Statistical properties and decoherence of two-mode photon-subtracted squeezed vacuum
aa r X i v : . [ qu a n t - ph ] M a y Statistical properties and decoherence of two-mode photon-subtracted squeezed vacuum
Li-yun Hu , , Xue-xiang Xu , and Hong-yi Fan College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China and Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China
We investigate the statistical properties of the photon-subtractions from the two-mode squeezedvacuum state and its decoherence in a thermal environment. It is found that the state can be consideredas a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state isthe Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) isalso derived analytically by using the Weyl ordered operators’ invariance under similar transformations.Especially, the nonclassicality is discussed in terms of the negativity of WF. The effect of decoherenceon this state is then discussed by deriving the analytical time evolution results of WF. It is shown thatthe WF is always positive for any squeezing parameter and photon-subtraction number if the decaytime exceeds an upper bound ( κt > ln n +22¯ n +1 ) . Key Words: photon-subtraction; nonclassicality; Wigner function; negativity; two-mode squeezed vacuum statePACS numbers: 03.65.Yz, 42.50.Dv
I. INTRODUCTION
Entanglement is an important resource for quantum information [1]. In a quantum optics laboratory, Gaussianstates, being characteristic of Gaussian Wigner functions, have been generated but there is some limitation in usingthem for various tasks of quantum information procession [2]. For example, in the first demonstration of continuousvariables quantum teleportation (two-mode squeezed vacuum state as a quantum channel), the squeezing is low, thusthe entanglement of the quantum channel is such low that the average fidelity of quantum teleportation is just more (8 ±
2) % than the classical limits. In order to increase the quantum entanglement there have been suggestions andrealizations to engineering the quantum state by subtracting or adding photons from/to a Gaussian field which areplausible ways to conditionally manipulate nonclassical state of optical field [3, 4, 5, 6, 7, 8, 9]. In fact, such methodsallowed the preparation and analysis of several states with negative Wigner functions, including one- and two-photonFock states [10, 11, 12, 13], delocalized single photons [14, 15], and photon-subtracted squeezed vacuum states(PSSV), very similar to quantum superpositions of coherent states with small amplitudes (a Schr¨odinger kitten state[16, 17, 18, 19]) for single-mode case.Recently, the two-mode PSSVs (TPSSVs) have been paid enough attention by both experimentalists and theoreti-cians [10, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Olivares et al [20, 21] considered the photon subtractionusing on–off photodetectors and showed the improvement of quantum teleportation depending on various parame-ters involved. Then they further studied the nonlocality of photon subtraction state in the presence of noise [22].Kitagawa et al [23], on the other hand, investigated the degree of entanglement for the TPSSV by using an on-offphotondetector. Using operation with single photon counts, Ourjoumtsev et al .[10, 11] have demonstrated experi-mentally that the entanglement between Gaussian entangled states, can be increased by subtracting only one photonfrom two-mode squeezed vacuum states. The resulted state is a complex quantum state with a negative two-modeWigner function. However, so far as we know, there is no report about the nonclassicality and decoherence of TPSSVfor arbitrary number PSSV in literature before.In this paper, we will explore theoretically the statistical properties and decoherence of arbitrary number TPSSV.This paper is arranged as follows: in Sect. II we introduce the TPSSV, denoted as a m b n S ( λ ) | i , where S ( λ ) istwo-mode squeezing operator with λ being squeezing parameter and m, n are the subtracted photon number from S ( λ ) | i for mode a and b , respectively. It is found that it is just a squeezed two-variable Hermite polynomialexcitation on the vacuum state, and then the normalization factor for a m b n S ( λ ) | i is derived, which turns outto be a Jacobi polynomial, a remarkable result. In Sec. III, the quantum statistical properties of the TPSSV, suchas distribution of photon number, squeezing properties, cross-correlation function and antibunching, are calculatedanalytically and then be discussed in details. Especially, in Sec. IV, the explicit analytical expression of Wigner function(WF) of the TPSSV is derived by using the Weyl ordered operators’ invariance under similar transformations, whichis related to the two-variable Gaussian-Hermite polynomials, and then its nonclassicality is discussed in terms of thenegativity of WF which implies the highly nonclassical properties of quantum states. Sec. V is devoted to studying theeffect of the decoherence on the TPSSV in a thermal environment. The analytical expressions for the time-evolutionof the state and its WF are derived, and the loss of nonclassicality is discussed in reference of the negativity of WFdue to decoherence. We find that the WF for TPSSV has no chance to present negative value for all parameters λ and m, n if the decay time κt > ln n +22¯ n +1 (see Eq.(46) below), where ¯ n denotes the average thermal photon number inthe environment with dissipative coefficient κ . II. TWO-MODE PHOTON-SUBTRACTED SQUEEZED VACUUM STATESA. TPSSV as the squeezed two-variable Hermite polynomial excitation state
