a good measure of entanglement for pure states X ð29Þ EðcÞ ¼ trðrLnrÞ ¼  ðli Lnli Þ,

i

0.0006

where the minimum is taken over all pure-state decompositions P of r ¼ pi jci S/ci j. However, Wootters has found an explicit i expression for entanglement formation of a two qubit mixed state r as a function of quantity called concurrence as [48] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 þ 1C 2 ðrÞ E½CðrÞ ¼ h , ð31Þ 2

0.0004

0.0002

0.0000 0

1

2

3

4

5

where hðxÞ ¼ xLnðxÞð1xÞLnð1xÞ,

t

ð32Þ

The concurrence CðrÞ is defined by CðrÞ ¼ maxð0, l1 l2 l3 l4 Þ,

SW (t)−Ln()−1

0.0008

ð33Þ

0.0006

in which the li are the square roots of eigenvalues in decreasing pffiffiffiffi pffiffiffiffi order of rr~ r with r~ ¼ ðsy #sy Þr ðsy #sy Þ. For the pure state jcS ¼ a11 j11S þ a12 j12Sþ a21 j21S þa22 j22S, the concurrence can be written as

0.0004

~ Sj ¼ 2ja a a a j: CðcÞ ¼ j/cjc 11 22 12 21

ð34Þ

Recently, Albeverio and Fei generalized the notion of concurrence by using invariants of local unitary transformations as [49] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N CðcÞ ¼ ð35Þ ðI2 I2 Þ, N1 0 1

0.0002

0.0000 0

1

2

3

4

5

t Fig. 1. The time evolution of the Wehrl entropy for: (a) g ¼2, k ¼ 0:2, g ¼ 0; (b) g¼ 6, k ¼ 0:4, g ¼ 0:3; (c) g ¼ 11, k ¼ 0:9, g ¼ 0:82, where we assumed e ¼ 0:03, and a ¼ 108 .

where I0 and I1 are two former invariants of the group of local unitary transformations. Furthermore, Akhtarshenas [50] has generalized definition of concurrence for an arbitrary bipartite N1 P N2 P pure state jcS ¼ aij jei  ej S as i¼1j¼2

By increasing g, k (see Fig. 1b) the Wehrl entropy decays too fast. This effect can be explained by the fact that the increase of the dissipation rates enhances the coupling between the system and the environment and consequently increases the decoherence. To have further insight, we plot in Fig.1c the Wehrl entropy for different value of parameters g. More oscillating and fast decaying of the Wehrl entropy appear in this case. This effect is also observed in the autocorrelation function [37]. By increasing the coupling constant g, the Rabi frequency increases. In other words when the coupling between exciton and photon is stronger, the exchange energy between the two systems occurs faster. In Fig. 2, we plotted the Wehrl entropy as function of t and g for numerical values of k ¼ 0:2, g ¼ 0:2, e ¼ 0:01, a ¼ 108 . For

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N1 N2 uX X jaik ajl ail ajk j2 , CðcÞ ¼ 2t

ð36Þ

ioj kol

where C is not normalized to unity. In other words, when jcS is a maximally entangled state, C takes its maximum value pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðN1Þ=N with N¼ min(N1,N2). Here we are interested to use normalized concurrence, hence we rewrite Eq.(36) as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiv u N1 N2 XX 2N u t ð37Þ ja a a a j2 : CðcÞ ¼ N1 i o j k o l ik jl il jk In this work, we deal with a pure state jcSA C3 #C3 so that, the Eq. (37) can be used to calculate the concurrence of the wavefunction Eq. (7); this is the subject of the following section.

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2095

SW (t)−Ln()−1

0.000 0.0002 0.0001 0.0000 0 2

4 5 t

g 6

8

10

Fig. 2. The time evolution of the Wehrl entropy as a function of t and g for k ¼ 0:2, g ¼ 0:2, e ¼ 0:01 and a ¼ 108 .

