Statistical properties and decoherence of two-mode photon-subtracted squeezed vacuumWork supported by the the National Natural Science Foundation of China under Grant Nos.10775097 and 10874174.

# Statistical properties and decoherence of two-mode photon-subtracted squeezed vacuum††thanks: Work supported by the the National Natural Science Foundation of China under Grant Nos.10775097 and 10874174.

Li-yun Hu, Xue-xiang Xu and Hong-yi Fan Corresponding author. Email: hlyun@sjtu.edu.cn; hlyun2008@126.com. College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China
###### Abstract

We investigate the statistical properties of the photon-subtractions from the two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state is the Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) is also 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 decoherence on this state is then discussed by deriving the analytical time evolution results of WF. It is shown that the WF is always positive for any squeezing parameter and photon-subtraction number if the decay time exceeds an upper bound (

Key Words: photon-subtraction; nonclassicality; Wigner function; negativity; two-mode squeezed vacuum state

PACS numbers: 03.65.Yz, 42.50.Dv

## I Introduction

Entanglement is an important resource for quantum information 1 (). In a quantum optics laboratory, Gaussian states, being characteristic of Gaussian Wigner functions, have been generated but there is some limitation in using them for various tasks of quantum information procession 2 (). For example, in the first demonstration of continuous variables quantum teleportation (two-mode squeezed vacuum state as a quantum channel), the squeezing is low, thus the entanglement of the quantum channel is such low that the average fidelity of quantum teleportation is just more than the classical limits. In order to increase the quantum entanglement there have been suggestions and realizations to engineering the quantum state by subtracting or adding photons from/to a Gaussian field which are plausible ways to conditionally manipulate nonclassical state of optical field 2a (); 2b (); 2c (); 2d (); 2e (); 2f (); 2g (). In fact, such methods allowed the preparation and analysis of several states with negative Wigner functions, including one- and two-photon Fock states 3 (); 4 (); 5 (); 6 (), delocalized single photons 7 (); 8 (), and photon-subtracted squeezed vacuum states (PSSV), very similar to quantum superpositions of coherent states with small amplitudes (a Schrödinger kitten state 9 (); 10 (); 11 (); 12 ()) for single-mode case.

Recently, the two-mode PSSVs (TPSSVs) have been paid enough attention by both experimentalists and theoreticians 3 (); 4 (); 13 (); 14 (); 15 (); 16 (); 17 (); 18 (); 19 (); 20 (); 21 (); 22 (). Olivares et al 13 (); 14 () considered the photon subtraction using on–off photodetectors and showed the improvement of quantum teleportation depending on various parameters involved. Then they further studied the nonlocality of photon subtraction state in the presence of noise 15 (). Kitagawa et al 16 (), on the other hand, investigated the degree of entanglement for the TPSSV by using an on-off photondetector. Using operation with single photon counts, Ourjoumtsev et al .3 (); 4 () have demonstrated experimentally that the entanglement between Gaussian entangled states, can be increased by subtracting only one photon from two-mode squeezed vacuum states. The resulted state is a complex quantum state with a negative two-mode Wigner function. However, so far as we know, there is no report about the nonclassicality and decoherence of TPSSV for 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 where is two-mode squeezing operator with being squeezing parameter and are the subtracted photon number from for mode and , respectively. It is found that it is just a squeezed two-variable Hermite polynomial excitation on the vacuum state, and then the normalization factor for is derived, which turns out to be a Jacobi polynomial, a remarkable result. In Sec. III, the quantum statistical properties of the TPSSV, such as distribution of photon number, squeezing properties, cross-correlation function and antibunching, are calculated analytically 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, which is related to the two-variable Gaussian-Hermite polynomials, and then its nonclassicality is discussed in terms of the negativity of WF which implies the highly nonclassical properties of quantum states. Sec. V is devoted to studying the effect of the decoherence on the TPSSV in a thermal environment. The analytical expressions for the time-evolution of the state and its WF are derived, and the loss of nonclassicality is discussed in reference of the negativity of WF due to decoherence. We find that the WF for TPSSV has no chance to present negative value for all parameters and if the decay time (see Eq.(46) below), where denotes the average thermal photon number in the environment with dissipative coefficient .

