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Statistical properties of photon-added coherent state in a dissipative channel

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2009 Phys. Scr. 79 035004

(http://iopscience.iop.org/1402-4896/79/3/035004)

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Phys. Scr. 79 (2009) 035004 (8pp) doi:10.1088/0031-8949/79/03/035004

Statistical properties of photon-addedcoherent state in a dissipative channelLi-Yun Hu1,2 and Hong-Yi Fan2

1 College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022,People’s Republic of China2 Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

E-mail: hlyun2008@126.com or hlyun@sjtu.edu.cn

Received 25 September 2008Accepted for publication 28 January 2009Published 3 March 2009Online at stacks.iop.org/PhysScr/79/035004

AbstractWe adopt a new approach (using the entangled state representation) to study time evolutionof the mixed state in a dissipative channel (DC) with decay rate κ . We then investigate thestatistical properties of the m-photon-added coherent state (PACS) in the DC, by deriving theanalytical expressions of the Wigner function (WF), which turns out to be theLaguerre–Gaussian function. The criterion time for the definite positivity of WF:κt > κtc ≡

12 ln 2, valid for arbitrary m, is revealed. Time evolution of the photocount

distribution is also derived analytically, which is related to two-variable Hermite polynomials.The tomogram of PACS is also calculated through the intermediate coordinate–momentumrepresentation.

PACS numbers: 03.65.−w, 42.50.Dv

(Some figures in this article are in colour only in the electronic version.)

1. Introduction

The photon-added coherent state (PACS), first introducedby Agarwal and Tara [1], is the result of successiveelementary one-photon excitations of a coherent state, andis an intermediate state between the Fock state and thecoherent state, since it exhibits the sub-Poissonian character.Theoretically, the PACS can be obtained by repeatedlyoperating the photon creation operator a† on a coherent state;its density operator is

ρ0 = Cα,ma†m|α〉 〈α| am, (1)

where Cα,m = [m!Lm(−|α|2)]−1 is the normalization factor,

|α〉 = exp(−(|α|2/2) + αa†)|0〉 is the coherent state [2, 3] and

Lm(x) is the mth-order Laguerre polynomial,

Lm(x) =

m∑l=0

(ml

)(−x)l

l!. (2)

The nonclassical quasi-probability distribution and theamplitude squeezing of the PACS have been discussedin detail [1, 4–6]. Later, Zavatta’s group [7] preparedexperimentally the single PACS by using parametric

down-conversion, and its nonclassical behavior was detectedthrough homodyne tomography technology [8]. Recently, ascheme was proposed by Kalamidas et al [9] to generate a twoPACS. For a recent review on photon addition and subtraction,we refer the reader to [10]. However, to our knowledge, theWigner function (WF) of the PACS in a dissipative channel(DC) has not been previously derived, neither exactly noranalytically, in the literature.

In the present paper, we shall adopt a new approach forderiving the photon distribution, WF and tomogram of thePACS in a dissipative environment. This work is organizedas follows. Instead of using the conventional techniques(such as solving the Langevin equation or the Fokker–Planckequation), in section 2, we solve the master equation byvirtue of the entangled state representation 〈η|; the meritof doing so lies in that the dissipative nature is manifestlyshown in the entangled state representation as 〈η| → 〈η e−κt

|,where κ is the rate of decay (see equation (19) below).In section 3, using the technique of integration within anordered product (IWOP) of operator [11, 12], we shall derivethe normal ordering form of the density operator, which isuseful for obtaining various phase space distributions. Insection 4 by using the coherent state representation of the

0031-8949/09/035004+08$30.00 1 © 2009 The Royal Swedish Academy of Sciences Printed in the UK

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

one-dimensional (1-D) Wigner operator, the WF [13–15]is obtained, which turns out to be the Laguerre–Gaussianfunction. The criterion time for the definite-positivity ofWF: κt > κtc ≡

12 ln 2, is revealed. In section 5, the time

evolution of photocounting distribution of the PACS isderived analytically, which is related to two-variable Hermitepolynomials (or the Legendre polynomials), and clearlyexhibits the manner in which the distribution of photoncounting depends on the photon-added number (or excitationnumber) and the intensity of the coherent field. Section 6 isdevoted to deriving the quantum tomogram, with the aid of theintermediate coordinate-momentum representation. It is foundthat the tomogram is analytical in the single-variable Hermitepolynomial.