The definition of the two-mode squeezed vacuum state is given by S ( λ ) | i = sech λ exp( a † b † tanh λ ) | i , (1)where S ( λ ) = exp[ λ ( a † b † − ab )] is the two-mode squeezing operator [30, 31, 32] with λ being a real squeezingparameter, and a , b are the Bose annihilation operators, [ a, a † ] = [ b, b † ] = 1 . Theoretically, the TPSSV can be obtainedby repeatedly operating the photon annihilation operator a and b on S ( λ ) | i , defined as | λ, m, n i = a m b n S ( λ ) | i , (2)where | λ, m, n i is an un-normalization state. Noticing the transform relations, S † ( λ ) aS ( λ ) = a cosh λ + b † sinh λ,S † ( λ ) bS ( λ ) = b cosh λ + a † sinh λ, (3)we can reform Eq.(2) as | λ, m, n i = S ( λ ) S † ( λ ) a m b n S ( λ ) | i = S ( λ )( a cosh λ + b † sinh λ ) m ( b cosh λ + a † sinh λ ) n | i = S ( λ ) sinh n + m λ m X l =0 m ! coth l λl ! ( m − l )! b † m − l a l a † n | i . (4)Further note that a † n | i = √ n ! | n i and a l | n i = √ n ! √ ( n − l )! | n − l i = √ n !( n − l )! a † n − l | i , leading to a l a † n | i = n !( n − l )! a † n − l | i , thus Eq.(4) can be re-expressed as | λ, m, n i = S ( λ ) sinh n + m λ min( m,n ) X l =0 m ! n ! coth l λl ! ( m − l )!( n − l )! a † n − l b † m − l | i = sinh ( n + m ) / λ (cid:0) i √ (cid:1) n + m S ( λ ) min( m,n ) X l =0 ( − l n ! m ! (cid:16) i √ tanh λb † (cid:17) m − l (cid:16) i √ tanh λa † (cid:17) n − l l !( n − l )! ( m − l )! | i = sinh ( n + m ) / λ (cid:0) i √ (cid:1) n + m S ( λ ) H m,n (cid:16) i √ tanh λb † , i √ tanh λa † (cid:17) | i , (5)where in the last step we have used the definition of the two variables Hermitian polynomials [33, 34], i.e., H m,n ( ǫ, ε ) = min( m,n ) X k =0 ( − k m ! n ! ǫ m − k ε n − k k !( m − k )!( n − k )! . (6)From Eq.(5) one can see clearly that the TPSSV | λ, m, n i is equivalent to a two-mode squeezed two-variable Hermite-excited vacuum state and exhibits the exchanging symmetry, namely, interchanging m ⇔ n is equivalent to a † ⇔ b † .It is clear that, when m = n = 0 , Eq.(5) just reduces to the two-mode squeezed vacuum state due to H , = 1 ;while for n = 0 and m = 0 , noticing H ,n ( x, y ) = y n , Eq.(5) becomes ( N − λ, ,n = n ! sinh n λ ) , see Eq.(11) below) | λ, , n i = S ( λ ) | n, i , which is just a squeezed number state, corresponding to a pure negative binomial state [35]. B. The normalization of | λ, m, n i To know the normalization factor N λ,m,n of | λ, m, n i , let us first calculate the overlap h λ, m + s, n + t | λ, m, n i . Forthis purpose, using the first equation in Eq.(5) one can express | λ, m, n i as | λ, m, n i = S ( λ ) min( m,n ) X l =0 m ! n ! sinh n + m λ coth l λl ! p ( m − l )!( n − l )! | n − l, m − l i , (7)which leading to h λ, m + s, n + t | λ, m, n i = m ! ( n + s )! δ s,t sinh n +2 m +2 s λ × min( m,n ) X l =0 ( m + s )! n ! coth l + s λl !( m − l )!( n − l )!( l + s )! , (8)where δ s,t is the Kronecker delta function. Without lossing the generality, supposing m < n and comparing Eq.(8)with the standard expression of Jacobi polynomials [36] P ( α,β ) m ( x ) = (cid:18) x − (cid:19) m m X k =0 (cid:18) m + αk (cid:19) (cid:18) m + βm − k (cid:19) (cid:18) x + 1 x − (cid:19) k , (9)one can put Eq.(8) into the following form h λ, m + s, n + t | λ, m, n i = m ! ( n + s )! δ s,t sinh n + s λ cosh s λP ( n − m,s ) m (cosh 2 λ ) , (10)which is just related to Jacobi polynomials. In particular, when s = t = 0 , the normalization constant N m,n,λ for thestate | λ, m, n i is given by N λ,m,n = h λ, m, n | λ, m, n i = m ! n ! sinh n λP ( n − m, m (cosh 2 λ ) , (11)which is important for further studying analytically the statistical properties of the TPSSV. For the case m = n , itbecomes Legendre polynomial of the squeezing parameter λ , because of P (0 , n ( x ) = P n ( x ) , P ( x ) = 1 ; while for n = 0 and m = 0 , noticing that P ( n, ( x ) = 1 then N λ, ,n = n ! sinh n λ. Therefore, the normalized TPSSV is || λ, m, n i ≡ h m ! n ! sinh n λP ( n − m, m (cosh 2 λ ) i − / a m b n S ( λ ) | i . (12)From Eqs. (2) and (11) we can easily calculate the average photon number in TPSSV (denoting τ = cosh 2 λ ), (cid:10) a † a (cid:11) = N λ,m,n h | S † ( λ ) a † m +1 b † n a m +1 b n S ( λ ) | i = ( m + 1) P ( n − m − , m +1 ( τ ) P ( n − m, m ( τ ) , (13) (cid:10) b † b (cid:11) = ( n + 1) sinh λ P ( n − m +1 , m ( τ ) P ( n − m, m ( τ ) . (14)In a similar way we have (cid:10) a † b † ab (cid:11) = ( m + 1) ( n + 1) sinh λ P ( n − m, m +1 ( τ ) P ( n − m, m ( τ ) . (15)Thus the cross-correlation function g (2)12 can be obtained by [37, 38, 39] g (2)12 ( λ ) = (cid:10) a † b † ab (cid:11) h a † a i h b † b i = P ( n − m, m +1 ( τ ) P ( n − m − , m +1 ( τ ) P ( n − m, m ( τ ) P ( n − m +1 , m ( τ ) . (16) (cid:79) (cid:79) g (2)12 g m=n=1 m=n=3 m=n=10 (b) (2) FIG. 1: (Color online) Cross-correlation function between the two modes a and b as a function of λ for different parameters ( m, n ).The number 1,2,3,4,5,6 in (a) denote that ( m, n ) are eauql to (1,2), (3,4), (2,4),(6,8),(3,6) and (7,10) respectively. Actually, the cross-correlation between the two modes reflects correlation between photons in two different modes,which plays a key role in rendering many two-mode radiations nonclassically. In Fig.1, we plot the graph of g (2)12 ( λ ) asthe function of λ for some different ( m, n ) values. It is shown that g (2)12 ( λ ) are always larger than unit, thus there existcorrelations between the two modes. We emphasize that the WF has negative region for all λ, and thus the TPSSV isnonclassical. In our following work, we pay attention to the ideal TPSSV. III. QUANTUM STATISTICAL PROPERTIES OF THE TPSSVA.