## Ii Two-mode photon-subtracted squeezed vacuum states

### ii.1 TPSSV as the squeezed two-variable Hermite polynomial excitation state

The definition of the two-mode squeezed vacuum state is given by

 S2(λ)|00⟩=sechλexp(a†b†tanhλ)|00⟩, (1)

where is the two-mode squeezing operator 23 (); 24 (); 25 () with being a real squeezing parameter, and , are the Bose annihilation operators, . Theoretically, the TPSSV can be obtained by repeatedly operating the photon annihilation operator and on , defined as

 |λ,m,n⟩=ambnS2(λ)|00⟩, (2)

where is an un-normalization state. Noticing the transform relations,

 S†2(λ)aS2(λ) =acoshλ+b†sinhλ, S†2(λ)bS2(λ) =bcoshλ+a†sinhλ, (3)

we can reform Eq.(2) as

 |λ,m,n⟩ =S2(λ)S†2(λ)ambnS2(λ)|00⟩ =S2(λ)(acoshλ+b†sinhλ)m(bcoshλ+a†sinhλ)n|00⟩ =S2(λ)sinhn+mλm∑l=0m!cothlλl!(m−l)!b†m−lala†n|00⟩. (4)

Further note that and , leading to , thus Eq.(4) can be re-expressed as

 |λ,m,n⟩ =S2(λ)sinhn+mλmin(m,n)∑l=0m!n!cothlλl!(m−l)!(n−l)!a†n−lb†m−l|00⟩ =sinh(n+m)/22λ(i√2)n+mS2(λ)min(m,n)∑l=0(−1)ln!m!(i√tanhλb†)m−l(i√tanhλa†)n−ll!(n−l)!(m−l)!|00⟩ =sinh(n+m)/22λ(i√2)n+mS2(λ)Hm,n(i√tanhλb†,i√tanhλa†)|00⟩, (5)

where in the last step we have used the definition of the two variables Hermitian polynomials 26 (); 27 (), i.e.,

 Hm,n(ϵ,ε)=min(m,n)∑k=0(−1)km!n!ϵm−kεn−kk!(m−k)!(n−k)!. (6)

From Eq.(5) one can see clearly that the TPSSV is equivalent to a two-mode squeezed two-variable Hermite-excited vacuum state and exhibits the exchanging symmetry, namely, interchanging is equivalent to . It is clear that, when Eq.(5) just reduces to the two-mode squeezed vacuum state due to ; while for and noticing Eq.(5) becomes (, see Eq.(11) below) which is just a squeezed number state, corresponding to a pure negative binomial state 28 ().

### ii.2 The normalization of |λ,m,n⟩

To know the normalization factor of , let us first calculate the overlap . For this purpose, using the first equation in Eq.(5) one can express as

 |λ,m,n⟩=S2(λ)min(m,n)∑l=0m!n!sinhn+mλcothlλl!√(m−l)!(n−l)!|n−l,m−l⟩, (7)

 ⟨λ,m+s,n+t|λ,m,n⟩ =m!(n+s)!δs,tsinh2n+2m+2sλ ×min(m,n)∑l=0(m+s)!n!coth2l+sλl!(m−l)!(n−l)!(l+s)!, (8)

where is the Kronecker delta function. Without lossing the generality, supposing and comparing Eq.(8) with the standard expression of Jacobi polynomials 29 ()

 P(α,β)m(x)=(x−12)mm∑k=0(m+αk)(m+βm−k)(x+1x−1)k, (9)

one can put Eq.(8) into the following form

 ⟨λ,m+s,n+t|λ,m,n⟩=m!(n+s)!δs,tsinh2n+sλcoshsλP(n−m,s)m(\allowbreakcosh2λ), (10)

which is just related to Jacobi polynomials. In particular, when , the normalization constant for the state is given by