2. Time evolution of mixed state in a DC derivedby the entangled state representation

For a lossy channel (or cavity at zero temperature) theevolution of the density operator is described by [16]

dρ(t)

dt= κ

[2aρ(t)a†

− a†aρ(t) − ρ(t)a†a], (3)

where [a, a†] = 1 and κ is the rate of decay. In theliterature, solving the master equation (such as (3)) is usuallydone by using the Langevin equation or the Fokker–Planckequation after recasting the density operators into somedefinite representation, e.g. particle number representation(Q-function), coherent state representation (P-representation)or the Wigner representation. Here, enlightened by the theoryof thermo field dynamics theory invented by Takahashiand Umezawa ([17]; see the memorial issue [18] andreferences therein; [19]), we introduce a fictitious mode a†

accompanying the real photon creation operator a†, andconstruct the two-mode entangled state representation

|η〉 = exp(−

12 |η|

2 + ηa†− η∗a† + a†a†

)|0, 0〉, (4)

where η = η1 + iη2, and |0〉 is annihilated by a, [a, a†] = 1. Itcan be seen that

|η〉 = D(η)|η = 0〉. (5)Here D(η) = exp(ηa†

− η∗a) is a displacement operator and

|η = 0〉 = ea†a†|0, 0〉 =

∞∑n=0

|n, n〉, (6)

where |n, n〉 =1n! (a

†a†)n|0, 0〉. Using the normally ordered

form of the vacuum projector |0, 0〉〈0, 0| = : exp(−a†a −

a†a) : (where : : stands for normal ordering) and the IWOPtechnique, we can show that |η〉 is complete and orthogonal,∫

d2η

π|η〉 〈η| = 1,

(7)⟨η′∣∣ η〉 = πδ

(η′

− η)δ(η′∗

− η∗).

This provides us with the possibility of expressing densityoperators in terms of |η〉. It is easily seen that

a|η = 0〉 = a†|η = 0〉,

a†|η = 0〉 = a|η = 0〉, (8)(

a†a)n

|η = 0〉 =(a†a

)n|η = 0〉.

At this point, it is convenient to introduce the state |ρ(t)〉 [20]for the density operator ρ(t),

|ρ(t)〉 = ρ(t)|η = 0〉. (9)

Multiplying both sides of (3) by |η = 0〉 yields

d

dt|ρ(t)〉 = κ

[2aρ(t)a†

− a†aρ (t) − ρ(t)a†a]|η = 0〉 .

(10)Using equation (8) and noting that the density operatorρ(t) is defined in the original space, which is commutativewith operators ( a†, a) in the tilde space, we can convertequation (10) into

d

dt|ρ(t)〉 = κ

(2aa − a†a − a†a

)|ρ(t)〉 . (11)

Thus the formal solution of equation (11) is

|ρ(t)〉 = exp{κt(2aa − a†a − a†a

)}|ρ0〉, (12)

where ρ0 is the initial density operator, |ρ0〉 = ρ0|η = 0〉. Todeal with equation (12), by noting that

2aa − a†a − a†a = −(a†

− a) (

a − a†)+ aa − a†a† + 1,

−2[−(a†

− a) (

a − a†)]=[aa − a†a†, −

(a†

− a) (

a − a†)](13)

and employing the operator identity

eλ(A+σ B)= eλA exp[σ(1 − e−λτ )B/τ ] (14)

(valid for [A, B] = τ B), we can rewrite equation (12) as

|ρ(t)〉 = eκt(aa−a†a†+1)

× exp[(

1 − e2κt) (a†− a

) (a − a†) /2

]|ρ0〉. (15)

Recall that exp[λ(aa − a†a†)] is the two-mode squeezingoperator and has its natural expression in the 〈η|

representation [24, 25],

exp[λ(aa − a†a†)]