Squeezing properties
For a two-mode system, the optical quadrature phase amplitudes can be expressed as follows: Q = Q + Q √ , P = P + P √ , [ Q, P ] = i , (17)where Q = ( a + a † ) / √ , P = ( a − a † ) / ( √ i ) , Q = ( b + b † ) / √ and P = ( b − b † ) / ( √ i ) are coordinate- andmomentum- operator, respectively. Their variances are (∆ Q ) = (cid:10) Q (cid:11) − h Q i and (∆ P ) = (cid:10) P (cid:11) − h P i . The phaseamplifications satisfy the uncertainty relation of quantum mechanics ∆ Q ∆ P ≥ / . By using Eqs.(10) and (11), it iseasy to see that h a i = (cid:10) a † (cid:11) = h b i = (cid:10) b † (cid:11) = 0 and (cid:10) a (cid:11) = (cid:10) a † (cid:11) = (cid:10) b (cid:11) = (cid:10) b † (cid:11) = 0 as well as (cid:10) ab † (cid:11) = (cid:10) a † b (cid:11) = 0 , whichleads to h Q i = 0 , h P i = 0 . Moreover, using Eq.(10) one can see (cid:10) a † b † (cid:11) = h ab i = n + 12 P ( n − m, m ( τ ) P ( n − m, m ( τ ) sinh 2 λ. (18)From Eqs.(13), (14) and (18) it then follows that (∆ Q ) = 12 ( (cid:10) a † a (cid:11) + (cid:10) b † b (cid:11) + h ab i + (cid:10) a † b † (cid:11) + 1)= 12 P ( n − m, m ( τ ) [( m + 1) P ( n − m − , m +1 ( τ ) + ( n + 1) P ( n − m +1 , m ( τ ) sinh λ + ( n + 1) P ( n − m, m ( τ ) sinh 2 λ + P ( n − m, m ( τ )] , (19)and (∆ P ) = 12 ( (cid:10) a † a (cid:11) + (cid:10) b † b (cid:11) − h ab i − (cid:10) a † b † (cid:11) + 1)= 12 P ( n − m, m ( τ ) [( m + 1) P ( n − m − , m +1 ( τ ) + ( n + 1) P ( n − m +1 , m ( τ ) sinh λ − ( n + 1) P ( n − m, m ( τ ) sinh 2 λ + P ( n − m, m ( τ )] . (20) (cid:311)(cid:312)(cid:312) (cid:311)(cid:312) (cid:311) ( ) P (cid:39) ( ) P (cid:39) (a) (b) (cid:79) (cid:79) FIG. 2: (Color online) The fluctuations variation of (∆ P ) with λ for several different parameters m, n values: (a) m = n =1 , , , from down to up; (b) (1) and (2) denote m = 0 and m = 1 , respectively, and n = 2 , , from down to up. Next, let us analyze some special cases. When m = n = 0 , corresponding to the two-mode squeezed state, Eqs. (19)and (20) becomes, respectively, to (∆ Q ) (cid:12)(cid:12) m = n =0 = 12 e λ , (∆ P ) (cid:12)(cid:12) m = n =0 = 12 e − λ , ∆ Q ∆ P = 12 , (21)which is just the standard squeezing case; while for m = 0 , n = 1 , Eqs. (19) and (20) reduce to (∆ Q ) (cid:12)(cid:12) m =0 ,n =1 = e λ , (∆ P ) (cid:12)(cid:12) m =0 ,n =1 = e − λ , ∆ Q ∆ P = 1 , (22)from which one can see that the state || λ, m, n i is squeezed at the “p-direction” when e − λ < , i.e., λ > ln 2 . Inaddition, when m = n = 1 , in a similar way, one can get (∆ Q ) (cid:12)(cid:12) m = n =1 = 12 e λ (cid:18) e λ − e λ + e − λ (cid:19) , (∆ P ) (cid:12)(cid:12) m = n =1 = 12 1 − e λ − e − λ (cid:0) − e − λ (cid:1) e λ + e − λ + 12 < , (23)which indicates that, for any squeezing parameter λ , there always exist squeezing effect for state | | λ, , i at the“p-direction”.In order to see clearly the fluctuations of (∆ P ) with other parameters m, n values, the figures are ploted in Fig.2.From Fig.2(a) one can see that the fluctuations of (∆ P ) are always less than when m = n, say, the state | | λ, m, m i is always squeezed at the “p-direction”; for given m values, there exist the squeezing effect only when the squeezingparameter exceeds a certain threshold value that increases with the increasement of n (see Fig.2(b)). B. Distribution of photon number
In order to obtain the photon number distribution of the TPSSV, we begin with evaluating the overlap betweentwo-mode number state h n a , n b | and | λ, m, n i . Using Eq.(1) and the un-normalized coherent state [37, 38], | z i =exp (cid:0) za † (cid:1) | i , leading to h n | = √ n ! ∂ n ∂z ∗ n h z | (cid:12)(cid:12) z ∗ =0 , it is easy to see that h n a , n b | λ, m, n i = sech λ h n a , n b | a m b n e a † b † tanh λ | i = ( m + n a )! √ n a ! n b ! sech λ tanh m + n a λδ m + n a ,n + n b . (24)It is easy to follow that the photon number distribution of | | λ, m, n i , i.e., P ( n a , n b ) = N − λ,m,n |h n a , n b | λ, m, n i| = (cid:2) ( m + n a )! sech λ tanh m + n a λδ m + n a ,n + n b (cid:3) n a ! n b ! m ! n ! sinh n λP ( n − m, m (cosh 2 λ ) . (25) n a n b n a n b P( ) , a b n n P( ) , a b n n (b)(a) n a n b n a n b P( ) , a b n n P( ) , a b n n (c) (d) FIG. 3: (Color online) Photon number distribution P ( n a , n b ) in the Fock space ( n a , n b ) for some given m = n values: (a) m = n = 0 , λ = 1 , (b) m = n = 1 , λ = 0 . , (c) m = n = 1 , λ = 1 , (d) m = 2 , n = 5 , λ = 1 . From Eq.(25) one can see that there exists a constrained condition, m + n a = n + n b , for the photon number distribution(see Fig. 3). In particular, when m = n = 0 , Eq.(25) becomes P ( n a , n b ) = (cid:26) sech λ tanh n a λ, n a = n b , n a = n b , (26)which is just the photon number distribution (PND) of two-mode squeezed vacuum state.In Fig. 3, we plot the distribution P ( n a , n b ) in the Fock space ( n a , n b ) for some given m, n values and squeezingparameter λ . From Fig. 3 it is found that the PND is constrained by m + n a = n + n b , resulting from the paired-present of photons in two-mode squeezed state. By subtracting photons, we have been able to move the peak fromzero photons to nonzero photons (see Fig.3 (a) and (c)). The position of peak depends on how many photons areannihilated and how much the state is squeezed initially. In addition, for example, the PND mainly shifts to the biggernumber states and becomes more “flat” and “wide” with the increasing parameter λ (see Fig.3 (b) and (c)). C. Antibunching effect of the TPSSV
