Higher nonclassical properties and entanglement of photon-added two-mode squeezed coherent states

Optics Communications 335 (2015) 108–115

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Higher nonclassical properties and entanglement of photon-added two-mode squeezed coherent states Shuai Wang a,n, Li-Li Hou a, Xue-Fen Xu b a b

School of Mathematics and Physics, Changzhou University, Changzhou 213164, P.R. China Department of fundamental course, Wuxi Institute of Technology, Wuxi 214121, P.R. China

art ic l e i nf o

a b s t r a c t

Article history: Received 14 July 2014 Received in revised form 3 September 2014 Accepted 7 September 2014 Available online 19 September 2014

We investigate how photon addition operations affect some higher-order nonclassical properties and the intermodal entanglement of a non-Gaussian entangled state generated by adding photons to both modes of a two-mode squeezed coherent state (TMSCS). We show that the photon addition operation can enhance the degrees of the sum squeezing and the difference squeezing for appropriate combinations of several parameters involved in the TMSCS. We consider the existing Hillery-Zubairy entanglement criterion, and find that the quantity of Hillery-Zubairy E gets more negative not only for a larger value of the squeezing parameter r but also for greater values of photon addition numbers (m, n). Our results may imply that the highest enhancement is obtained when same number of operations (i.e., m¼n) is applied to both modes. However, for the photon-added TMSCS (PA-TMSCS), the more violation to the HilleryZubairy inequality does not necessarily following strong sum squeezing and difference squeezing. This because their behaviors with three phases and squeezing parameter r involved in the TMSCS are somewhat different. & 2014 Published by Elsevier B.V.

Keywords: Non-Gaussian entangled state Photon-addition operation Sum squeezing Difference squeezing Entanglement

1. Introduction Non-Gaussian states generated by photon subtraction and photon addition have received increasing attention from both experimentalists and theoreticians recently [1–10]. This because that non-Gaussian states with highly nonclassical properties may constitute useful resources in quantum information with continuous variables, and overcome some limitations found in the use of Gaussian states for quantum information processing. Agarwal and Tara [2] first introduced photon-added coherent state (a nonGaussian state) and studied its nonclassical properties. In 2004, the photon addition operation was successfully demonstrated experimentally [9] via a nondegenerate parametric amplifier with small coupling strength. The photon subtraction was implemented [10] with a beam splitter of high transmissivity and was considered in enhancing not only the entanglement but also the performance of a quantum-noise-limited amplifier [11–14]. For a review about quantum state engineering by photon subtraction and addition, we refer to Refs. [15,16] (and references therein). Non-Gaussian two-mode squeezed thermal states were introduced and their nonclassical properties have been studied via

n

Corresponding author. E-mail address: [email protected] (S. Wang).

http://dx.doi.org/10.1016/j.optcom.2014.09.018 0030-4018/& 2014 Published by Elsevier B.V.

Mandel Q parameter, anti-bunching effects and Wigner functions [17–19]. Ourjoumtsev et al. [20] demonstrated experimentally that the entanglement between Gaussian entangled states can be increased by subtracting one photon from two-mode squeezed vacuum states. Yang and Li [21] investigated the entanglement properties of those non-Gaussian squeezed vacuum states, and showed that the partial von Neumann entropy of all the resulting states is greater than that of the original two-mode squeezed vacuum state. For multiple-photon addition and subtraction, the optimal enhancement is obtained when the same number of operations is applied to both modes, where both addition and subtraction give the same entanglement enhancement [22]. Lee et al. [23] proposed a coherent superposition of photon subtraction and addition to enhance quantum entanglement of two-mode squeezed vacuum sate. For two-mode squeezed coherent state (TMSCS), Wang et al. [24] firstly introduced the non-Gaussian TMSCS which is generated by single-mode photon addition, and they studied those nonclassical properties of resulting states in terms of Mandel's Q parameter, the cross-correlation function and Wigner functions. Different from the work in Ref.[24], we consider the photon-added TMSCS (PA-TMSCS) generated by repeatedly adding different photons to both modes of the TMSCS, and showed that the photon statistical properties are sensitive to the compound phase involved in the TMSCS [25]. And the photon subtraction and addition applied to both modes can enhance the

