Time evolution of Wigner function in laser process derived by entangled state representation

Optics Communications 282 (2009) 4379–4383

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Time evolution of Wigner function in laser process derived by entangled state representation Li-yun Hu a,b,*, Hong-yi Fan b a b

College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, PR China Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, PR China

a r t i c l e

i n f o

Article history: Received 5 April 2009 Received in revised form 29 July 2009 Accepted 3 August 2009

a b s t r a c t Evaluating the Wigner function of quantum states in the entangled state representation is introduced, based on which we present a new approach for deriving time evolution formula of Wigner function in laser process. Application of this formula to photon number measurement in laser process is also presented, as an example, the case when the initial state is a photon-added coherent state is discussed. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction One of the major topics in Quantum Statistical Mechanics is the evolution of pure states into mixed states [1,2]. Such evolution usually happens when a system is immersed in a thermal environment, or a signal (a quantum state) passes through a quantum channel, and is described by a master equation. Alternately, description of evolution of density matrices q can be replaced by its Wigner function’s evolution in phase space [3,4]. The partial negativity of Wigner function can be considered as an indicator of non-classicality of quantum state. To describe system-environment evolution Takahashi–Umezawa introduced thermo field dynamics (TFD) [5–7], with which one may convert the calculations of ensemble averages at finite temperature

hAi ¼ trðAqÞ;

q ¼ ebH =Z ðbÞ

ð1Þ

to equivalent expectation values with a pure state j 0ðbÞi, i.e.,

hAi ¼ h0ðbÞjAj0ðbÞi;

ð2Þ

where b ¼ 1=kT; k is the Boltzmann constant, and ZðbÞ ¼ trebH is the partition function; H is the system’s Hamiltonian. The TFD approach has the advantage of transforming mixed state average to pure state average, so it has become a self-consistent and widely accepted theory in quantum statistical physics and quantum field theory. The worthwhile convenience in Eq. (2) is at the expense of introducing a fictitious field (or called a tilde-conjugate field, de~y ) in the extended Hilbert space, i.e., the original noted as operator a optical field state j ni in the Hilbert space H is accompanied by a f A similar rule holds for operators: every Bose ~ i in H. tilde state j n ~ acting on annihilation operator a acting on H has an image a f ½a ~; a ~y  ¼ 1. H; * Corresponding author. Address: College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, PR China. Fax: +61 2162932080. E-mail addresses: [email protected], [email protected] (L.-y. Hu). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.08.004

For hxay a; j ni ¼ pffiffiffiffiffi a harmonic oscillator the Hamiltonian is  ayn = n! j 0i, Takahashi–Umezawa obtained the explicit expression of j 0ðbÞi in doubled Fock space,

 E  y  E ~ ; ~ ¼ SðhÞ00 ~ tanh h 00 j0ðbÞi ¼ sechh exp ay a

ð3Þ

which is named thermo vacuum state, and SðhÞ thermo operator, ~Þ (where h is a parameter related to the tem~ y  aa SðhÞ  exp½hðay a  hx  ), which is similar in form to the perature by tanh h ¼ exp  2kT two-mode squeezing operator. Based on Eq. (3) in the limiting case tanh h ! 1, Fan et al. have introduced so-called thermo entangled state j gi (or named as coherent thermal state) [8–10] (see Eq. (4) below). In Ref. [11], we have adopted the entangled state approach for treating the time evolution of density operator qðtÞ in dissipative channel. The merits of this approach are in three aspects: 1. Using the eigenstate equation for j gi, the master equation of qðtÞ can be easily converted into C-number equation of hg j qðtÞi; j qðtÞi  qðtÞ j g ¼ 0i. This coincides with Dirac’s teaching that choosing an appropriate representation can save much mental labour for solving dynamic problems in quantum mechanics. 2. In our approach the dissipation of the mixed state qðtÞ in hgj representation turns out to be the evolution of initial state q0 in the decayed entangled state hgejt j representation, accom2 1e2jt panying with a Gaussian damping factor e 2 jgj . This exhibits dissipation in an intuitive manner. Moreover, the transition hg j! hgejt j is governed by a two-mode squeezing operator, thus dissipation of a system immersed in a thermal environment can be described by squeezing in the thermo entangled state representation, a fresh view. 3. By depriving j g ¼ 0i from j qðtÞi we can identify the explicit infinite-dimensional Kraus operators of qðtÞ, say, for the amplitude-damping channel and for the laser process.