Next we will discuss the antibunching for the TPSSV. The criterion for the existence of antibunching in two-moderadiation is given by [40] R ab ≡ (cid:10) a † a (cid:11) + (cid:10) b † b (cid:11) h a † ab † b i − < . (27)In a similar way to Eq.(13) we have (cid:10) a † a (cid:11) = ( m + 1) ( m + 2) P ( n − m − , m +2 ( τ ) P ( n − m, m ( τ ) , (28) (cid:16) (cid:16) (cid:16) (cid:16) (cid:16) (cid:16) (cid:16) (cid:311) (b) m = n =0 m = n =1 m = n =2 m = n =8 m = n =26 (cid:311) (cid:312)(cid:311)(cid:312)(cid:312) (a) (cid:79)(cid:79) FIG. 4: (Color online) The R ab as a function of λ and m, n . (1) and (2) in Fig.4 (a) denote m = 0 and m = 1 respectively, and n = 2 , , from down to up. and (cid:10) b † b (cid:11) = ( n + 1) ( n + 2) sinh λ P ( n − m +2 , m ( τ ) P ( n − m, m ( τ ) , (29)For the state | | λ, m, n i , substituting Eqs.(15), (28)and (29) into Eq.(27), we can recast R ab to R ab = ( m + 1) ( m + 2) P ( n − m − , m +2 ( τ ) + ( n + 1) ( n + 2) sinh λP ( n − m +2 , m ( τ )2 ( m + 1) ( n + 1) sinh λP ( n − m, m +1 ( τ ) − . (30)In particular, when m = n = 0 (corresponding to two-mode squeezed vacuum state), Eq.(30) reduces to R ab,m = n =0 = − sech 2 λ < , which indicates that there always exist antibunching effect for two-mode squeezed vacuum state. Inaddition, when m = n, the TPSSV is always antibunching. However, for any parameter values m, n ( m = n ) , the caseis not true. The R ab as a function of λ and m, n is plotted in Fig. 4. It is easy to see that, for a given m the TPSSVpresents the antibunching effect when the squeezing parameter λ exceeds to a certain threshold value. For instance,when m = 0 and n = 2 then R ab = − λ λ ) csch λ may be less than zero with λ > . about. IV. WIGNER FUNCTION OF THE TPSSV
The Wigner function (WF)[32, 41, 42] is a powerful tool to investigate the nonclassicality of optical fields. Its partialnegativity implies the highly nonclassical properties of quantum states and is often used to describe the decoherenceof quantum states, e.g., the excited coherent state in both photon-loss and thermal channels [43, 44], the single-photon subtracted squeezed vacuum (SPSSV) state in both amplitude decay and phase damping channels [6], and soon [11, 17, 45, 46, 47]. In this section, we derive the analytical expression of WF for the TPSSV. For this purpose, wefirst recall that the Weyl ordered form of single-mode Wigner operator [48, 49, 50], ∆ ( α ) = 12 :: δ ( α − a ) δ (cid:0) α ∗ − a † (cid:1) :: , (31)where α = ( q + ip ) / √ and the symbol :: :: denotes Weyl ordering. The merit of Weyl ordering lies in the Weylordered operators’ invariance under similar transformations proved in Ref.[48], which means S :: ( ◦ ◦ ◦ ) :: S − = :: S ( ◦ ◦ ◦ ) S − :: , (32)as if the “fence” :: :: did not exist, so S can pass through it.Following this invariance and Eq.(3) we have S † ( λ ) ∆ ( α ) ∆ ( β ) S ( λ )= 14 S † ( λ ) :: δ ( α − a ) δ (cid:0) α ∗ − a † (cid:1) δ ( β − b ) δ (cid:0) β ∗ − b † (cid:1) :: S ( λ )= 14 :: δ (cid:0) α − a cosh λ − b † sinh λ (cid:1) δ (cid:0) α ∗ − a † cosh λ − b sinh λ (cid:1) δ (cid:0) β − b cosh λ − a † sinh λ (cid:1) δ (cid:0) β ∗ − b † cosh λ − a sinh λ (cid:1) ::= 14 :: δ (¯ α − a ) δ (cid:0) ¯ α ∗ − a † (cid:1) δ (cid:0) ¯ β − b (cid:1) δ (cid:0) ¯ β ∗ − b † (cid:1) ::= ∆ ( ¯ α ) ∆ (cid:0) ¯ β (cid:1) , where ¯ α = α cosh λ − β ∗ sinh λ, ¯ β = β cosh λ − α ∗ sinh λ, and β = ( q + ip ) / √ . Thus employing the squeezed two-variable Hermite-excited vacuum state of the TPSSV in Eq.(5) and the coherent state representation of single-modeWigner operator [51], ∆ ( α ) = e | α | Z d z π | z i h− z | e − z α ∗ − αz ∗ ) , (33)where | z i = exp (cid:0) z a † − z ∗ a (cid:1) | i is Glauber coherent state [37, 38], we finally can obtain the explicit expression ofWF for the TPSSV (see Appendix A), W ( α, β ) = 1 π sinh n + m λ n + m N λ,m,n e − | ¯ α | − | ¯ β | m X l =0 n X k =0 × [ m ! n !] ( − tanh λ ) l + k l ! k ! [( m − l )! ( n − k )!] | H m − l,n − k ( B, A ) | , (34)where we have set A = − i ¯ α √ tanh λ, B = − i ¯ β √ tanh λ. Obviously, the WF W ( α, β ) in Eq.(34) is a real function andis non-Gaussian in phase space due to the presence of H m − l,n − k ( B, A ) , as expected.In particular, when m = n = 0 , Eq.(34) reduces to W ( α, β ) = π e − | ¯ α | − | ¯ β | = π e α ∗ β ∗ + αβ ) sinh 2 λ − αα ∗ + ββ ∗ ) cosh 2 λ corresponding to the WF of two-mode squeezed vacuum state; while forthe case of m = 0 and n = 0 , noticing H ,n ( x, y ) = y n and N λ, ,n = n ! sinh n λ, Eq.(34) becomes W ( α, β ) = ( − n π e − | ¯ α | − | ¯ β | L n (cid:16) | ¯ α | (cid:17) , (35)where L n is m -order Laguerre polynomials and Eq.(35) is just the WF of the negative binomial state S ( λ ) | n, i [35].In Figs. 5-7, the phase space Wigner distributions are depicted for several different parameter values m, n , and λ . Asan evidence of nonclassicality of the state, squeezing in one of quadratures is clear in the plots. In addition, thereare some negative region of the WF in the phase space which is another indicator of the nonclassicality of the state.For the case of m = 0 and n = 1 , it is easily seen from (35) that at the center of the phase space ( α = β = 0 ),the WF is always negative in phase space. Fig.5 shows that the negative region becomes more and visible as theincreasement of photon number subtracted m (= n ) , which may imply the nonclassicality of the state can be enhanceddue to the augment of photon-subtraction number. For a given value m and several different values n ( = m ), the WFdistributions are presented in Fig.7, from which it is interseting to notice that there are around | m − n | wave valleysand | m − n | + 1 wave peaks. V. DECOHERENCE OF TPSSV IN THERMAL ENVIRONMENTS