= eλ

∫d2η

π

∣∣eλη⟩〈η| , (16)

so

〈η| exp{κt[aa − a†a†]}

= eκt∫

d2η′

π〈η| eκtη′

⟩ ⟨η′∣∣

= e−κt ⟨η e−κt∣∣ . (17)

By noting the eigenvector equations

〈η|(a − a†)

= η∗〈η| ,

(18)〈η|(a†

− a)= η〈η|,

we see that the wave function of |ρ(t)〉 in the 〈η|

representation is

〈η| ρ(t)〉 =⟨ηe−κt

∣∣ exp[(

1 − e2κt) (a†− a

) (a − a†) /2

]|ρ0〉

= e−12 T |η|

2 ⟨η e−κt

∣∣ ρ0〉, (19)

2

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

where T = 1 − e−2κt . This is a new relationship between |ρ0〉

and |ρ(t)〉, which clearly shows that the wave function ofthe mixed state ρ(t) in the 〈η| representation is proportionalto that of the initial state ρ0 in the decayed entangledstate 〈ηe−κt

|, accompanied by a Gaussian damping factore−(T/2)|η|

2. Thus we can clearly see that the DC plays a role

in time evolution. Multiplying∫ d2η

π|η〉 by the left-hand side

of equation (19) and using the IWOP technique as well as theoperator identity exp(λa†a) = : exp[(eλ

− 1)a†a] :, we have

|ρ(t)〉 =

∫d2η

π|η〉 〈η| ρ(t)〉

=

∫d2η

πe−

12 T |η|

2|η〉⟨ηe−κt

|ρ0〉

=

∫d2η

π: exp

[− |η|

2 + η(a†

− e−κt a)

+η∗(e−κt a − a†)+ a†a† + aa − a†a − a†a

]: |ρ0〉

= : exp[(

e−κt−1)(

a†a+a†a)+(1 − e−2κt) aa

]: |ρ0〉

= exp[−κt

(a†a + a†a

)]exp

[T aa

]|ρ0〉 . (20)

Using equation (8), we can rewrite equation (20) as

|ρ(t)〉 = e−κta†a∞∑

n=0

e−κt a†a T n

n!an anρ0 |η = 0〉

= e−κta†a∞∑

n=0

T n

n!e−κt a†aanρ0a†n

|η = 0〉

= e−κta†a∞∑

n=0

T n

n!anρ0a†ne−κt a†a

|η = 0〉

= e−κta†a∞∑

n=0

T n

n!anρ0a†ne−κta†a

|η = 0〉 , (21)

which indicates that

ρ(t) =

∞∑n=0

T n

n!e−κta†aanρ0a†n e−κta†a . (22)

This is the density operator for a lossy channel [21], whichis just the Kraus operator sum representation [22]. For moreapplications of the entangled state representation, we refer thereader to [23].

3. The normal ordered form of ρ(t) for PACS

When the initial ρ0 is the PACS in (1), ρ0 = Cα,ma†m|α〉〈α|am ,

substituting it into equation (22) yields

ρα,m(t) = Cα,m

∞∑n=0

T n

n!Gm,n

(a, a†

; t), (23)

where

Gm,n(a, a†

; t)≡ e−κta†aana†m

|α〉 〈α| ama†n e−κta†a . (24)

It is useful to derive the normal ordered form of ρα,m(t)for seeking different phase space distributions, such as

Q-function, P-representation and the WF. Using the operatoridentity

ana†m= (−i)m+n : Hm,n

(ia†, ia

): , (25)

where Hm,n(x, y) is the two-variable Hermite polynomial[26, 27], whose generating function is

Hm,n(x, y) =∂m+n

∂tm∂t ′nexp

[−t t ′ + t x + t ′y

]∣∣t=t ′=0 , (26)

and the vacuum state projector |0〉〈0| = : exp(−a†a) :, we cantranslate equation (24) into

Gm,n(a, a†

; t)

= (−1)m+n ∂m+n

∂λm∂λ′n

∂m+n

∂τ n∂τ ′m

× exp[−λλ′ + iαλ′

− ττ ′ + iα∗τ]