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Thus the thermo entangled state representation is beneficial to quantum decoherence theory. In this work, we shall reveal another merit of this approach, that is, the Wigner function of density operator can be expressed as an overlap between two pure states (see Eq. (13) below). This brings much convenience for calculating time evolution of Wigner functions when quantum decoherence happens, the worthwhile convenience is at the expense of introducing the j gi state by introducing a fictitious mode, which people should accept without difficulty, since the construction of j gi in this work is similar to that of Einstein–Podolsky–Rosen (EPR) eigenstate which is the common eigenvector of two particles’ relative coordinate and total momentum [8]. In the following, as the continuation of our preceding Communication [11], we present a new formula for deriving time evolution of the Wigner function for quantum decoherence, as examples, we apply the formula to the cases of amplitude-damping channel and laser process, respectively. (So far as we know, the time evolution of Wigner function in laser process has not been reported in the literature before.) Application of this formula to photon number measurement in laser process is also presented, as an example, the case when the initial state is a photon-added coherent state is discussed.

Note that density operators qðay ; aÞ are defined in the real space ~Þ in the tilde space. ~y ; a which are commutative with operators ða Next, we shall derive a new expression of Wigner function in the TESR. According to the definition of Wigner function [15,16] of density operator q,

W ða; a Þ ¼ Tr½Dða; a Þq;

where Dða; a Þ is the single-mode Wigner operator [15], whose explicit normally ordered form is [17]

Dða; a Þ ¼

1

p

or

jgi ¼ DðgÞjg ¼ 0i;

y



DðgÞ ¼ ega g a ;

ð5Þ

~y is a fictitious mode where DðgÞ is the displacement operator, a ~ ¼ accompanying the real photon creation operator ay ; j 0; 0i ~ and j 0i ~ is annihilated by a ~ ; ½a ~; a ~y  ¼ 1. The structure of j gi j 0i j 0i, is similar to that of the EPR eigenstate shown in Ref. [8]. Operating ~ on j gi in Eq. (4) we obtain the eigen-equations of j gi, a and a

~y Þjgi ¼ gjgi; ðay  a ~Þjgi ¼ g jgi; ða  a ~y Þ ¼ ghgj: ~Þ ¼ g hgj; hgjða  a hgjðay  a

ð6Þ

~y Þ; ðay  a ~Þ ¼ 0, thus j gi is the common eigenvector Note that ½ða  a ~  ay Þ. Using the normally ordered form of vacuum ~y Þ and ða of ða  a ~ ~ j:¼ expðay a  a ~y a ~Þ :, and the technique of inteprojector j 0; 0ih0; 0 gration within an ordered product (IWOP) of operators [12–14], we can easily prove that j gi is complete and orthonormal,

Z

2

d g

p

jgihgj ¼ 1;

0

0

0



hg jgi ¼ pdðg  gÞdðg  g Þ:

ð7Þ

ð8Þ

n¼0

~ , and n ~ denotes the number in the fictitious Hilbert (where n ¼ n space) and

~y

ajg ¼ 0i ¼ a jg ¼ 0i; ~jg ¼ 0i; ay jg ¼ 0i ¼ a  y n  y n ~ jg ¼ 0i: ~a a a jg ¼ 0 i ¼ a

ð12Þ

1 X

W ða; a Þ ¼

~ jDða; a Þqjm; m ~i hn; n

m;n

1

p 1

p

y

hg ¼ 0jDð2aÞð1Þa a jqi hg ¼ 2ajð1Þ

ay a

jqi ¼

1

p

hn ¼ 2ajqi;