In this section, we next consider how this state evolves at the presence of thermal environment. (cid:11) (cid:12)
W p p (cid:11) (cid:12)
W p p (cid:11) (cid:12)
W p p (c)(b)(a) m=n=0 m=n=5m=n=1
FIG. 5: (Color online) The Wigner function W( α, β ) in phase space ( , , p , p ) for several different parameter values m = n with λ = 0 . . (a) m=n=0; (b) m=n=1 and (c) m=n=5. q qW (cid:168) (cid:184)(cid:168) (cid:184)(cid:169) (cid:185) p p (cid:167) (cid:183)(cid:16)(cid:16) (cid:11) (cid:12) , ,0,01 2 W q q (cid:11) (cid:12)
W p p (a) (c)(b) p pq qW (cid:167) (cid:183)(cid:16)(cid:16)(cid:168) (cid:184)(cid:168) (cid:184)(cid:169) (cid:185) (cid:11) (cid:12)
W p p (cid:11) (cid:12) , ,0,01 2
W q q (d) (f)(e)
FIG. 6: (Color online) The Wigner function W( α, β ) in three different phase spaces for m = 0 , n = 1 with λ = 0 . (first row) and λ = 0 . (second low). A. Model
When the TPSSV evolves in the thermal channel, the evolution of the density matrix can be described by thefollowing master equation in the interaction picture[52] ddt ρ ( t ) = ( L + L ) ρ ( t ) , (36)where L i ρ = κ (¯ n + 1) (cid:16) a i ρa † i − a † i a i ρ − ρa † i a i (cid:17) + κ ¯ n (cid:16) a † i ρa i − a i a † i ρ − ρa i a † i (cid:17) , ( a = a, a = b ) , (37)and κ represents the dissipative coefficient and ¯ n denotes the average thermal photon number of the environment.When ¯ n = 0 , Eq.(36) reduces to the master equation (ME) describing the photon-loss channel. The two thermalmodes are assumed to have the same average energy and coupled with the channel in the same strength and have thesame average thermal photon number ¯ n . This assumption is reasonable as the two-mode of squeezed state are in thesame frequency and temperature of the environment is normally the same [53, 54]. By introducing two entangled (0, 0, , )1 2 W p p (0, 0, , )1 2
W p p (0, 0, , )1 2
W p p (b) (c)(a)
FIG. 7: (Color online) The Wigner function W( α, β ) in phase space ( , , p , p ) for several parameter values m, n with λ = 0 . . (a) m1,=n=2; (b) m=1,n=3 and (c) m=1,n=5. ρ ( t ) = ∞ X i,j,r,s =0 M i,j,r,s ρ M † i,j,r,s , (38)where ρ denotes the density matrix at initial time, M i,j,r,s and M † i,j,r,s are Hermite conjugated operators (Krausoperator) with each other, M i,j,r,s = 1¯ nT + 1 s ( T ) r + s ( T ) i + j r ! s ! i ! j ! a † r b † s e ( a † a + b † b ) ln T a i b j , (39)and we have set T = 1 − e − κt , as well as T = ¯ nT ¯ nT + 1 , T = e − κt ¯ nT + 1 , T = (¯ n + 1) T ¯ nT + 1 . (40)It is not difficult to prove the M i,j,r,s obeys the normalization condition P ∞ i,j,r,s =0 M † i,j,r,s M i,j,r,s = 1 by using theIWOP technique. B. Evolution of Wigner function
By using the thermal field dynamics theory [55, 56] and thermal entangled state representation, the time evolutionof Wigner function at time t to be given by the convolution of the Wigner function at initial time and those of twosingle-mode thermal state (see Appendix C), i.e., W ( α, β, t ) = 4(2¯ n + 1) T Z d ζd ηπ W ( ζ, η, e − | α − ζe − κt | | β − ηe − κt | n +1) T . (41)Eq.(41) is just the evolution formula of Wigner function of two-mode quantum state in thermal channel. Thus the WFat any time can be obtained by performing the integration when the initial WF is known.In a similar way to deriving Eq.(34), substituting Eq.(34) into Eq.(41) and using the generating function of two-variable Hermite polynomials (A2), we finally obtain W ( α, β, t ) = N − λ,m,n ( E sinh 2 λ ) m + n π n + m (2¯ n + 1) T D e − | α − β ∗ | e − λ − κt +(2¯ n +1) T − | α + β ∗ | e λ − κt +(2¯ n +1) T × n X l =0 m X k =0 [ m ! n !] (cid:0) − FE tanh λ (cid:1) l + k l ! k ! [( m − k )! ( n − l )!] (cid:12)(cid:12)(cid:12) H m − k,n − l (cid:16) G/ √ E, K/ √ E (cid:17)(cid:12)(cid:12)(cid:12) , (42)where we have set C = e − κt (2¯ n + 1) T , D = (cid:0) Ce − λ (cid:1) (cid:0) Ce λ (cid:1) ,E = e κt D (2¯ nT + 1) C , F = C − D ,G = Ce κt D (cid:0) ¯ B + B ∗ C (cid:1) , ¯ B = i √ tanh λ ( β ∗ cosh λ + α sinh λ ) ,K = Ce κt D (cid:0) ¯ A + A ∗ C (cid:1) , ¯ A = i √ tanh λ ( α ∗ cosh λ + β sinh λ ) . (43)Eq.(42) is just the analytical expression of WF for the TPSSV in thermal channel. It is obvious that the WF loss itsGaussian property due to the presence of two-variable Hermite polynomials.1In particular, at the initial time ( t = 0 ) , noting E → , (2¯ n + 1) T D → , FE → and C D → , CD → as well as K → A ∗ , G → B ∗ , Eq.(42) just dose reduce to Eq.(34), i.e., the WF of the TPSSV. On the other hand, when κt → ∞ , noticing that C → , D → , E → , F → − , and G/ √ E → , K/ √ E → , as well as H m,n (0 ,