 Nλ,m,n=⟨λ,m,n|λ,m,n⟩=m!n!sinh2nλP(n−m,0)m(cosh2λ), (11)

which is important for further studying analytically the statistical properties of the TPSSV. For the case , it becomes Legendre polynomial of the squeezing parameter , because of ; while for and noticing that then Therefore, the normalized TPSSV is

 ||λ,m,n⟩≡[m!n!sinh2nλP(n−m,0)m(cosh2λ)]−1/2ambnS2(λ)|00⟩. (12)

From Eqs. (2) and (11) we can easily calculate the average photon number in TPSSV (denoting ),

 ⟨a†a⟩ =N2λ,m,n⟨00|S†2(λ)a†m+1b†nam+1bnS2(λ)|00⟩ =(m+1)P(n−m−1,0)m+1(τ)P(n−m,0)m(τ), (13) ⟨b†b⟩ =(n+1)sinh2λP(n−m+1,0)m(τ)P(n−m,0)m(τ). (14)

In a similar way we have

 ⟨a†b†ab⟩=(m+1)(n+1)sinh2λP(n−m,0)m+1(τ)P(n−m,0)m(τ). (15)

Thus the cross-correlation function can be obtained by 30 (); 31 (); 32 ()

 g(2)12(λ) =P(n−m,0)m+1(τ)P(n−m−1,0)m+1(τ)P(n−m,0)m(τ)P(n−m+1,0)m(τ). (16)

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 as the function of for some different () values. It is shown that are always larger than unit, thus there exist correlations between the two modes. We emphasize that the WF has negative region for all and thus the TPSSV is nonclassical. In our following work, we pay attention to the ideal TPSSV.

## Iii Quantum statistical properties of the TPSSV

### iii.1 Squeezing properties

For a two-mode system, the optical quadrature phase amplitudes can be expressed as follows:

 Q=Q1+Q2√2, P=P1+P2√2, [Q,P]=i, (17)

where , , and are coordinate- and momentum- operator, respectively. Their variances are and . The phase amplifications satisfy the uncertainty relation of quantum mechanics . By using Eqs.(10) and (11), it is easy to see that and as well as which leads to , . Moreover, using Eq.(10) one can see

 ⟨a†b†⟩=⟨ab⟩=n+12P(n−m,1)m(τ)P(n−m,0)m(τ)sinh2λ. (18)

From Eqs.(13), (14) and (18) it then follows that

 (ΔQ)2 =12(⟨a†a⟩+⟨b†b⟩+⟨ab⟩+⟨a†b†⟩+1) =12P(n−m,0)m(τ)[(m+1)P(n−m−1,0)m+1(τ)+(n+1)P(n−m+1,0)m(τ)sinh2λ +(n+1)P(n−m,1)m(τ)sinh2λ+P(n−m,0)m(τ)], (19)

and

 (ΔP)2 =12(⟨a†a⟩+⟨b†b⟩−⟨ab⟩−⟨a†b†⟩+1) =12P(n−m,0)m(τ)[(m+1)P(n−m−1,0)m+1(τ)+(n+1)P(n−m+1,0)m(τ)sinh2λ −(n+1)P(n−m,1)m(τ)sinh2λ+P(n−m,0)m(τ)]. (20)

Next, let us analyze some special cases. When corresponding to the two-mode squeezed state, Eqs. (19) and (20) becomes, respectively, to

 (ΔQ)2∣∣m=n=0=12\allowbreake2λ,(ΔP)2∣∣m=n=0=12\allowbreake−2λ,ΔQΔP=12, (21)

which is just the standard squeezing case; while for Eqs. (19) and (20) reduce to

 (ΔQ)2∣∣m=0,n=1=\allowbreake2λ,(ΔP)2∣∣m=0,n=1=\allowbreak\allowbreake−2λ,ΔQΔP=1, (22)

from which one can see that the state is squeezed at the “p-direction” when i.e., . In addition, when in a similar way, one can get

 (ΔQ)2∣∣m=n=1 =12e2λ(1+2e2λ−2e2λ+e−2λ), (ΔP)2∣∣m=n=1 =121−e2λ−3e−2λ(1−\allowbreake−2λ)e2λ+e−2λ+12<12, (23)

which indicates that, for any squeezing parameter , there always exist squeezing effect for state at the “p-direction”.