S. Wang et al. / Optics Communications 335 (2015) 108–115

cross-correlation and anti-bunching effects of resulting states. Very recently, Truong et al. [26] showed that the non-Gaussian TMSCS, which is generated by single-mode photon addition, can also possess higher-order nonclassical properties such as sum squeezing, the difference squeezing and the antibunching. Besides the entanglement, squeezing has attracted considerable attention in quantum optics and quantum information processing. In this work, different from the work in Refs. [22,26], we will further study the higher-order nonclassical properties of the PA-TMSCS, as well as its entanglement properties. This paper is organized as follows. In Section 2, we make a brief review about the PA-TMSCS generated by simultaneously adding different photons to each mode of the TMSCS and derive the expectation h †j value of a general product of the operator ak a†l b b . In Section 3, we derive the general analytic expressions for the sum squeezing and the difference squeezing, and explore numerically how photon addition operations affect the squeezing properties of the PA-TMSCS. In Section 4, we prove that the PA-TMSCS is two-mode entangled as characterized by Hillery-Zubairy entanglement criterion. Our results may imply that the entanglement can be increased with the number of photon additions, and the optical enhancement is obtained when the same number of operations is applied to both modes. Our main results are summarized in Section 5.

109

the same approach as deriving Eq. (4)  D E
 C 1 h †j ak a†l b b ¼ ξjam bn ak a†l bh b†j a†m b†n
ξ ¼ m þ k;m þ l;n þ h;n þ j ; C m;n C m;n ð7Þ where we set C m þ k;m þ l;n þ h;n þ j ¼ eiðk  lÞϕ1 þ iðh  jÞϕ2 jλ1 jk þ l jλ2 jh þ j

min½m þ k;m þ l min½n þ h;n þ j

∑

p¼0

∑

q¼0

2p þ 2q

r ðm þ kÞ!ðn þ hÞ!ðm þlÞ!ðn þ jÞ!



p!q!
λ1
2p  2m
λ2
2q  2n

f f minðm þ k  p;n þ h  qÞ ð2
λ1 λ2
eiχ Þ ðsinh 2r Þ 1  ∑ ðf !Þ ðm þ k  p  f Þ!ðn þ h  q  f Þ! f ¼0

s s minðm þ l  p;n þ j  qÞ ð2
λ1 λ2
e  iχ Þ ð sinh 2r Þ : ðs!Þ  1  ∑ ðm þl p  sÞ!ðn þj  q  sÞ! s¼0 

cosh

ð8Þ

Eqs. (4) and (7) are important for further studying higher-order nonclassical properties of the PA-TMSCS. When k ¼l and h¼j are satisfied, Eq. (8) actually reduces to Eq. (4), i.e., C m þ k;m þ l;n þ h;n þ j  C m þ k;n þ h .

3. Sum and difference squeezing 2. The PA-TMSCS and its normalization factor Let us firstly make a brief review about the PA-TMSCS and its normalization factor. Theoretically, the PA-TMSCS can be obtained † by repeatedly operating photon creation operators a† and b on both modes of a TMSCS [25], i.e.,

   1=2 †m †n 
ξ ¼ C m;n a b D λ1 ; λ2 S2 ðξÞ
0a ; 0b ; ð1Þ pa where m and n are the numbers of adding photons to each mode of the TMSCS, respectively. And S2 ðξÞ and Dðλ1 ; λ2 Þ are the twomode squeezing and displacement operators, respectively, defined by h i n † S2 ðξÞ ¼ exp ξa† b  ξ ab ; h i   n n † D λ1 ; λ2 ¼ exp λ1 a†  λ1 a þ λ2 b  λ2 b ; ð2Þ with













λ1 ¼
λ1
exp½iϕ1 ; λ2 ¼
λ2
exp ½iϕ2 :

ξ ¼ r exp½iθ;

ð3Þ

The corresponding normalization factor C m;n in Eq. (1) is [See Appendix in detail] 2p þ 2q

ðm!n!Þ2 cosh r





2p  2m
λ2
2q  2n p ¼ 0 q ¼ 0 p!q! λ1


minðm  p;n  qÞ ð2jλ λ jeiχ Þ  f ðsinh 2rÞf
2

1 2 
∑

f !ðm  p  f Þ!ðn  q  f Þ!
f ¼0 m

C m;n ¼ ∑

n

∑

ð4Þ

Although there are three phases ϕ1, ϕ2 and θ in the TMSCS, Eq. (4) shows that C m:n is actually a periodic function of the compound phase χ. The compound phase χ is defined by