ð13Þ

where the state vector j ni is defined as y

jni ¼ ð1Þa a jg ¼ ni

 E 1 ~ ~y  ay a ~y 0; 0 ¼ exp  jnj2 þ nay þ n a 2  E y ay a ~ : ¼ DðnÞe ~ 0; 0

ð14Þ

Eq. (13) is the Wigner function formula in thermo entangled state representation, with which the Wigner function of density operator is simplified as an overlap between two ‘‘pure states” in enlarged Fock space, rather than using ensemble average in the system-mode space. This will brings much convenience for calculating time evolution of Wigner functions when quantum decoherence happens (see the next section). The overlap between hg j and j ni can be calculated (using the R 2 20 completeness of coherent states d pzd2 ~z j z; ~z0 ihz; ~z0 j¼ 1),

hgjni ¼

  1 ng  n g ; exp 2 2

ð15Þ

since ng  n g is pure imaginary, hg j ni is actually a Fourier transformation kernel, and j ni is conjugate to j gi. Obviously, j ni possesses orthonormal and complete properties

Z

2

d n

p

hn0 jni ¼ pdðn0  nÞdðn0  n Þ:

jnihnj ¼ 1;

ð16Þ

For example, for the number state j nihn j, noticing j nihn j g ¼ ~ i, and the generating function of two-dimensional La0i ¼j n; n guerre polynomials [18,19] Hm;n ðx; yÞ,

It is easily seen that j g ¼ 0i has the properties 1  E X y y ~ ¼ ~ i; jg ¼ 0i ¼ ea a~ 0; 0 jn; n

ð11Þ

(Here one should understand the single-mode density operator q in the left of Eq. (12) as the direct product q  eI when q acts onto the y y ~ where eI is the identity operatwo-mode state j g ¼ 0i ¼ ea a~ j 0; 0i, tor in the auxiliary mode.) We can reform Eq. (10) as

¼

ð4Þ

p

jqi  qjg ¼ 0i:

¼

  E 1 2 ~ ~y þ ay a ~y 0; 0 jgi ¼ exp  jgj þ gay  g a 2

y y  1 : e2ða a ÞðaaÞ :¼ Dð2aÞð1Þa a :

~ j mi ~ ¼ dn;m ðn ¼ n ~ ; m ¼ mÞ ~ and introducing the notation By using hn

2. Wigner function formula in thermo entangled state representation We begin with briefly reviewing the thermo entangled state representation (TESR). On the basis of Takahash–Umezawa thermo field dynamics (TFD) [5–7] we have constructed the TESR in the doubled Fock space [9,10],

ð10Þ

1 X tm t0n Hm;n ðx; yÞ ¼ exp ½tt0 þ tx þ t 0 y m!n! m;n

from Eq. (13) we see that the Wigner function of j nihn j is

1 1jnj2 e 2 Hn;n ðn; n Þ n!p

ð1Þn 2jaj2 ¼ e Ln 4jaj2 ;

W jnihnj ða; a Þ ¼ ð9Þ

ð17Þ

1

p

~i ¼ hn ¼ 2ajn; n

p

ð18Þ

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where in the last step we have used the relation between Hm;n ðx; yÞ and Laguerre polynomial Lm ðxÞ [20],

Ln ðxyÞ ¼

ð1Þn Hn;n ðx; yÞ: n!

ð19Þ

Similarly, for coherent state j zihz j ðj zi ¼ expð j zj2 =2 þ zay Þ j 0i) ~ ¼j z; ~z i, we have e  Þ j 00i [21,22], due to j zihz j g ¼ 0i ¼ DðzÞ Dðz



1 D ~  ~  aa ~ jz; ~z i W jzihzj ða; a Þ ¼ 0; 0 exp 2jaj2 þ 2a ay þ 2aa p

1 ¼ exp 2ja  zj2 :

p

ð20Þ Further, using Eq. (15) and the completeness of hg j in Eq. (7), we can reform Eq. (13) as

W ða; a Þ ¼

Z

2

d g

p2

hn ¼ 2ajgihgjqi ¼

Z

2

d g a gag e hgjqi: 2p 2

ð21Þ

Once hg j qi is known, one can calculate the Wigner function by taking the Fourier transform of hgjqi. Eqs. (13) and (21) are two ways accessing to Wigner function, one can use either one to derive Wigner functions. 3. Evolution formula of Wigner function for amplitudedamping channel In this section, we consider Wigner function’s time evolution in the amplitude decay channel (dissipation in a lossy cavity) described by the following master equation [23]

  dq ¼ j 2aqay  ay aq  qay a ; dt

ð22Þ

where j is the rate of decay. In Ref. [11] we have reformed Eq. (22) as

  d ~ jqi; ~  ay a  a ~y a jqi ¼ j 2aa dt

ð23Þ

where Wðb; b ; 0Þ ¼ h2b j q0 i=p is the Wigner function at initial time, and we have used the following integral formula [20]