In order to see clearly the fluctuations of with other parameters values, the figures are ploted in Fig.2. From Fig.2(a) one can see that the fluctuations of are always less than when say, the state is always squeezed at the “p-direction”; for given values, there exist the squeezing effect only when the squeezing parameter exceeds a certain threshold value that increases with the increasement of (see Fig.2(b)).

### iii.2 Distribution of photon number

In order to obtain the photon number distribution of the TPSSV, we begin with evaluating the overlap between two-mode number state and Using Eq.(1) and the un-normalized coherent state 30 (); 31 (), , leading to , it is easy to see that

 ⟨na,nb|λ,m,n⟩ =sechλ⟨na,nb|ambnea†b†tanhλ|00⟩ =(m+na)!√na!nb!sechλtanhm+naλδm+na,n+nb. (24)

It is easy to follow that the photon number distribution of , i.e.,

 P(na,nb) =N−1λ,m,n|⟨na,nb|λ,m,n⟩|2 =[(m+na)!sechλtanhm+naλδm+na,n+nb]2na!nb!m!n!sinh2nλP(n−m,0)m(cosh2λ). (25)

From Eq.(25) one can see that there exists a constrained condition, for the photon number distribution (see Fig. 3). In particular, when Eq.(25) becomes

 P(na,nb)={sech2λtanh2naλ,na=nb0,na≠nb, (26)

which is just the photon number distribution (PND) of two-mode squeezed vacuum state.

In Fig. 3, we plot the distribution in the Fock space () for some given values and squeezing parameter . From Fig. 3 it is found that the PND is constrained by resulting from the paired-present of photons in two-mode squeezed state. By subtracting photons, we have been able to move the peak from zero photons to nonzero photons (see Fig.3 (a) and (c)). The position of peak depends on how many photons are annihilated and how much the state is squeezed initially. In addition, for example, the PND mainly shifts to the bigger number states and becomes more “flat” and “wide” with the increasing parameter (see Fig.3 (b) and (c)).

### iii.3 Antibunching effect of the TPSSV

Next we will discuss the antibunching for the TPSSV. The criterion for the existence of antibunching in two-mode radiation is given by 33 ()

 (27)

In a similar way to Eq.(13) we have

 ⟨a†2a2⟩=(m+1)(m+2)P(n−m−2,0)m+2(τ)P(n−m,0)m(τ), (28)

and

 ⟨b†2b2⟩=(n+1)(n+2)sinh4λP(n−m+2,0)m(τ)P(n−m,0)m(τ), (29)

For the state , substituting Eqs.(15), (28)and (29) into Eq.(27), we can recast  to

 Rab=(m+1)(m+2)P(n−m−2,0)m+2(τ)+(n+1)(n+2)sinh4λP(n−m+2,0)m(τ)2(m+1)(n+1)sinh2λP(n−m,0)m+1(τ)−1. (30)

In particular, when (corresponding to two-mode squeezed vacuum state), Eq.(30) reduces to which indicates that there always exist antibunching effect for two-mode squeezed vacuum state. In addition, when the TPSSV is always antibunching. However, for any parameter values , the case is not true. The as a function of and is plotted in Fig. 4. It is easy to see that, for a given the TPSSV presents the antibunching effect when the squeezing parameter exceeds to a certain threshold value. For instance, when and then csch may be less than zero with about.