χ ¼ ϕ1 þ ϕ2  θ:

ð5Þ

For example, when m ¼n ¼1     2 C 1;1 ¼ jλ1 j2 þ jλ2 j2 þ cosh 2r cosh r þ jλ1 λ2 j jλ1 λ2 j þ cos χ sinh 2r ; ð6Þ Obviously, C 1;1 is a periodic function of χ involved in the TMSCS with a period 2π . For the PA-TMSCS state given by Eq. (1), we can obtain the h †j expectation value of a general product of the operator ak a†l b b by

In this section, with the help of the normalization factor we shall discuss how the photon addition affects the higher-order nonclassical statistical properties of the PA-TMSCS in terms of the sum squeezing and the difference squeezing. Especially, we shall present how the compound phase χ and photon addition operations affect those nonclassical properties. 3.1. Sum squeezing Sum and difference squeezing are both higher-order, two-mode squeezing effects [27,28]. In quantum optics, squeezing is an earliest studied nonclassical phenomenon. For two arbitrary modes a and b, the sum squeezing is associated with a so-called two-mode quadrature operator V φ of the form [27]  1 † V φ ¼ eiφ a† b þ e  iφ ab ð9Þ 2 where φ is an angle made by V φ with the real axis in the complex plane. A state is said to be sum squeezed for a φ if D E  E 1D ΔV φ 2 o a† a þb† bþ 1 : ð10Þ 4 D 2 E D 2 E 2 ¼ V φ  V φ . From Eq. (10), one can define where ΔV φ the degree of the sum squeezing S in the form of antinormally ordered operators as following D E 2 E D † †  aa þbb  1 4 ΔV φ D E S¼ : ð11Þ † aa† þ bb  1 Substituting Eq. (9) into Eq. (11), we obtain S as S¼

B  2A ; A

where D E † A ¼ aa† þ bb  1 ; D E  D E   † 2  4 Re2 e  iφ ab : B ¼ 2 aa† bb þ 2 Re e  2iφ a2 b

ð12Þ

ð13Þ

Then its negative value in the range ½  1; 0 indicates sum squeezing (a higher-order nonclassicality). It is clear that S has a lower

110

S. Wang et al. / Optics Communications 335 (2015) 108–115

bound equal to  1. Hence, the closer the value of S is to 1 the higher the degree of the sum squeezing is. Noting Eq. (7), we can † 2 obtain the expected values of operators, aa† ; bb ; ab; a2 b and † † aa bb , respectively, D E C D E C m þ 1;n † m;n þ 1 ; bb ¼ ; aa† ¼ C m;n C m;n D E  C m þ 1;m;n þ 1;n C m þ 2;m;n þ 2;n 2 ab ¼ ; a2 b ¼ ; ð14Þ C m;n C m;n and D E C † m þ 1;n þ 1 : aa† bb ¼ C m;n

1

1.0

0.5

0.0

0.5 0

2

ð15Þ

Let us first consider the angle dependence of the sum squeezing. From Eq. (4), one can see that C m þ 1;n , C m;n þ 1 and C m þ 1;n þ 1 are all periodic functions of the compound phase χ with a period 2π . However, C m þ 1;m;n þ 1;n (C m þ 2;m;n þ 2;n ) is not only related to the phase χ , but also to the phase ϕ1 þ ϕ2 ð2ϕ1 þ 2ϕ2 Þ, which can be seen from Eq. (8). Then the degree of the sum squeezing S is actually related to three phases, i.e., θ, ϕ1 þ ϕ2 and φ. Thus, the dependence on these phases of S appears to be complicated. And the optimal degree of the sum squeezing depends on the appropriate combination of three phases ϕ1 þ ϕ2 , θ and φ. For simplicity, we considerthe case of φ ¼ 0, i.e., the two-mode † quadrature operator is V 1 ¼ a† b þ ab =2. In Fig. 1, we plot the sum squeezing degree S as a function of the phase ϕ1 þ ϕ2 for different λ1 and λ2. It shows that the degree of the sum squeezing can be improved by photon addition operation when the value of phase ϕ1 þ ϕ2 approaches to π. Further numerical anaylsis indicates that the sum squeezing degree is optimized at ϕ1 þ ϕ2 ¼ π or θ ¼ π . In Fig. 2, we present how both displacement parameters affect the sum squeezing degree. Fig. 2 shows that for symmetric operation
on
both
modes, i.e., m ¼n, the S is symmetrical along the diagonal
λ1
¼
λ2 j, and decreases with the increasement of displacement parameters. In addition, the sum squeezing degree