0) = ( − m m ! δ m,n , aswell as the definition of Jacobi polynomials in Eq.(9), then Eq.(42) becomes W ( α, β, ∞ ) = 1 π (2¯ n + 1) e − n +1 ( | α | + | β | ) , (44)which is independent of photon-subtraction number m and n and corresponds to the product of two thermal stateswith mean thermal photon number ¯ n . This implies that the two-mode system reduces to two-mode thermal state aftera long time interaction with the environment. Eq.(44) denotes a Gaussian distribution. Thus the thermal noise causesthe absence of the partial negative of the WF if the decay time κt exceeds a threshold value. In addition, for the caseof m = n = 0 , corresponding to the case of two-mode squeezed vacuum, Eq.(42) just becomes W m = n =0 ( α, β, t ) = N − e − ED ( | α | + | β | ) + FD ( αβ + α ∗ β ∗ ) , (45)where N = π (2¯ n + 1) T D is the normalization factor, D = (2¯ n + 1) T D, E = 2 (2¯ n + 1) T + e − κt cosh 2 λ, and F = 2 e − κt sinh 2 λ . Eq.(45) is just the result in Eq.(14) of Ref. [54].In Fig.8, the WFs of the TPSSV for ( m = 0 , n = 1 ) are depicted in phase space with λ = 0 . and ¯ n = 1 for severaldifferent κt. It is easy to see that the negative region of WF gradually disappears as the time κt increases. Actually,from Eq.(43) one can see that D > and E > , so when F < leading to the following condition: κt > κt c ≡
12 ln 2¯ n + 22¯ n + 1 , (46)we know that the WF of TPSSV has no chance to be negative in the whole phase space when κt exceeds a thresholdvalue κt c . Here we should point out that the effective threshold value of the decay time corresponding to the transitionof the WF from partial negative to fully positive definite is dependent of m and n. When κt = κt c , it then follows fromEq.(42) that W ( α, β, t c ) = tanh m + n λ sech λ π N m,n,λ e − κt c e − e κtc [ | α | + | β | − ( α ∗ β ∗ + αβ ) tanh λ ] × (cid:12)(cid:12)(cid:12) H m,n (cid:16) i √ tanh λβ ∗ e κt c , i √ tanh λα ∗ e κt c (cid:17)(cid:12)(cid:12)(cid:12) , (47)which is an Hermite-Gaussian function and positive definite, as expected.In Figs. 9 and 10, we have presented the time-evolution of WF in phase space for different ¯ n and λ, respectively.One can see clearly that the partial negativity of WF decreases gradually as ¯ n (or λ ) increases for a given time. Thiscase is true for a given ¯ n (or κt ) as the increasement of κt (or ¯ n ). The squeezing effect in one of quadratures is shownin Fig.10. In principle, by using the explicit expression of WF in Eq.(42), we can draw its distributions in phase space.For the case of m = 0 , n = 2 , there are two negative regions of WF, which is different from the case of m = 0 , n = 1 (see Fig.11). The absolute value of the negative minimum of the WF decreases as κt increases, which leads to the fullabsence of partial negative region. VI. CONCLUSIONS
In summary, we have investigated the statistical properties of two-mode photon-subtracted squeezed vacuum state(TPSSV) and its decoherence in thermal channelwith average thermal photon number ¯ n and dissipative coefficient κ . For arbitrary number TPSSV, we have for the first time calculated the normalization factor, which turns out tobe a Jacobi polynomial of the squeezing parameter λ , a remarkable result. We also show that the TPSSV can betreated as a squeezed two-variable Hermite polynomial excitation vacuum. Based on Jacobi polynomials’ behaviorthe statistical properties of the field, such as photon number distribution, squeezing properties, cross-correlationfunction and antibunching, are also derived analytically. Especially, the nonclassicality of TPSSV is discussed in termsof the negativity of WF after deriving the explicit expression of WF. Then the decoherence of TPSSV in thermalchannel is also demonstrated according to the compact expression for the WF. The threshold value of the decay timecorresponding to the transition of the WF from partial negative to completely positive is presented. It is found thatthe WF has no chance to present negative value for all parameters λ and any photon-subtraction number ( m, n ) if κt > ln n +22¯ n +1 for TPSSV. The technique of integration within an ordered product of operators brings convenience inour derivation.2 (cid:11) (cid:12) , ,0,01 2 W q q (cid:11) (cid:12) , ,0,01 2
W q q (a) (b) (cid:11) (cid:12) , ,0,01 2
W q q (cid:11) (cid:12) , ,0,01 2
W q q (d)(c)
FIG. 8: (Color online) The time evolution of WF ( m = 0 , n = 1) at ( q , q , , phase space for ¯ n = 1 , λ = 0 . . (a) κt = 0 . , (b) κt = 0 . , (c) κt = 0 . , (d) κt = 0 . . (cid:11) (cid:12) , ,0,01 2 W q q (cid:11) (cid:12) , ,0,01 2
W q q (b)(a) (cid:11) (cid:12) , ,0,01 2
W q q (cid:11) (cid:12) , ,0,01 2
W q q (d)(c)
FIG. 9: (Color online) The time evolution of WF ( m = 0 , n = 1) in ( q , q , , phase space for λ = 0 . and κt = 0 . with (a) ¯ n = 0 , (b) ¯ n = 1 , (c) ¯ n = 2 , (d) ¯ n = 7 . (cid:11) (cid:12) , ,0,01 2 W q q (cid:11) (cid:12) , ,0,01 2
W q q (a) (b) (cid:11) (cid:12) , ,0,01 2
W q q (cid:11) (cid:12) , ,0,01 2
W q q (d)(c)
FIG. 10: (Color online) The time evolution of WF ( m = 0 , n = 1) in ( q , q , , phase space for ¯ n = 1 , and κt = 0 . with (a) λ = 0 . , (b) λ = 0 . , (c) λ = 0 . , (d) λ = 1 . . (cid:11) (cid:12) , ,0,01 2 W q q (cid:11) (cid:12) , ,0,01 2
W q q (a) (b) (cid:11) (cid:12) , ,0,01 2
W q q (cid:11) (cid:12) , ,0,01 2
W q q (d)(c)
FIG. 11: (Color online) The time evolution of WF for m = 0 , n = 2 in ( q , q , , phase space with (a) κt = 0 , (b) κt = 0 . , (c) κt = 0 . , (d) κt = 0 . . Acknowledgments
Work supported by the the National Natural Science Foundation of China under GrantNos.10775097 and 10874174.