× : e−|α|2+ia† e−κt λ+a† e−κt α+α∗a e−κt +ia e−κt τ ′

−a†a :

= (−1)m+n: exp[− |α|

2 +(a†α + aα∗

)e−κt]

× Hm,n(ia† e−κt , iα

)Hm,n

(ia e−κt , iα∗

): . (27)

Thus the normal ordering form of ρα,m(t) is

ρα,m(t) =(−T )m

m!Lm(− |α|2)

: e−(a−α e−tκ)(a†−α∗ e−tκ)

× Hm,m

[i(T α∗ + a† e−κt

)T −1/2,

i(T α + a e−κt

)T −1/2

]: , (28)

where we have used an identity about the two-variableHermite polynomial

∞∑n=0

zn

n!Hm,n (x, y) Hm,n

(x ′, y′

)= (−z)m ezyy′

×Hm,m

[i(

√zy′

−x

√z

), i(

√zy −

x ′

√z

)],

(29)

which can be directly proved by using equation (26).

4. The WF of ρα,m(t)

In this section, we calculate the WF of ρα,m(t) for PACS (andits marginal distributions) in a DC. Recall that the coherentstate representation of the Wigner operator is [28, 29]

1(β) =

∫d2z

π2|β + z〉 〈β − z| eβz∗

−zβ∗

, (30)

where β = (q + ip)/√

2 and |β〉 is a coherent state, so using(28) and (26) we see that the WF W (β, t) of ρα,m(t) is

W (β, t) = Tr[ρα,m (t) 1(β)

]=

∫d2z

π2〈β − z| ρα,m(t) |β + z〉 eβz∗

−zβ∗

=(−T )m

πm!Lm(− |α|2)

e−2|β−αe−tκ |2

3

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

×∂2m

∂λm∂τmexp

[−T −1 (T − e−2κt) λτ

+ iλ(T α∗

− α∗e−2κt + 2β∗e−κt) T −1/2

+iτ(T α − αe−2κt + 2βe−κt) T −1/2]

λ,τ=0 . (31)

Further, by using (26) and the relation between the Hermitepolynomial and the Laguerre polynomial,

Lm(xy) =(−1)m

m!Hm,m(x, y), (32)

equation (31) is transformed into

W (β, t) =

(1 − 2e−2κt

)m

π Lm(− |α|2)

e−2|β−α e−κt |2

× Lm

[−

∣∣α (1 − 2e−2κt)

+ 2β e−κt∣∣2

1 − 2e−2κt

], (33)

so the WF is the Laguerre–Gaussian function, and thisanalytical result is appealing. In particular, at the initial timet = 0,

W (β, 0) =(−1)m

π Lm(− |α|

2)e−2|β−α|

2Lm

(|2β − α|

2) , (34)

which agrees with equation (3.8) of [1]. For a long timeinterval, κt → ∞, the WF becomes W (β, ∞) = 1/πe−2|β|

2,

which corresponds to the vacuum state.Further, due to Lm(−|α|

2) > 0, the WF becomes positivefor arbitrary m values if 1 − 2e−2κt > 0, which leads to thefollowing criterion time for the definite positivity of WF:

κt > κtc ≡12 ln 2, (35)

where tc is the critical time for a fixed κ and this result isα independent. On the contrary, when κt < κtc, the WF hasthe chance to exhibit negativity for different m values, whichexhibits the nonclassicality of PACS.

Using equation (33), we plot in figures 1 and 2 thevariation of the WF W (β, t) in the phase space for severaldifferent values of κt and m, where α = (1 + i)/4. Fromfigure 1, it can be seen that the WF shows the partial negativeregion and its minimum in the negative region becomes largerwith the increasing of m. On the other hand, for a given κt,the WF in figure 2(a) presents the obvious negative regionand the one in figure 2(b) has almost no negative region,whereas there is no negative value in figure 2(c). These implythat the ‘effective time’ for the WF to become fully positive isdependent on the parameter m.