S

S

1

2

3

4

1

5

6

2

S

1

2

S

1

2

Fig. 2. The sum squeezing degree S varies with two displacement parameters for given ϕ1 þ ϕ2 ¼ π; θ ¼ 0 and r¼ 0.5 : (a) m ¼ n ¼ 0; (b) m ¼ 1, n¼ 0; (c) m ¼ n ¼ 1.

is optimized along the diagonal for any values of (m; n). From Figs. 1 and 2, we can see that for more photon addition operations and larger displacment parameters, the stronger degree of the sum squeezing of the PA-TMSCS is. Finally, in order to see clearly the effects of photon addition operations and the squeezing parameter r on the sum squeezing,

we draw Fig. 3 in the case of
λ1 j ¼
λ2 j and ϕ1 þ ϕ2 ¼ π , as well as φ ¼ θ ¼ 0. We can see that the degree of the sum squeezing is first increasing (i.e., S is getting more negative) until a critical value, but afterwards it is becoming smaller and smaller, and eventually disappears (i.e., S turns out to be positive). Particularly, for large values of both displacement parameters, the degree of sum squeezing can be enhanced by photon addition operations, and increased with the number of photon additions as shown in Fig. 3(b).

1.0

3.2. Difference squeezing

S

0.5

Another interesting kind of multi-mode squeezing is the socalled difference squeezing [27]. In the two-mode situation, it is associated with a “collective” operator W ϕ in the form

0.0

0.5

0

1

2

3

4 1

5

6

2

Fig. 1. The sum squeezing degree S as a function of the phase ϕ1 þ ϕ2 with r ¼0.5 and θ ¼0 for different values of (m, n): (a) jλ1 j ¼ 2 and jλ2 j ¼ 5, (From top to bottom lines ð0; 0Þ, ð1; 1Þ, ð2; 2Þ, ð1; 0Þ, ð2; 1Þ, and ð2; 0Þ); (b) jλ1 j ¼ 5 and jλ2 j ¼ 2, (From top to bottom lines ð2; 0Þ, ð1; 0Þ, ð0; 0Þ, ð2; 1Þ, ð1; 1Þ, and ð2; 2Þ).

Wϕ ¼

 1  iϕ †  iϕ e ab þ a† be 2

ð16Þ

where ϕ is an angle made by W ϕ with the real axis in the complex plane. A state is said to be difference squeezed for a ϕ if  2 1

D E
†
o

aa†  bb

ΔW ϕ ð17Þ 4

S. Wang et al. / Optics Communications 335 (2015) 108–115

 D E h  D Ei2 †2 † B0 ¼ 2 Re e2iϕ a2 b  4 Re eiϕ ab D E D E D E † † þ2 aa† bb  aa†  bb

2.0 1.5

0.5 0.0 0.0

0.2

0.4

0.0

0.2

0.4

r

0.6

0.8

1.0

0.2 0.0

S

0.2 0.4 0.6

ð20Þ

Noting Eqs. (4) and (7), we can obtain the expected values of † †2 operators ab and a2 b , respectively, D E C D E C † m þ 1;m;n;n þ 1 †2 m þ 2;m;n;n þ 2 ; a2 b : ð21Þ ¼ ab ¼ C m;n C m;n

1.0

S

111

r

0.6

0.8

1.0

Fig. 3. The sum squeezing degree S as a function of squeezing parameter r with ϕ1 þ ϕ2 ¼ π, φ ¼ θ ¼ 0 for different values of (m; n): (a) jλ1 j ¼ jλ2 j ¼ 1:0, (From top lines to bottom (2; 2), (2; 1), (1; 1), (2; 0), (1; 0) and (0; 0)); (b) jλ1 j ¼ jλ2 j ¼ 3:0, (From top lines to bottom (0; 0), (1; 0), (2; 0), (1; 1), (2; 1) and (2; 2)).