## Iv Wigner function of the TPSSV

The Wigner function (WF)25 (); 34 (); 35 () is a powerful tool to investigate the nonclassicality of optical fields. Its partial negativity implies the highly nonclassical properties of quantum states and is often used to describe the decoherence of quantum states, e.g., the excited coherent state in both photon-loss and thermal channels 36 (); 37 (), the single-photon subtracted squeezed vacuum (SPSSV) state in both amplitude decay and phase damping channels 2d (), and so on 4 (); 10 (); 38 (); 39 (); 40 (). In this section, we derive the analytical expression of WF for the TPSSV. For this purpose, we first recall that the Weyl ordered form of single-mode Wigner operator 41 (); 42 (); 43 (),

 Δ1(α)=12::δ(α−a)δ(α∗−a†)::, (31)

where and the symbol denotes Weyl ordering. The merit of Weyl ordering lies in the Weyl ordered operators’ invariance under similar transformations proved in Ref.41 (), which means

 S::(∘∘∘)::S−1=::S(∘∘∘)S−1::, (32)

as if the “fence” did not exist, so can pass through it.

Following this invariance and Eq.(3) we have

 S†2(λ)Δ1(α)Δ2(β)S2(λ) =14S†2(λ)::δ(α−a)δ(α∗−a†)δ(β−b)δ(β∗−b†)::S2(λ) =14::δ(α−acoshλ−b†sinhλ)δ(α∗−a†coshλ−bsinhλ) =14::δ(¯α−a)δ(¯α∗−a†)δ(¯β−b)δ(¯β∗−b†):: =Δ1(¯α)Δ2(¯β),

where and . Thus employing the squeezed two-variable Hermite-excited vacuum state of the TPSSV in Eq.(5) and the coherent state representation of single-mode Wigner operator 44 (),

 Δ1(α)=e2|α|2∫d2z1π2|z1⟩⟨−z1|e−2(z1α∗−αz∗1), (33)

where is Glauber coherent state 30 (); 31 (), we finally can obtain the explicit expression of WF for the TPSSV (see Appendix A),

 W(α,β) =1π2sinhn+m2λ2n+mNλ,m,ne−2|¯α|2−2∣∣¯β∣∣2m∑l=0n∑k=0 ×[m!n!]2(−tanhλ)l+kl!k![(m−l)!(n−k)!]2|Hm−l,n−k(B,A)|2, (34)

where we have set Obviously, the WF in Eq.(34) is a real function and is non-Gaussian in phase space due to the presence of , as expected.

In particular, when Eq.(34) reduces to corresponding to the WF of two-mode squeezed vacuum state; while for the case of and noticing and Eq.(34) becomes

 (35)

where is -order Laguerre polynomials and Eq.(35) is just the WF of the negative binomial state 28 (). In Figs. 5-7, the phase space Wigner distributions are depicted for several different parameter values , and . As an evidence of nonclassicality of the state, squeezing in one of quadratures is clear in the plots. In addition, there are some negative region of the WF in the phase space which is another indicator of the nonclassicality of the state. For the case of and it is easily seen from (35) that at the center of the phase space (), the WF is always negative in phase space. Fig.5 shows that the negative region becomes more and visible as the increasement of photon number subtracted , which may imply the nonclassicality of the state can be enhanced due to the augment of photon-subtraction number. For a given value  and several different values (), the WF distributions are presented in Fig.7, from which it is interseting to notice that there are around wave valleys and 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.