Z

2

d z

p

 n o 1 ng exp fjzj2 þ nz þ gz ¼  exp  ; f f

ð29Þ Eq. (28) is the expression of time evolution of Wigner function for amplitude-damping channel. For example, for the photon-added coherent state C m aym j zi, where C m ¼ ½m!Lm ð j zj2 Þ1=2 is the normalization factor, the initial Wigner function Wðb; b ; 0Þ is given by [24]

W ðb; b ; 0Þ ¼

jqðtÞi ¼ e

e

jq0 i:

ð24Þ

hgj exp



~y y

jt aa~  a a



jt

¼e





jt 

ge

ð25Þ

as well as Eq. (6), we obtain

 2 1 hgjqðtÞi ¼ e2T jgj gejt q0 i;

ð26Þ

where T ¼ 1  e2jt . Substituting Eq. (26) into Eq. (21), we derive the Wigner function at time t

W ða; a ; t Þ ¼

Z

2 d g a gag 1T jgj2  jt  2 e ge q0 i: 2p2

ð27Þ

Inserting the completeness relation Eq. (16) into Eq. (27) and noticing Eqs. (13) as well as (15), we can reform Eq. (27) as

Z

2 0

Z

2

d g a gag 1T jgj2  jt 0 0 2 e ge jn ihn jq0 i 2p 2  Z 2 2   d bd g T n0 ¼2b exp  jgj2 þ g a  b ejt ¼ 2 p2   þ g bejt  a W ðb; b ; 0Þ  Z 2 2 2 d b 2 ¼ exp  a  bejt  W ðb; b ; 0Þ; T T p

W ða; a ; t Þ ¼

d n

2

2

pLm ðjzj Þ

Hm;n ðx; yÞ ¼

Lm ðj2b  zj2 Þ:

ð30Þ

  @ mþn 0 0  exp ½  ss þ s x þ s y   0 ; m 0n @s @s s¼s ¼0

ð31Þ

we have

W ða; a ;t Þ 2 1 2 2 e2ðjzj þT jaj Þ @ 2m 0  0 ¼ ess szz s T pm!Lm ðjzj2 Þ @ sm @ s0m 

 Z 2 d b 2 a  exp  jbj2 þ 2b z þ etj þ s T p T a

i  tj 0 þ2b z þ e þ s T s¼s0 ¼0 2 2m 2jazejt j   e @ ¼ exp 1  2e2tj ss0 pm!Lm ðjzj2 Þ @ sm @ s0m        þ 1  2e2tj z þ 2aetj s þ 1  2e2tj z þ 2a etj s0 s¼s0 ¼0 : ð32Þ Performing a scaled transformation in the right-hand part of Eq. (32) we finally obtain



~a ~y ay Þ Then projecting Eq. (24) on hg j, and noticing exp½jtðaa being the two-mode squeezing operator,



ð1Þm e2jbzj

Substituting Eq. (30) into Eq. (28) and using Eq. (19) as well as the alternate form of the generating function of Hm;n ðx; yÞ,

thus the formal solution of Eq. (23) is jtðaa~a~y ay þ1Þ ð1e2jt Þðay a~Þðaa~y Þ=2

ReðfÞ < 0:



W ða; a ; t Þ ¼

1  2e2jt

m

jt e2jaze j

2

pLm ðjzj2 Þ "    # 2aejt þ z 1  2e2jt 2

 Lm 

1  2e2jt

;

ð33Þ

which is the analytical expression of the time evolution of Wigner function for any number (m) photon-added coherent state in a photon loss channel [25]. In particular, when t ¼ 0, Eq. (33) just reduces to Eq. (30). 4. Evolution formula of Wigner function for laser process The mechanism of laser is described by the following master equation:

  dqðtÞ ¼ g 2ay qðt Þa  aay qðt Þ  qðt Þaay dt   þ j 2aqðtÞay  ay aqðt Þ  qðt Þay a ;

ð34Þ

where g and j represent the cavity gain and loss, respectively. Eq.  and (34) reduces to Eq. (22) when g ¼ 0; while for g ! jn j ! jðn þ 1Þ, Eq. (34) becomes

p

  dq  þ 1Þ 2aqay  ay aq  qay a ¼ jðn dt    2ay qa  aay q  qaay ; þ jn ð28Þ

ð35Þ

which corresponds to the master equation in thermal environment [23].