For the TMSCS, our numerical analysis shows that there is no difference squeezing at all (i.e., D is always positive). From Eqs. (8) and (21), we can see that C m þ 1;m;n;n þ 1 ðC m þ 2;m;n;n þ 2 Þ is not only related to the phase χ, but also to the phase ϕ1  ϕ2 ð2ϕ1  2ϕ2 Þ, which is different from C m þ 1;m;n þ 1;n ðC m þ 2;m;n þ 2;n Þ in Eq. (14). Then the degree of difference squeezing D is actually related to four phases, i.e., θ, ϕ1, ϕ2 and ϕ. Thus, compared with the case of the sum squeezing the dependence on these phases of D appears to be more complicated. In the following, for simplicity we set ϕ ¼ 0. Our numerical analysis show that the variation tendency of D with ϕ1 and ϕ2 is similar, and the optimal degree of the difference squeezing depends on the appropriate combination of three phases (ϕ1 ; ϕ2 and θ). For example, in the case of ϕ1 ¼ π and ϕ2 ¼ θ ¼ 0, the degree of the difference squeezing is optimal as shown in Fig. 4. Obviously, the degree of the difference squeezing is increased with the values of (m; n). Based on Eq. (19), in Fig. 5 we present how
the difference
squeezing varies with two absolute displacements
λ1 j and
λ2 j for different (m; n). In Fig. 5, the positive part of each plot is clipped off

4 1

3

D

2

D

1 0 0

1

2

3

4

5

2

6

1 1

Fig. 4. The difference squeezing degree D as a function of the phase ϕ1 with r¼ 0.5, ϕ2 ¼ θ ¼ ϕ ¼ 0, jλ1 j ¼ 3, and jλ2 j ¼ 8 for different values of (m; n). (From top to bottom lines (0; 0), (1; 1), (2; 2), and (5; 5)).

D Like in the case of the sum squeezing, we can make use of Eq. (16) to define the degree of the difference squeezing as  2

D E
†
4 ΔW ϕ 

aa†  bb


D ð18Þ D¼ E



aa†  bb†


Hence, a state is difference squeezed if  1 r D o 0. Taking Eq. (16) into account in the definition Eq. (18), we readily obtain D for the PA-TMSCS D¼

B0  A0 ; A0

2

1

D

ð19Þ 2

where
D E

†
A0 ¼

aa†  bb



Fig. 5. The difference squeezing degree D varies with two displacement parameters for given ϕ1 ¼ π; ϕ2 ¼ θ ¼ 0 and r¼ 0.5: (a) m ¼ 1; n ¼ 0; (b) m ¼ n ¼ 1; (c) m ¼ n ¼ 3. (The positive part of each plot is clipped off to reveal more clearly difference squeezing region where the higher nonclassical properties exist.

112

S. Wang et al. / Optics Communications 335 (2015) 108–115

realizability and their recent success in detecting entanglement in various optical, atomic and optomechanical systems [34,35]. Here for our purpose, it is convenient to employ the criterion given by Hillery and Zubairy in Ref. [32]. Thus, according to the Hillery-Zubairy entanglement criterion, the two modes of a twomode state are mutually entangled if the following inequality is satisfied, i.e., D ED E

† E ¼ aa†  1 bb  1 
ab
2 o 0: ð22Þ

1.0 0.8 0.6

D

0.4 0.2 0.0 0.2 0.0

0.2

0.4

r

0.6

0.8

1.0

1.5

This inequality represents sufficient condition for the entanglement, but if it is not satisfied, we cannot say whether the total state is entangled or not. For the PA-TMSCS, we can easily obtain the expectation values † of the operators aa† , bb , and ab by using Eq. (7). Thus, for the PATMSCS we have  E¼

1.0

D 0.5 0.0 0.0

0.2

0.4

r

0.6

0.8

1.0

Fig. 6. The difference squeezing degree D as a function of squeezing parameter r with ϕ1 ¼ π, ϕ2 ¼ θ ¼ 0 for different values of (m; n): (a) jλ1 j ¼ 2; jλ2 j ¼ 10, (From top lines to bottom (1; 1), (1; 0), (2; 2), (2; 1) and (2; 0)); (b) jλ1 j ¼ 10; jλ2 j ¼ 2, (From top lines to bottom (2; 0), (1; 0), (2; 1), (1; 1) and (2; 2)).