### v.1 Model

When the TPSSV evolves in the thermal channel, the evolution of the density matrix can be described by the following master equation in the interaction picture45 ()£º

 ddtρ(t)=(L1+L2)ρ(t), (36)

where

 Liρ =κ(¯n+1)(2aiρa†i−a†iaiρ−ρa†iai) +κ¯n(2a†iρai−aia†iρ−ρaia†i), (a1=a,a2=b), (37)

and represents the dissipative coefficient and denotes the average thermal photon number of the environment. When Eq.(36) reduces to the master equation (ME) describing the photon-loss channel. The two thermal modes are assumed to have the same average energy and coupled with the channel in the same strength and have the same average thermal photon number . This assumption is reasonable as the two-mode of squeezed state are in the same frequency and temperature of the environment is normally the same 46 (); 47 (). By introducing two entangled state representations and using the technique of integration within an ordered product (IWOP) of operators, we can obtain the infinite operator-sum expression of density matrix in Eq.(36) (see Appendix B):

 ρ(t)=∞∑i,j,r,s=0Mi,j,r,sρ0M†i,j,r,s, (38)

where denotes the density matrix at initial time, and are Hermite conjugated operators (Kraus operator) with each other,

 Mi,j,r,s=1¯nT+1√(T1)r+s(T3)i+jr!s!i!j!a†rb†se(a†a+b†b)lnT2aibj, (39)

and we have set , as well as

 T1=¯nT¯nT+1,T2=e−κt¯nT+1,T3=(¯n+1)T¯nT+1. (40)

It is not difficult to prove the obeys the normalization condition by using the IWOP technique.

### v.2 Evolution of Wigner function

By using the thermal field dynamics theory 48 (); 49 () and thermal entangled state representation, the time evolution of Wigner function at time to be given by the convolution of the Wigner function at initial time and those of two single-mode thermal state (see Appendix C), i.e.,

 (41)

Eq.(41) is just the evolution formula of Wigner function of two-mode quantum state in thermal channel. Thus the WF at 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−1λ,m,n(Esinh2λ)m+nπ22n+m(2¯n+1)2T2De−∣∣α−β∗∣∣2e−2λ−2κt+(2¯n+1)T−∣∣α+β∗∣∣2e2λ−2κt+(2¯n+1)T ×n∑l=0m∑k=0[m!n!]2(−FEtanhλ)l+kl!k![(m−k)!(n−l)!]2∣∣Hm−k,n−l(G/√E,K/√E)∣∣2, (42)

where we have set

 C =e−2κt(2¯n+1)T, D=(1+Ce−2λ)(1+Ce2λ), E =\allowbreake4κtD(2¯nT+1)2C2, F=C2−1D, G K =CeκtD(¯A+A∗C), ¯A=i2√tanhλ(α∗coshλ+βsinhλ). (43)

Eq.(42) is just the analytical expression of WF for the TPSSV in thermal channel. It is obvious that the WF loss its Gaussian property due to the presence of two-variable Hermite polynomials.

In particular, at the initial time () noting , , and as well as , , Eq.(42) just dose reduce to Eq.(34), i.e., the WF of the TPSSV. On the other hand, when noticing that and as well as as well as the definition of Jacobi polynomials in Eq.(9), then Eq.(42) becomes

 W(α,β,∞)=1π2(2¯n+1)2e−22¯n+1(|α|2+|β|2), (44)

which is independent of photon-subtraction number and and corresponds to the product of two thermal states with mean thermal photon number . This implies that the two-mode system reduces to two-mode thermal state after a long time interaction with the environment. Eq.(44) denotes a Gaussian distribution. Thus the thermal noise causes the absence of the partial negative of the WF if the decay time exceeds a threshold value. In addition, for the case of , corresponding to the case of two-mode squeezed vacuum, Eq.(42) just becomes

 Wm=n=0(α,β,t)=N−1e−ED(|α|2+|β|2)+FD(αβ+α∗β∗), (45)

where is the normalization factor, and . Eq.(45) is just the result in Eq.(14) of Ref. 47 ().

In Fig.8, the WFs of the TPSSV for () are depicted in phase space with and for several different It is easy to see that the negative region of WF gradually disappears as the time increases. Actually, from Eq.(43) one can see that and , so when leading to the following condition:

 κt>κtc≡12ln2¯n+22¯n+1, (46)

we know that the WF of TPSSV has no chance to be negative in the whole phase space when