We now integrate the WF either over the variable p orthe variable q and find the probability distribution of positionor momentum—the marginal distributions, respectively. Byusing equations (33) (denoting α = α1 + iα2), we derive

P(q, t) ≡

∫∞

−∞

W (q, p, t) dp

=e−2mκt e−2(q/

√2−α1 e−κt )2

2m√

π Lm(− |α|2)

×

m∑l=0

(2T e2κt

)lm!

l! [(m − l)!]2|Hm−l(A)|2 , (36)

(a)

(b)

(c)

Figure 1. WFs for several different values m = 1, 2, 5 at κt = 0.

where Hm(x) is the single-variable Hermite polynomialand A = (α eκt

− 2α1 e−κt +√

2q/√

2. At initial time t = 0,T = 0, A = q − α∗/

√2, then the marginal distribution in the

‘q-direction’ becomes

P(q, 0) =2−m e−2(q/

√2−α1)

2

√πm!Lm(− |α|

2)|Hm(q − α/

√2)|2. (37)

If time t is large enough, equation (36) reduces to P(q, ∞) =

e−q2/√

π , which indicates that the PACS shall reduce to thevacuum state after a long-time decoherence.

On the other hand, performing the integration over dqyields

P (p, t) =e−2mκt e−2(p/

√2−α2 e−κt )2

2m√

π Lm(− |α|2)

×

m∑l=0

m!(2T e2κt

)ll! [(m − l)!]2

∣∣Hm−l(iA′∗

)∣∣2 , (38)

where A′= (α eκt

− 2iα2 e−κt +√

2ip)/√

2. Equation (38) isthe other marginal distribution (in the ‘p-direction’) of theWF.

4

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

(b) m=3

(c) m=6

(a) m=1

Figure 2. WFs for several different values m = 1, 3, 6 at κt = 0.3.

5. Photocount distribution of PACS in a dissipativereservoir

Now, by using equation (28), we calculate the photocountdistribution of the optical field. The quantum mechanicalphoton counting formula was first derived by Kelley andKleiner [30]. As shown in [30–34] the probability distributionp(n) of registering n photoelectrons in time interval 1τ isgiven by

p(n) = Tr

{ρ:

(ξa†a

)n

n!e−ξa†a :

}, (39)

where ρ is the single-mode density operator of the lightfield concerned and ξ = σ(hω1τ/2ε0υ) (with σ beingthe quantum efficiency which depends on the detectorparameters) is usually called the quantum efficiency (ameasure) of the detector. In [35], on the basis of the IWOPtechnique, we have reformed (39) as

p(n)=

(ξ

ξ − 1

)n∫ d2β

πe−ξ |β|

2Ln(|β|

2) Q(√

1 − ξβ)

, (40)

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

p(n)

n=4

p(n)

kt

0.00

0.05

0.10

0.15

0.20

0.25

m=2

m=2

n=7

m=7

m=4

m=4m=3

m=1

m=1

Figure 3. Time evolution of the probability distribution p fordifferent values of m and n with α = 1.5 and ξ = 1.

where Q(z) = 〈z|ρ(t)|z〉 is the Q-function. Once theQ-function of ρ(t) is known, it is easy to calculate photo-count distribution by using equation (40). Using the normallyordered ρ(t) in equation (28), we directly obtain theQ-function

Q(z) =T m

Lm(− |α|2)

e−|z−α e−κt |2

Lm

(−∣∣T α + z e−κt

∣∣2 /T)

.

(41)

Then substituting equation (41) into equation (40) and usingequation (26), we have

p(n) =ξ n

(1 − ξ)n

(−T )m

m!n!Lm(− |α|2)

∂2n

∂λn∂λ′n

∂2m

∂υm∂υ ′m

× exp[−λλ′

− υυ ′− αα∗ e−2κt + iυ

√T α + i

√T υ ′α∗

]×

∫d2β

πexp

{− |β|

2 +β

[λ+√

1 − ξ e−κt

(α∗+

iυ√

T

)]}× exp

{β∗

[λ′ +

√1 − ξ e−tκ

(α +

iυ ′

√T

)]}λ=λ′=υ=υ ′=0

=ξ n (−T )m

(1 − ξ)n

e−|α|2ξ e−2κt

m!n!Lm(− |α|2)

∂m+n

∂λn∂υ ′m

∂m+n

∂λ′n∂υm

× exp[(

1 − ξ e−2κt) ( iυ√

T

iυ ′

√T

+iυ ′

√T

α∗ +iυ

√T

α

)+(λ

iυ ′

√T

+ λ′iυ

√T

+ λα + λ′α∗

)√1 − ξ e−κt

]λ=λ′=υ=υ ′=0.