to reveal more clearly the difference squeezing region where the non-classical properties exist. It is clear that the degree of the difference squeezing is increased with the values of (m; n). In the case of m ¼n there are two regions which are symme

negative



trical along the diagonal D (c). E


λ1 ¼ λ2 j as shown in Fig. 4(b)  and † When m ¼n and
λ1
¼
λ2 j are all fulfilled, one has aa† ¼ bb which leads to A0 ¼ 0, and thus D is always positive which is very different from the case of the sum squeezing (as shown in Fig. 2). In addition, when m ¼ 0 (or n¼ 0), there is only one negative region as shown in Fig. 5(a). In Fig. 6,
we
show the dependence on the squeezing parameter
r of D with
λ1
a
λ2 j in the case of ϕ1 ¼ π and ϕ2 ¼ θ ¼ 0. Similar to the sum squeezing, the degree of D is first increasing until a critical value, but afterwards it is becoming smaller and smaller, and eventually disappear. The difference between Fig. 6(a) and (b) reflects that when m a
n,
the
negative regions are not symmetrical along the diagonal
λ1
¼
λ2 j as shown in Fig. 5. Thus, we can see the degree of the difference squeezing can be also enhanced by the photon addition and large displacement paramters. In this section, we demonstrate that photon addition operations can enhance both sum squeezing and difference squeezing of the PA-TMSCS in some appropriate combination of several parameters involving in the TMSCS, i.e., squeezing parameter, absolute displacement parameters, as well as three corresponding phases.

C m þ 1;n 1 C m;n




C m þ 1;m;n þ 1;n
2 C m;n þ 1
 1 


C m;n C m;n

ð23Þ

Combining Eqs. (4), (8) and our numerical analysis, we can prove that E is a periodic function of the compound phase χ ¼ ϕ1 þ ϕ2  θ although there are three independent phases, θ, ϕ1, and ϕ2 involved in the TMSCS. When m¼ n ¼0, i.e., for the case of the TMSCS, Eq. (23) reduces to










2 E0 ¼ 
λ1 λ2
sinh 2r cos χ þ
λ1
2 þ
λ1
2 1 sinh r; ð24Þ is always negative for any
values

of the squeezing parameter r with conditions χ ¼ 0 and
λ1
¼
λ2 j. To see whether E o0 we analyze it with regard to the parameters involved. We firstly plot E as a function of r in Fig. 7. It is clear from Fig. 7 that E is negative when the squeezing parameter r is larger than a certain threshold. More concretely, E gets more negative not only for larger values of r but also for greater values of (m; n). The latter implies that adding more photons to both modes of the TMSCS may improve the degree of entanglement. Our numerical analysis shows that the minimum value of E is optimized at χ ¼ 0 and jλ1 j ¼ jλ2 j as shown in Figs. 8 and 9. When m ¼n (the symmetric operation), the E is symmetrical along the diagonal jλ1 j ¼ jλ2 j as shown in Fig. 9(c). And the minimum value of E is also along the diagonal and decrease with the increasement of the absolute values of jλ1 j ¼ jλ2 j. It has been proven that the Gaussian states are extremal for entanglement in continuousvariable systems according to the Giedke-Wolf-Cirac theorem [36]. Therefore, non-Gaussian states, such as photon-added or -subtracted states, are always more entangled than Gaussian states. From Hillery-Zubairy entanglement criterion, our results indicate that the more violation will occur when adding more photons to each mode of the TMSCS, especially when same

2 1 0

E 4. Intermodal entanglement



1 2 3 4

Entanglement is proven to be an important resource in quantum information processing. There exist several inseparability criteria [29–33] that are expressed in terms of expectation values of field operators. Among these criteria, Duan's criterion and Hillery-Zubairy criterion have received more attention because of various reasons, such as computational simplicity, experimental

5

0.0

0.2

0.4

0.6

r

0.8

1.0

1.2

1.4

Fig. 7. The entanglement parameter E as a function of r with jλ1 j ¼ jλ2 j ¼ 1:0, and χ ¼ 0 for different values of (m; n). (From top to bottom lines (0; 0), (1; 0), (1; 1), (3; 0), and (3; 3).).

S. Wang et al. / Optics Communications 335 (2015) 108–115 10

the same entanglement enhancement [22]. Thus, for the PATMSCS, our results may be indicate that the entanglement also increases with the number of photon additions, and the greatest enhancement is obtained when same number of operations is applied to both modes.

5

E

0

5. Conclusions

5

10

113

0

2

4

6

8

10

12

Fig. 8. The entanglement parameter E as a function of the compound phase χ with r ¼0.6 and jλ1 j ¼ jλ2 j ¼ 2 for different values of (m; n). (From top to bottom lines (0; 0), (1; 0), (1; 1) and (3; 3).).