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L.-y. Hu, H.-y. Fan / Optics Communications 282 (2009) 4379–4383

Similar to the way of obtaining Eq. (26), we have derived in Ref. [11]

  ~a ~y ay þ 1 ðj  g Þt aa      ðj þ g Þ 1  e2ðjgÞt  y ~ aa ~y jq0 i: a a  exp 2ðj  g Þ

jqðtÞi ¼ exp



ð36Þ

Thus the matrix element hg j qðtÞi is given by

 A 2  hgjqðtÞi ¼ exp  jgj geðjgÞt q0 i; 2

ð37Þ

where

 jþg 1  e2ðjgÞt : jg

ð38Þ

According to Eq. (27) the Wigner function’s evolution for Laser process is given by

  2  ðjgÞt  d g A 2   q i exp  g j þ a g  ag ge j 0 2 2p2 Z 2 2  d nd g Ajgj2 þa gag  ðjgÞt  ¼ n¼2b W ðb; b ; 0Þ e 2 ge 2p2 Z 2 2 d nd g A2jgj2 þgða b eðjgÞt Þþg ðbeðjgÞt aÞ e W ðb; b ; 0Þ ¼ 2

W ða; a ; t Þ ¼

Z

2 ¼ A

Z

p

2

d b

p

 2 2 exp  a  beðjgÞt  W ðb; b ; 0Þ; A

Z

2

d b

p

the Wigner function (42) is always positive-definite. Thus we emphasize that for any values of m, once the condition (44) is satisfied, the Wigner function has no chances to be negative.

Next we consider photon number (PN) measurement in laser process. According to the TFD, we can reform the PN pðnÞ ¼ tr½q j nihn j as

pðnÞ ¼ hnjqjni ¼

1 X

~ jqjm; m ~ i ¼ hn; n ~ jqjg ¼ 0i ¼ hn; n ~ jqi hn; n

ð45Þ

m¼0

~ jqi in the conthus the PN is converted to the matrix element hn; n text of thermo dynamics. Then using the completeness of hn j and Eq. (13) as well as Eq. (18), we see

pðnÞ ¼

Z

2

d n

p

Z

~ jnihnjqi hn; n

2

~ jniW ða ¼ n=2; a ¼ n =2Þ d nhn; n Z 2 ¼ 4p d aW jnihnj ða; a ÞW ða; a Þ; ¼

ð46Þ

ð39Þ

where we have used Eq. (29). In particular, when g ¼ 0, Eq. (39) re þ 1Þ, leading to  and j ! jðn duces to Eq. (28). For g ! jn  þ 1ÞT (where T ¼ 1  e2jt ), Eq. (39) becomes A ¼ ð2n

2 W ða; a ; t Þ ¼  þ 1ÞT ð2n

ð44Þ

5. Photon number measurement in laser process



A¼

 þ 1Þ 2ðn

1

jt P jtc ¼ ln  2 2n þ 1

2

jabejt j W ðb; b ; 0Þe2 ð2nþ1ÞT ;

ð40Þ

one can see this formula also in [1,26]. Thus one can calculate the PN by combining Eqs. (39) and (46). Now we evaluate the PN for the above decoherence model in Eq. (34). Substituting Eq. (39) into Eq. (46), we see

pðnÞ ¼

8 A

Z

2

d bW ðb; b ; 0ÞGðb; b Þ;

ð47Þ

where

or

W ða; a ; t Þ ¼ 2e2jt

Z



pffiffiffi 2 d bW T ðb; b ÞW ða  T bÞejt ; 0 ;