(42)

By making some scaled transformations in the right-hand sideof equation (42), we finally obtain

p (n) =m!(ξ e−2κt

)n

n!Lm(− |α|2)

m∑j=0

(1 − ξ e−2κt

)m−ne−|α|

2ξ e−2κt

j!(m − j)!(m − j)!

×

∣∣∣Hn,m− j

(i√

1 − ξ e−2κtα, i√

1 − ξ e−2κtα∗

)∣∣∣2 ,

(43)

which is positive definite, as expected. From equation (43) onecan see that the photocount distribution is related to the decayof quantum efficiency ξ ′

= ξ e−2κt .

5

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

(a) m=n=1

p

Re Im

(b) m=4, n=1

p

Re Im

(c) m=1, n=2

p

Re Im

(d) m= n=5

p

Re Im

Figure 4. Plot of the probability distribution p in the space phasefor different values of m, n with given κt = 0.15 and ξ = 0.99.

Using equation (43), we plot the variation of distributionp with time κt in figure 3 for given α and ξ . It is seen thatfor a given n, the larger the photon-added number m is, thelonger it takes to reach p’s maximum; whereas for a givenm, the smaller the parameter n is, the longer it takes to reachp’s maximum. In figure 4, for given κt and ξ , we show theplot of p in the phase space (α = Re α + i Im α), as one cansee that: (i) when m = n, the shape of the plots loses theGaussian form with the increase of m and n (figures 4(a) and(d)); in particular, when n = 1, the distribution of p always

keeps the Gaussian form for any m. (ii) for a given m, the‘collapse’ appears at the center region of the space phase andit becomes larger with the increase of n (figures 4(a) and (c));in addition, the distribution region of p becomes larger withthe augment of n. (iii) For a given n, the distribution regionbecomes smaller with the increase of m (figures 4(a) and (b)).These imply that the modulation of p with the variation of mand n is in the opposite direction.

The special cases of formula (43) are as follows:

p(n) = 0, κt → ∞, n 6= 0, (44)

p(n) =e−ξ |α|

2

n!

(ξ |α|

2)n, t = 0, m = 0, (45)

p(n) =n!e−|α|

2

m!Lm(− |α|2)

|α|2(n−m)

[(n − m)!]2 (t = 0, ξ = 1, n > m).

(46)

Equation (44) implies that after a long-time decoherence,one cannot detect photons anymore, whereas equation (45)is a Poisson distribution of the coherent state in agreementwith the results of [31–34]; equation (46) is just the numberdistribution of the PACS, which is zero for n < m [1].

6. Tomogram of ρα,m(t)

In [36], Vogel and Risken pointed out that the WF canbe obtained by tomographic inversion of a set of measuredprobability distributions Pφ(xφ) of the quadrature amplitude.Later, Smithy et al [37] also pointed out that once thedistributions Pφ(xφ) are obtained, one can use the inverseRadon transformation familiar in tomographic imaging toobtain the WF and density matrix. A direct description ofthe quantum states by means of the quantum tomogram forthe system observable is interesting from both the theoreticaland the experimental points of view. In this section, wederive the tomogram of ρα,m(t). For the single-mode case, wehave proved in [38] that the Radon transform of the Wigneroperator is just a pure state density operator,∫