E

2

1

E

2

1

In summary, we have investigated how photon addition operations affect the sum squeezing, the difference squeezing and the intermodal entanglement of the PA-TMSCS. By the generating function of two-variable Hermite polynomials, we derive the compact expression of normalization factor of the PA-TMSCS, which is a periodic function of the compound phase χ ¼ ϕ1 þ ϕ2  θ involved in squeezing and displacement parameters. Using the same approach, we obtain the expectation value of a general product of D E h †j the operator ak a†l b b . Then, based on the normalization factor D E h †j and the expectation value ak a†l b b , we numerically analyze in detail the behaviors of those higher nonclassical properties and the entanglement depending on the parameters involved. The behaviors of both sum squeezing and difference squeezing are almost similar in some aspects. Both kinds of squeezing can be improved by photon addition operations when the value of phase ϕ1 approaches to π (where we set ϕ2 ¼ θ ¼ 0). And the degree of both sum squeezing and difference squeezing increases with the values of photon addition number (m; n).
However,
their dependence on both displacement parameters
λ1 j and
λ2 j are somewhat different. For example, when the sum squeezing is

m ¼n,
symmetrical along the diagonal
λ1
¼
λ2 j and show maximum squeezing along
the
diagonal. But for the difference squeezing,
when m ¼ n and
λ1
¼
λ2 j are all fulfilled, there is no difference squeezing at all. In addition, both kinds of squeezing exist only for small values of r. The degree of both sum squeezing and difference squeezing increase with the squeezing paramter r first until a threshold value, but afterwards it is becoming smaller and smaller, and eventually disappears. Finally, with the help of the HilleryZubairy entanglement criterion, we show that the quantity of Hillery-Zubairy E gets more negative not only for a larger value of r but also for greater values of photon addition numbers (m; n) at the optimal compound phase χ ¼ 0. While for both sum squeezing and difference squeezing, the corresponding optimal phase dependents on some appropriate combinations of three phases (ϕ1 ; ϕ2 and θ). Our results indicate that both sum squeezing and difference squeezing, as well as the more violation to Hillery-Zubairy inequality can be enhanced by photon addition operations, especially for large displacement parameters.

E Acknowledgments

2

1

Fig. 9. The entanglement parameter E as a function of both displacement parameters with χ ¼ 0 and r ¼0.5 for different values of (m; n): (a) m ¼ n ¼ 0; (b) m ¼ 3, n¼ 0; (c) m ¼ n ¼ 3.

We would like to thank the referees for their helpful suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11404040 and 11174114), and the Natural Science Foundation of Jiangsu Province of China (Gant No. BK20140253).

Appendix A. Derivation of Eq. (4) number of operations is applied to both modes. For multiplephoton added and subtracted two-mode squeezed vacuum states, the optimal enhancement (characterized by von Neumann entropy) is obtained when the same number of operations is applied to both modes, where both addition and subtraction give

The explicit representation of the TMSCS in two-mode Fock space is [19]

  1 1 n n † † exp  λ1 μ1 þ λ2 μ2 þa† μ1 þ b μ2 þ a† b eiθ tanh r j0; 0〉: jξ〉 ¼ cosh r 2 ðA:1Þ

114

S. Wang et al. / Optics Communications 335 (2015) 108–115

where n iθ

μ1 ¼ λ1  λ2 e tanh r; μ2 ¼ λ2  λn1 eiθ tanh r:

ðA:2Þ

Substituting Eq. (A.1) into Eq. (1) and inserting the completeness relation of two-mode coherent states, the normalization factor is 
n  †n C m;n ¼ Tr a†m b jξ〉 ξ
b am D

n †n
¼ ξ
am a†m b b
ξ〉 Z ¼

2 2
D d z1 d z2

m n
†n
〈ξ
a b jz1 z2 i z1 z2
a†m b
ξ〉 2

π

Using the integral formula

 Z 2 d z 1 ξη ; exp½ζ jzj2 þ ξz þ ηzn  ¼  exp 

π

ζ

ζ

ðA:3Þ

ðA:4Þ

whose convergent condition is Reðζ Þ o0, after doing straightforward calculation, we obtain

 ∂2m ∂2n 1 iθ 2 C m;n ¼ m m n n exp ðfs þt τÞ cosh r þ fte þ sτe  iθ sinh 2r 2 ∂f ∂s ∂t ∂τ i n n þ f λ1 þ sλ1 þ t λ2 þ τλ2 : ðA:5Þ f ;s;t;τ ¼ 0

h i 2 Expanding the partial exponential item exp ðfs þ t τÞcosh r in Eq. (A.5), then it becomes 2ðl þ jÞ