0.0010

0.0008 0.0006 C (t)

C (t)

0.0008 0.0006

0.0004

0.0004 0.0002

0.0002 0.0000

0.0000 0

2

4

6

8

10

0

2

4

6

8

10

t

0.00030

0.00012

0.00025

0.00010

0.00020

0.00008

0.00015

0.00006

ct

C (t)

t

0.00010

0.00004

0.00005

0.00002 0.00000

0.00000 0

2

4

6 t

8

10

0

2

4

6

8

10

t

Fig. 3. The time evolution of the generalized concurrence for: (a) g ¼ 1, k ¼ 0:3, g ¼ 0; (b) g ¼ 1, k ¼ 0:45, g ¼ 0:4; (c) g ¼6,k ¼ 0:45, g ¼ 0:4; (d) g ¼ 15, k ¼ 0:92, g ¼ 0:82, where we assumed e ¼ 0:03, and a ¼ 108 .

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5.2. Dynamical evolution of generalized concurrence In this section, we derive an explicit expression governing the dynamical behavior of the generalized concurrence of a semiconductor cavity QED with a quantum well, where the system is weakly pumped with resonant laser light. After substituting Eq.(7) into Eq.(37) with N1 ¼3, N2 ¼ 3, we obtain the following expression for the concurrence pffiffiffiffiffiffiffiffiffiffi CðcÞ ¼ 3IðtÞ, ð38Þ with IðtÞ ¼ jA20 ðtÞA02 ðtÞj2 þjA20 ðtÞA11 ðtÞj2 þ jA20 ðtÞA01 ðtÞj2 þ jA11 ðtÞA02 ðtÞj2 þ jA10 ðtÞA02 ðtÞj2 þ jA11 ðtÞA10 ðtÞA01 ðtÞj2 :

ð39Þ

The analytical expression for the generalized concurrence can be computed by Eqs. (12) and (22) into Eq. (39). Unfortunately, this expression is too long and complicated, hence we restrict our investigation to the numerical calculations. The generalized concurrence CðcÞ can be evaluated numerically for different values of g, k as function of t. Figs. 3a–d illustrates the temporary evolution of the generalized concurrence for various set of parameters g, k, g. Fig. 3a and b show the generalized concurrence versus time for g¼1. These figures exhibit that the concurrence decays slowly to zero. On the other hand, for large coupling between the exciton and the cavity mode, a rapid damped oscillation is observed. This behavior of concurrence is totaly similar to the Wehrl entropy.

6. Conclusion In this paper we have explored the time evolution of the quantum entanglement for a semiconductor microcavity containing a quantum well. The system is excited by a coherent light in the weak excitation regime. In particular, we have studied the influence of the system parameters on the quantum entanglement. Two measures of the entanglement were calculated namely, the Wehrl entropy and the generalized concurrence. Both measures show that a large coupling between the system and the environment induces a rapid destruction of the quantum entanglement after few damped oscillations. A rapid destruction of the quantum entanglement is also observed for large exciton–photon coupling inside the cavity after large number of high frequency damped oscillations. References ¨ ¨ [1] E. Schrodinger, Naturwissenschaften 23 (1935) 807. [2] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) 777.