δ(q − f q ′

− gp′)1(β) dq ′ dp′

= |q〉 f,g f,g 〈q| , (47)

where β = (q + ip)/√

2 and ( f, g) are real,

|q〉 f,g = C ′ exp

[√2qa†

f − ig−

f + ig

2 ( f − ig)a†2

]|0〉, (48)

and C ′= [π( f 2 + g2)]−1/4exp{−q2/[2( f 2 + g2)]}. Equation

(48) satisfies the eigenvector equation

( f Q + g P) |q〉 f,g = q |q〉 f,g, (49)

where Q and P are related to a, a† by Q = (a + a†)√

2 andP = (a − a†)(i

√2). Equation (49) becomes the coordinate

(momentum) eigenstate, respectively, when f = 1, g = 0( f = 0, g = 1). Thus we name it the intermediate coordinate-momentum representation [38]. Since Tr[1(β)ρ(t)] is the WF

6

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

of ρ(t), it then follows that the tomogram is given by,

R(q) f,g ≡

∫δ(q − f q ′

− gp′)

Tr [1(β)ρ(t)] dq ′ dp′

= Tr[|q〉 f,g f,g 〈q| ρ(t)

]= f,g 〈q| ρ(t) |q〉 f,g . (50)

From equation (50) one can see that the tomogram of quantumstate ρ(t) can be considered as the quantum average ofρ(t) in the intermediate coordinate-momentum representation|q〉 f,g . This provides us with an alternative way to calculatethe tomogram of quantum states. For the calculation ofthe tomogram in the two-mode case, we refer the readerto [39].

Substituting the normal ordered form ρα,m(t) in equation(28) into equation (50) and then inserting the completenessrelation of the coherent state, we have

R(q) f,g =

∫d2z′

πf,g 〈q

∣∣z′⟩ ⟨

z′∣∣ ρ(t)α,m

∫d2z

π|z〉 〈z |q〉 f,g

=C ′2T m

Lm(− |α|2)

(−1)m

m!exp

[e−tκRe

(√

2q2α

D

− e−tκ Dα2

D∗

)− αα∗e−2tκ

]∂2m

∂τ ′m∂τm

× exp[−τ ′τ +

i√

T

(τ ′ B + τ B∗

)+

e−2κt

2T

(D

D∗τ ′2 +

D∗

Dτ 2

)]∣∣∣∣τ=τ ′=0

, (51)

where

B = T α∗ +

√2q

De−κt

−Dα

D∗e−2tκ

D = f − ig.

Then using the generation function of the single-variableHermite polynomial

Hn(x) =∂n

∂tnexp

(2xt − t2)∣∣

t=0 , (52)

and the well-known recurrence relations

d

dx lHn(x) =

2ln!

(n − l)!Hn−l(x), (53)

we can further reform equation (51) as

R(q) f,g =m!C ′2e−2mκt

2m Lm(− |α|2)

exp[2e−tκRe

(√

2qα

D−

Dα2

2D∗e−tκ

)

− |α|2 e−2κt

] m∑l=0

(2e2κt T

)ll! [(m − l)!]2

×

∣∣∣∣∣Hm−l

(√D∗

2DBeκt

)∣∣∣∣∣2

, (54)

which is the positive-definite tomogram of ρα,m(t), asexpected.

In summary, in the two-mode entangled staterepresentation, the dissipation of the density operator for thePACS in the dissipative channel can be clearly seen. We haveshown that the corresponding WF is the Laguerre–Gaussianfunction. The criterion time for the definite-positivity of WF:κt > κtc ≡

12 ln 2, is revealed. The time evolution of the

photocount distribution is also derived analytically, whichclearly exhibits in the manner in which the distribution ofphoton counting depends on the photon-added number andthe intensity of the coherent field. The tomogram of thePACS in DC has also been calculated with the analyticalexpression. These results may be comparable with those ofthe experimental measurement of the quadrature-amplitudedistribution. For discussions on density matrices in thermalnoise, we refer the reader to [40].

Acknowledgments

We sincerely thank the referee for his useful suggestion.This work was supported by the National Natural ScienceFoundation of China under grant numbers 10775097 and10874174.

References

[1] Agarwal G S and Tara K 1991 Phys. Rev. A 43 492[2] Glauber R J 1963 Phys. Rev. 130 2529

Glauber R J 1963 Phys. Rev. 131 2766[3] Klauder J R and Skargerstam B S 1985 Coherent States

(Singapore: World Scientific)[4] Sivakumar S 2000 J. Opt. B: Quantum Semiclass. Opt.