ð41Þ

n o 2 where W T ðb; b Þ ¼ pð2n1þ1Þ exp  22jbj  þ1 is the Wigner function of the n . thermal state with mean photon number n Similar to the way of deriving Eq. (33), when the initial state is C m aym j zi, substituting Eq. (30) into Eq. (39) we have

!  þ 1ÞT m ½ðn Am jBj2 ; W ðb; b ; t Þ ¼ Lm  A pLm ðjzj2 Þ ð2n T þ 1Þ ðn T þ 1Þm

Z

ð48Þ Using Eqs. (29) and (31) we can evaluate Eq. (48) as

2



  2 2 2 d aW jnihnj ða; a Þ exp  a  beðjgÞt   2jaj2 A   Z 2

2 d a 2 Ln 4jaj2 exp  a  beðjgÞt   2jaj2 : ¼ ð1Þn A p

Gðb; b Þ 

eC2jbj

ð42Þ

Z 2 1 @ nþn ss0 2jbj2 e2ðjgÞt d a A e n 0n n! @ s @ s p 



 Aþ1 2 b b jaj þ 2a s þ ðjgÞt þ 2a s0 þ ðjgÞt  exp 2 A Ae Ae s¼s0 ¼0

Gðb; b Þ ¼

where T ¼ 1  e2jt and

e2jt =T ;  þ 1Þðn  þ 1Þ ð2T n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jt  z ejt rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  T þ 1e n 2b  nTþ1  þ 1ÞT  ðn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z þ B¼ T þ 1 n   þ 1ÞT ð2nT þ 1Þ ðn

    1 3nT þ 2  jt 2  2 jbj2 C¼ þ 4T 2 n ze T þ 1 T n þ1 2n jt  2e ðT n þ 1Þ  ðzb þ bz Þ: T þ 1 2n A¼1

2e2ðjg Þt

2

Ae Aþ1 jbj @ nþn 2ðA þ 1Þn! @ sn @ s0n  1A 0 2b eðjgÞt 0 2beðjgÞt  exp  s sþ s þs : 1þA Aþ1 Aþ1 s¼s0 ¼0

¼

ð43Þ

ð49Þ

Performing some scaled transformations, we finally obtain 2tj

 ¼ 0, leading to A ¼ 12eT ; B ¼ In particular, when n 2 2 p1ffiffi ðð1  2e2jt Þz þ 2ejt b Þ, and C  2 j bj ¼ 2 j b  zetj j , thus T Eq. (42) reduces to Eq. (33). Eq. (42) manifestly shows that the Wigner function of C m aym j zi in thermal environment is closely related to the Laguerre polynomials. In addition, due to Lm ð j xj2 Þ > 0, so C m > 0, it is easily seen that when A > 0, which means the condition

  nþn 2e2ðjgÞt 2 @ exp  j b j Aþ1 @ sn @ s0n 2ð1 þ AÞnþ1 n! ( ) 2b eðjgÞt 0 2beðjgÞt  0  exp s s þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s þ s pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  A2 1  A2 s¼s0 ¼0 !   AðA  1Þn 2e2ðjgÞt 2 4jbj2 e2ðjgÞt j b j L : ¼ exp  n Aþ1 2ð1 þ AÞnþ1 1  A2

Gðb; b Þ ¼

AðA  1Þn

ð50Þ

L.-y. Hu, H.-y. Fan / Optics Communications 282 (2009) 4379–4383

 jt In particular, when g ¼ p 0,ffiffiffiffiffiffiffiffiffiffiffi leading to A ¼ x ¼ T; r ¼ peffiffiffiffiffiffiffiffi2 ; ffi 1T l ¼ 2e22jt ; kr ¼ 1, and k ¼ 1 þ T , thus