[3] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [4] A. Ekert, R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. ¨ -, T. Yu, J.H. Eberly, J. Phys. B 39 (2006) S621. [5] M. Yonac [6] L. Dio´si, L.N.P. 622 (2003) 157. [7] S. Chan, M.D. Reid, Z. Ficek, J. Phys. B 42 (2009) 0953-4075. [8] K. Barrada, A. Chafik, H. Eleuch, Y. Hassouni, Int. J. Mod. Phys. B 23 (2009) 2021. [9] K. Barrada, A. Chafik, H. Eleuch, Y. Hassouni, Quantum Inform. Process 9 (2010) 13. [10] E. Alebachew, Phys. Rev. A 76 (2007) 023808. [11] E. Alebachew, Opt. Commun. 280 (2007) 133. [12] E.A. Sete, Opt. commun. 281 (2008) 257. [13] W.H. Zurek, Phys. Today 10 (1991) 36. [14] E. Joos, H.D. Zeh, Z. Phys. B 49 (1985) 223. [15] W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 (1993) 1187. [16] A.O. Caldeira, A.J. Legett, Physica (Amsterdam). 121A (1983) 587. [17] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 77 (1996) 4887. [18] Q.A. Turchette, C.J. Myatt, B.E. King, C.A. Sackett, D. Kielpinski, W.M. Itano, C. Monroe, D.J. Wineland, Phys. Rev. A 62 (2000) 053807. [19] A. Wehrl, Rev. Mod. Phys. 50 (1978) 221. [20] A. Wehrl, Rep. Math. Phys. 30 (1991) 119. [21] V. Buzˇek, C.H. Keitel, P.L. Knight, Phys. Rev. A 51 (1995) 2575. [22] A. Anderson, J.J. Halliwell, Phys. Rev. D 48 (1993) 2753. [23] V. Buzˇek, C.H. Keitel, P.L. Knight, Phys. Rev. A 51 (1995) 2594. [24] J.B. Watson, C.H. Keitel, P.L. Knight, K. Burnett, Phys. Rev. A 54 (1996) 729. [25] A. Orlowski, H. Paul, G. Kastelewicz, Phys. Rev. A 52 (1995) 1621. [26] M. Abdel-Aty, Prog. Quant. Electron. 31 (2007). [27] A. Miranowicz, J. Bajer, M.R.B. Wahiddin, N. Imoto, J. Phys. A 34 (2001) 3887. [28] C. Ciuti, et al., Phys. Rev. B 58 (1998) R10123. [29] F. Tassone, Y. Yamamoto, Phys. Rev. B 59 (1999) 10830. [30] H. Haug, Z. Phys. B 24 (1976) 351. [31] E. Hanamura, J. Phys. Soc. Japan 37 (1974) 1545. [32] A.B. Naguyen, Phys. Rev. B 48 (1993) 11732. [33] H. Eleuch, Appl. Math. Inform. Sci. 3 (2009) 185. [34] A. Baas, J.Ph. Karr, H. Eleuch, E. Giacobino, Phys. Rev. A 69 (2004) 23809. [35] E. Giacobino, J.Ph. Karr, G. Messin, H. Eleuch, C. R. Phys. 3 (2002) 41. [36] C. Ciuti, P. Schwendimann, B. Deveaud, A. Quattropani, Phys. Rev. B 62 (2000) R4825. [37] H. Eleuch, J. Phys. B 41 (2008) 055502. [38] G. Messin, J.Ph. Karr, H. Eleuch, J.M. Courty, E. Giacobino, J. Phys. 11 (1999) 6069. [39] H. Eleuch, J.M. Courty, G. Messin, C. Fabre, E. Giacobino, J. Opt. B 1 (1999). [40] W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. [41] H. Eleuch, Euro. Phys. J. D 49 (2008) 391. [42] H. Eleuch, Euro. Phys. J. D 48 (2008) 139. [43] H.J. Carmichael, Statistical Methods in Quantum Optics2, Springer, Berlin, 2007. [44] H.J. Carmichael, R.J. Brecha, P.R. Rice, Opt. Commun. 82 (1991) 73. [45] R.J. Brecha, P.R. Rice, M. Xiao, Phys. Rev. A 59 (1991) 2392. [46] H. Jabri, H. Eleuch, T. Djerad, Laser Phys. Lett. 5 (2005) 253. [47] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [48] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 3824. [49] S. Albeverio, S.M. Fei, J. Opt. B Quantum Semiclass. Opt. 3 (2001). [50] S.J. Akhtarshenas, J. Phys. A Math. Gen. 38 (2005) 6777.