Appendix A: Deriviation of Wigner function Eq.(34) of TPSSV
The definite of the WF of two-mode quantum state | Ψ i is given by W ( α, β ) = h Ψ | ∆ ( α ) ∆ ( β ) | Ψ i , thus by uisngEqs.(5), and (33) the WF of TPSSV can be calculated as W ( α, β ) = h λ, m, n | | ∆ ( α ) ∆ ( β ) | | λ, m, n i = sinh n + m λ n + m N λ,m,n h | H m,n (cid:16) − i √ tanh λb, − i √ tanh λa (cid:17) ∆ ( ¯ α ) ⊗ ∆ (cid:0) ¯ β (cid:1) H m,n (cid:16) i √ tanh λb † , i √ tanh λa † (cid:17) | i = sinh n + m λ n + m N λ,m,n e | ¯ α | +2 | ¯ β | Z d z d z π e −| z | −| z | − z ¯ α ∗ − ¯ αz ∗ ) − z ¯ β ∗ − ¯ βz ∗ ) × H m,n (cid:16) − i √ tanh λz , − i √ tanh λz (cid:17) H m,n (cid:16) − i √ tanh λz ∗ , − i √ tanh λz ∗ (cid:17) . (A1)Further noticing the generating function of two variables Hermitian polynomials, H m,n ( ǫ, ε ) = ∂ m + n ∂t m ∂t ′ n exp [ − tt ′ + ǫt + εt ′ ] | t = t ′ =0 , (A2)Eq.(A1) can be further rewritten as W ( α, β ) = sinh n + m λ n + m N λ,m,n e | ¯ α | +2 | ¯ β | ∂ m + n ∂t m ∂τ n ∂ m + n ∂t ′ m ∂τ ′ n e − tτ − t ′ τ ′ × Z d z π e −| z | + ( − α ∗ − i √ tanh λτ ) z + ( α − i √ tanh λτ ′ ) z ∗ (cid:12)(cid:12)(cid:12) t = τ =0 × Z d z π e −| z | + ( − β ∗ − i √ tanh λt ) z + ( β − i √ tanh λt ′ ) z ∗ (cid:12)(cid:12)(cid:12) t ′ = τ ′ =0 = sinh n + m λ n + m N λ,m,n e − | ¯ α | − | ¯ β | ∂ m + n ∂t m ∂τ n ∂ m + n ∂t ′ m ∂τ ′ n × e − tτ − t ′ τ ′ + A ∗ τ ′ + B ∗ t ′ + Aτ + Bt − ( tt ′ + ττ ′ ) tanh λ (cid:12)(cid:12)(cid:12) t = τ = t ′ = τ ′ =0 , (A3)where we have set B = − i ¯ β √ tanh λ, A = − i ¯ α √ tanh λ, (A4)4and have used the following integration formula Z d zπ e ζ | z | + ξz + ηz ∗ = − ζ e − ξηζ , Re ( ζ ) < . (A5)Expanding the exponential term exp [ − ( tt ′ + τ τ ′ ) tanh λ ] , and using Eq.(A2), we have W ( α, β ) = sinh n + m λ n + m N λ,m,n e − | ¯ α | − | ¯ β | ∞ X l =0 ∞ X k =0 ( − tanh λ ) l + k l ! k ! × ∂ l + k ∂B l ∂A k ∂ l + k ∂B ∗ l ∂A ∗ k ∂ m ∂t m ∂τ n × ∂ n ∂t ′ m ∂τ ′ n e − tτ + Aτ + Bt − t ′ τ ′ + A ∗ τ ′ + B ∗ t ′ (cid:12)(cid:12)(cid:12)(cid:12) t = τ = t ′ = τ ′ =0 = sinh n + m λ n + m N λ,m,n e − | ¯ α | − | ¯ β | ∞ X l =0 ∞ X k =0 ( − tanh λ ) l + k l ! k ! × ∂ l + k ∂B l ∂A k ∂ l + k ∂B ∗ l ∂A ∗ k H m,n ( B, A ) H m,n ( B ∗ , A ∗ ) . (A6)Noticing the well-known differential relations of H m,n ( ǫ, ε ) ,∂ l + k ∂ǫ l ∂ε k H m,n ( ǫ, ε ) = m ! n ! H m − l,n − k ( ǫ, ε )( m − l )! ( n − k )! , (A7)we can further recast Eq.(A6) to Eq.(34). Appendix B: Derivation of solution of Eq.(36)
To solve the ME in Eq.(36), we first introduce two entangled state representations [57]: | η a i = exp (cid:20) − | η a | + η a a † − η ∗ a ˜ a † + a † ˜ a † (cid:21) (cid:12)(cid:12) (cid:11) , (B1) | η b i = exp (cid:20) − | η b | + η b b † − η ∗ b ˜ b † + b † ˜ b † (cid:21) (cid:12)(cid:12) (cid:11) , (B2)which satisfy the following eigenvector equations, for instance, ( a − ˜ a † ) | η a i = η a | η a i , ( a † − ˜ a ) | η a i = η ∗ a | η a i , h η a | ( a † − ˜ a ) = η ∗ a h η a | , h η a | ( a − ˜ a † ) = η a h η a | . (B3)which imply operators ( a − ˜ a † ) and ( a † − ˜ a ) can be replaced by number η a and η ∗ a , (cid:2) ( a − ˜ a † ) , ( a † − ˜ a ) (cid:3) = 0 . Operatingtwo-side of Eq.(36) on the vector | I a , I b i ≡ | η a = 0 i ⊗ | η b = 0 i , (denote | ρ ( t ) i ≡ ρ ( t ) | I a , I b i ) , and noticing thecorresponding relation: a | I a , I b i = ˜ a † | I a , I b i , a † | I a , I b i = ˜ a | I a , I b i ,b | I a , I b i = ˜ b † | I a , I b i , b † | I a , I b i = ˜ b | I a , I b i , (B4)we can put Eq.(36) into the following form: ddt | ρ ( t ) i = (cid:2) κ (¯ n + 1) (cid:0) a ˜ a − a † a − ˜ a † ˜ a (cid:1) + κ ¯ n (cid:0) a † ˜ a † − aa † − ˜ a ˜ a † (cid:1) + κ (¯ n + 1) (cid:16) b ˜ b − b † b − ˜ b † ˜ b (cid:17) + κ ¯ n (cid:16) b † ˜ b † − bb † − ˜ b ˜ b † (cid:17)i | ρ ( t ) i . (B5)It’s formal solution is given by | ρ ( t ) i = exp (cid:2) κt (¯ n + 1) (cid:0) a ˜ a − a † a − ˜ a † ˜ a (cid:1) + κt ¯ n (cid:0) a † ˜ a † − aa † − ˜ a ˜ a † (cid:1) + κt (¯ n + 1) (cid:16) b ˜ b − b † b − ˜ b † ˜ b (cid:17) + κt ¯ n (cid:16) b † ˜ b † − bb † − ˜ b ˜ b † (cid:17)i | ρ i , (B6)5where | ρ i ≡ ρ | I a , I b i . In order to solve Eq.