Substituting Eq. (50) into Eq. (47) yields

  4ðA  1Þn 2e2ðjgÞt 2 2 d pðnÞ ¼ b exp  j b j Aþ1 ðA þ 1Þnþ1  2ðjgÞt  4e  Ln jbj2 W ðb; b ; 0Þ; 1  A2 Z

pðnÞ ¼ ð51Þ

which is a new formula for calculating the photon number distribution of the open system in an environment. From Eq. (51) it is easily seen that once the Wigner function of initial state is known, one can obtain its photon number distribution by performing the integration in Eq. (51). In particular, when g ¼ 0; A ¼ 1  e2jt ¼ T, Eq. (51) reduces to

pðnÞ ¼

4ð1Þn e2jt nþ1

ð2e2jt  1Þ  Z 2  d bW ðb; b ; 0Þ exp 

 ( 2jt 2 ) 4e jbj jbj Ln ; 2e2jt  1 2e2jt  1 2

2

ð52Þ which corresponds to the photon number of density operator in the amplitude-damping quantum channel.  þ 1Þ, Eq. (51) becomes to  and j ! jðn While for g ! jn

  2e2jt 2 2 pðnÞ ¼ d b exp  jbj nþ1 Aþ1 ðA þ 1Þ  2jt  4e  Ln jbj2 W ðb; b ; 0Þ; 1  A2 4ðA  1Þn

Z

ð53Þ

 þ 1ÞT ¼ ð2n  þ 1Þð1  e2jt Þ. Eq. (53) corresponds to where A ¼ ð2n the photon number of system interacting with thermal bath. For example, we still consider the photon-added coherent state field. Substituting Eq. (30) into Eq. (51) and using Eqs. (29) and (31) yields

Z



2

4e2ðjgÞt 2 Lm ðj2b  zj2 ÞLn jbj 1  A2

  e2ðjgÞt jbj2  exp 2ðzb þ bz Þ  2 1 þ Aþ1

pðnÞ ¼ Ne2jzj

2

d b



p

2

Ne2jzj ð1Þmþn @ 2m @ 2n 0 exp ftt0  z t0 g  eztss m 0m m!n! @ t @ t @ sn @ s0n Z 2 h d b  exp 2ljbj2 þ 2ðrs þ z þ tÞb p   Nð1Þmþn 2  2l 2 þ2ðrs0 þ z þ t0 Þb t¼t0 ¼s¼s0 ¼0 ¼ exp j zj 2lm!n! l

¼

@ 2m @ 2n exp ½xtt0 þ ðkr  1Þss0 þ kðst0 þ ts0 Þ m 0m @ t @ t @ sn @ s0n ð54Þ þxðz t0 þ ztÞ þ kðzs þ z s0 Þt¼t0 ¼s¼s0 ¼0 ; 

where we have set

x¼

2l

l

;

k¼

2r

l

eðjgÞt

r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

;

ð55Þ

1A

and

N¼

4ðA  1Þn ð A þ 1Þ

nþ1

ð1Þm 2

Lm ðjzj Þ

;

l¼1þ

e2ðjgÞt : Aþ1

ð56Þ

Further expanding the exponential item exp½xtt0 þ ðkr  1Þss0 , we finally obtain

pðnÞ ¼

 k 22l 2 m;n m!n! xðkr  1Þ=k2 Nk2n e l jzj X 2lðxÞnm l;k¼0 l!k!½ðm  lÞ!ðn  kÞ!2   pffiffiffiffiffi pffiffiffiffiffi 2  Hml;nk i xz; i xz  :

4383

ð57Þ

m m! ð1  xÞn X xmn n! Lm ðjzj2 Þ l¼0 l!½ðm  lÞ!2 n o  pffiffiffiffiffi pffiffiffiffiffi 2  exp e2jt jzj2 Hml;n i xz; i xz  ;

ð58Þ

which coincides with Eq. (43) with ideal detection efficiency in Ref. [25]. In sum, by virtue of the thermo entangled state representation we have presented a new approach for deriving time evolution formula of the Wigner functions when a quantum system interacts with an environment, during the time quantum decoherence, damping and/or amplification may happen. As an application of the formula, the time evolution formula of photon number measurement is also derived. In our approach the Wigner function of density operator is simplified as an overlap between two ‘‘pure states” in enlarged Fock space, rather than using ensemble average in the system-mode space, in this aspect, our approach seems selfconsistent and concise. Acknowledgements We sincerely thank the referees for their constructive suggestion. Work supported by the National Natural Science Foundation of China under Grants 10775097 and 10874174, and the Research Foundation of the Education Department of Jiangxi Province.

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