2 R61[5] Sivakumar S 1999 J. Phys. A: Math. Gen. 32 2441[6] Dodonov V V, Marchiolli M A, Korennoy Y A, Man’ko V I

and Moukhin Y A 1998 Phys. Rev. A 58 4087[7] Zavatta A, Viciani S and Bellini M 2004 Science 306 660[8] Zavatta A, Viciani S and Bellini M 2005 Phys. Rev. A 72

023820[9] Kalamidas D, Gerry C C and Benmoussa A 2008 Phys. Lett.

A 372 1937[10] Kim M S 2008 J. Phys. B: At. Mol. Opt. Phys. 41 133001

Marek P, Jeong H and Kim M S 2008 Phys. Rev. A 78063811

[11] Fan H Y, Lu H L and Fan Y 2006 Ann. Phys. 321 480Fan H Y and Jiang N Q 2005 Phys. Scr. 71 277Fan H Y and Guo Q 2007 Phys. Scr. 76 533

[12] Wünsche A 1999 J. Opt. B: Quantum Semiclass. Opt. 1 R11[13] Wigner E P 1932 Phys. Rev. 40 749[14] O’Connell R F and Wigner E P 1981 Phys. Lett. A 83 145[15] Schleich W P 2001 Quantum Optics in Phase Space (Berlin:

Wiley)[16] Gardiner C and Zoller P 2000 Quantum Noise (Berlin:

Springer)[17] Takahashi Y and Umezawa H 1975 Collect. Phenom. 2 55[18] Umezawa H 1996 Int. J. Mod. Phys. B 10 1695[19] Umezawa H 1993 Advanced Field Theory—Micro, Macro, and

Thermal Physics (New York: AIP)[20] Fan H Y and Fan Y 1998 Phys. Lett. A 246 242[21] Fan H Y and Hu L Y 2008 Opt. Commun.

doi:10.1016/j.optcom.2008.08.002Fan H Y and Hu L Y 2008 Mod. Phys. Lett. B 22 2435

[22] Fan H Y and Hu L Y 2008 Opt. Commun.doi:10.1016/j.optcom.2008.11.029

7

Phys. Scr. 79 (2009) 035004 L-Y Hu and H-Y Fan

[23] Zhou N R, Hu L Y and Fan H Y 2008 Phys. Scr. 78 035003Hu L Y and Fan H Y 2008 J. Mod. Opt. 55 2429

[24] Fan H Y and Klauder J R 1994 Phys. Rev. A 49 704[25] Fan H Y and Fan Y 1996 Phys. Rev. A 54 958[26] Wünsche A 2001 J. Comput. Appl. Math. 133 665[27] Wünsche A 2000 J . Phys. A: Math. Gen. 33 1603[28] Fan H Y and Ruan T N 1984 Commun. Theor. Phys. 3 345[29] Fan H Y and Zaidi H R 1987 Phys. Lett. A 124 343[30] Kelley P L and Kleiner W H 1964 Phys. Rev. 136 316[31] Scully M O and Lamb Jr W E 1969 Phys. Rev. 179 368[32] Mollow B R 1968 Phys. Rev. 168 1896

[33] Orszag M 2000 Quantum Optics (Berlin: Springer)[34] Loudon R 1983 The Quantum Theory of Light 2nd edition

(Oxford: Oxford University Press)[35] Fan H Y and Hu L Y 2008 Opt. Lett. 33 443

Hu L Y and Fan H Y 2008 J. Opt. Soc. Am. B 25 1955[36] Vogel K and Risken H 1989 Phys. Rev. A 40 2847[37] Smithey D T et al 1993 Phys. Rev. Lett. 70 1244[38] Fan H Y and Chen H L 2001 Chin. Phys. Lett. 18 850[39] Hu L Y and Fan H Y 2008 J. Mod. Opt. 55 2011–24[40] Vourdas A 1989 Phys. Rev. A 39 206

Vourdas A 1988 Phys. Rev. A 37 3890

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