∂2m m m ∂f ∂s ∂λ λ λ λ

  2n ∂ 1 iθ fte þ sτe  iθ sinh 2r  n n exp 2 ∂t ∂τ i n n þ f λ1 þ sλ1 þ t λ2 þ τλ2 : 1

cosh l!j! l;j ¼ 0

C m;n ¼ ∑

r

∂2l ∂2j

l nl j nj 1∂ 1 ∂ 2∂ 2

f ;s;t;t ¼ 0

ðA:6Þ

Further using the generating function of the two-variable Hermite polynomials  pffiffiffi ∂m ∂n  Aft þ Bf þ Ct B C ðA:7Þ jf ;t ¼ 0 ¼ ð AÞm þ n H m;n pffiffiffi; pffiffiffi ; m ne ∂f ∂t A A

Eq. (A.5) reduces to the compact forms   jλ1 j2 2m C m;0 ¼ m!cosh rLm ; 2 cosh r

ðA:13Þ

and  2n

C 0;n ¼ n!cosh rLn

 jλ2 j2 2

cosh r

;

ðA:14Þ

respectively. Eqs. (A13) and (A14) are just those results of Ref. [24], and are not dependent on three phases involved in such states. While λ1 ¼ λ2 ¼ 0, Eq. (A.5) reduces to h ∂2m ∂2n 2 C m;n ¼ m m n n exp ðfs þt τÞ cosh r ∂f ∂s ∂t ∂τ   1  iθ ; ðA:15Þ þ fte þ sτe  iθ sinh 2r 2 f ;s;t;τ ¼ 0 which leads to C m;n ¼

 r sin 2r 2k : 2 ðm kÞ!ðn kÞ!ðk!Þ2

min½m;n ðm!n!Þ2

∑ k

cosh

2ðm þ n  2kÞ

ðA:16Þ

In the special case that the four quantities n, n þ α, n þ β , and n þ α þ β are nonnegative integers, the Jacobi polynomial can be written as !  nþα nþβ 1 n ðA:17Þ ðx  1Þn  k ðx þ 1Þk : P nðα;βÞ ðxÞ ¼ n ∑ k 2 k¼0 n k Then, we can further obtain the normalized factor of the photonadded two-mode squeezed vacuum state (without loss of generality assuming m Z n) C m;n ¼ m!n! cosh

2m

 nÞ rP ð0;m ðcosh 2rÞ; n

ðA:18Þ

where we have used the property of the Jacobi polynomials P nðα;βÞ ð  xÞ ¼ ð 1Þn P nðβ;αÞ ðxÞ. References

and the relation ∂l þ k m!n! H H m;n ðx; yÞ ¼ ðx; yÞ; ðm  lÞ!ðn  kÞ! m  l;n  k ∂xl ∂yk we can finally obtain  m þ n m n sinh 2r ðcoth rÞðl þ jÞ ðm!n!Þ2 ∑ ∑ C m;n ¼ 2 2 l ¼ 0 j ¼ 0 l!j!½ðm  lÞ!ðn  jÞ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi !
2

2e  iθ λ1 2e  iθ λ2


H m  l;n  j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

: i sinh 2r i sinh 2r

ðA:8Þ

ðA:9Þ

In the last step we have used the following relation H m;n ðξ; ηÞ ¼ ð  1Þm þ n H m;n ð  ξ;  ηÞ;

ðA:10Þ

which can be obtained easily from the two-variable Hermite polynomials, i.e., H m;n ðξ; ηÞ ¼

minðm;nÞ

∑

l¼0

ð  1Þl m!n! ξm  l ηn  l : l!ðm  lÞ!ðn  lÞ!

ðA:11Þ

By using Eq. (A.11), we can immediately obtain Eq. (4) which is a periodic function of the compound phase χ ¼ ϕ1 þ ϕ2  θ involved in the TMSCS. In case of m ¼0 or n¼ 0, used the generating function of the Laguerre polynomials   ð  1Þn ∂2n   Ln zzn ¼ exp  tt 0 þ tz þ t 0 zn jt;t 0 ¼ 0 : n! ∂t n ∂t 0n

ðA:12Þ

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