(B6), noticing that, for example, a ˜ a − a † a − ˜ a † ˜ a = − (cid:0) a † − ˜ a (cid:1) (cid:0) a − ˜ a † (cid:1) + ˜ aa − ˜ a † a † , (B7)we have | ρ ( t ) i = exp (cid:2)(cid:0) a ˜ a − ˜ a † a † + 1 (cid:1) κt (cid:3) × exp (cid:20) n + 12 (cid:0) − e κt (cid:1) (cid:0) a † − ˜ a (cid:1) (cid:0) a − ˜ a † (cid:1)(cid:21) × exp h(cid:16) b ˜ b − ˜ b † b † + 1 (cid:17) κt i × exp (cid:20) n + 12 (cid:0) − e κt (cid:1) (cid:16) b † − ˜ b (cid:17) (cid:16) b − ˜ b † (cid:17)(cid:21) | ρ i , (B8)where we have used the identity operator, exp[ λ ( A + σB )] = e λA exp[ σB (1 − e − λτ ) /τ ] valid for [ A, B ] = τ B.
Thus the element of ρ ( t ) between h η a , η b | and | I a , I b i is h η a , η b | ρ ( t ) i = exp (cid:20) − n + 12 T | (cid:0) η a | + | η b | (cid:1)(cid:21) (cid:10) η a e − κt , η b e − κt (cid:12)(cid:12) ρ i , (B9)from which one can see clearly the attenuation due to the presence of environment.Further, using the completeness relation of | η a , η b i , R d η a d η b π | η a , η b i h η a , η b | = 1 and the IWOP technique [58, 59],we see | ρ ( t ) i = Z d η a d η b π | η a , η b i h η a , η b | ρ ( t ) i = 1(¯ nT + 1) exp h T (cid:16) a † ˜ a † + b † ˜ b † (cid:17)i × exp h(cid:16) a † a + b † b + ˜ a † ˜ a + ˜ b † ˜ b (cid:17) ln T i × exp h T (cid:16) a ˜ a + b ˜ b (cid:17)i ρ | I a , I b i , (B10)where T , T and T are defined in Eq.(40). Noticing Eq.(B4), we can reform Eq.(B10) as ρ ( t ) = P ∞ i,j,r,s =0 M i,j,r,s ρ M † i,j,r,s , where M i,j,r,s and M † i,j,r,s are defined in Eq.(39). Appendix C: Deriviation of Eq.(41) by using thermo field dynamics and entangled state representation
In this appendix, we shall derive the evolution formula of WF, i.e., the relation between the any time WF and theinitial time WF. According to the definition of WF of density operator ρ : W ( α ) = Tr [∆ ( α ) ρ ] , where ∆ ( α ) is thesingle-mode Wigner operator, ∆ ( α ) = π D (2 α ) ( − a † a . By using h ˜ n | ˜ m i = δ m,n we can reform W ( α ) as [60] W ( α ) = ∞ X m,n h n, ˜ n | ∆ ( α ) ρ | m, ˜ m i = 1 π h ξ =2 α | ρ i , (C1)where h ξ | is the conjugate state of h η | , whose overlap is h η | ξ i = exp (cid:2) ( ξη ∗ − ξ ∗ η ) (cid:3) , a Fourier transformation kernel.In a similar way, thus for two-mode quantum system, the WF is given by W ( α, β ) = Tr [∆ a ( α ) ∆ b ( β ) ρ ] = 1 π h ξ a =2 α , ξ b =2 β | ρ i . (C2)Employing the above overlap relation, Eq.(C2) can be recast into the following form: W ( α, β, t ) = Z d η a d η b π e α ∗ η a − αη ∗ a + β ∗ η b − βη ∗ b h η a , η b | ρ ( t ) i . (C3)Then substituting Eq.(B9) into Eq.(C3) and using the completeness of h ξ | , R d ξπ | ξ i h ξ | = 1 , we have W ( α, β, t ) = Z d η a d η b π e − n +12 T | ( η a | + | η b | ) × e α ∗ η a − αη ∗ a + β ∗ η b − βη ∗ b (cid:10) η a e − κt , η b e − κt (cid:12)(cid:12) ρ i = Z d ξ a d ξ b π W ( ζ, η, Z d η a d η b π e − n +12 T | ( η a | + | η b | ) × e α ∗ η a − αη ∗ a + β ∗ η b − βη ∗ b (cid:10) η a e − κt , η b e − κt (cid:12)(cid:12) ξ a =2 ζ , ξ b =2 η i . (C4)6Performing the integration in Eq.(C4) over d η a d η b then we can obtain Eq.(41).Making variables replacement, α − ζe − κt √ T → ζ, β − ηe − κt √ T → η, Eq.(41) can be reformed as W ( α, β, t ) = 4 e κt Z d ζd ηW tha ( ζ ) W thb ( η ) × W n e κt (cid:16) α − √ T ζ (cid:17) , e κt (cid:16) β − √ T η (cid:17) , o , (C5)where W th ( ζ ) is the Wigner function of thermal state with average thermal photon number ¯ n : W th ( ζ ) = π (2¯ n +1) e − | ζ | n +1 . Eq.(C5) is another expression of the evolution of WF and is actually agreement with that inRefs.[53, 54]. [1] D. Bouwmeester, A. Ekert and A. Zeilinger,
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