1 Introduction

Nonclassical field state has been a great interesting topic in quantum optics and quantum information [1]. In recent years, both experimentalists and theoreticians have proposed non-Gaussian operations such as photon-addition and photon-subtraction to obtain some nonclassical states [2–11]. These states exhibit numerous nonclassical properties and provide access to a complete engineering of quantum states and fundamental quantum phenomena [12–18]. For example, the superposition operation ta+ra † was studied with possible applications to quantum-state engineering [19] and entanglement (nonlocality) concentration [20]. Some other nonlocal coherent operations, such as a+b [8], a 2+b 2 [21]and a †+b † [22], were also investigated. In addition, an experimental scheme to implement a second-order nonlocal superposition operation was proposed [23]. Several years ago, Agarwal and Tara have introduced a single-mode photon added thermal state [24]. They also construct two-mode photon added thermal state by making the phase-insensitive amplification of N00N states [25]. Moreover, these states have been studied experimentally [4, 6].

On the other hand, many interesting theoretical works were focused on the non-local effects of entangled states [26–31]. Followed by the seminal contribution of Bell [32], the meaning of quantum reality and quantum nonlocality has become a central issue of the modern interpretation and understanding of quantum phenomena [33, 34]. If two subsystem are in an entangled state, the Bell’s inequality may be violated [35]. Non-Gaussian transformation can be useful for implementing nonlocality test [36] and enhancing nonlocality [37, 38].

Enlighten by these ideas, we shall study the superposition of high-order (N) photon addition (a †N+b †N) to two-mode thermal state (SPA-TMTS) and discuss the nonlocality of the nonclassical state in this paper. The paper is organized as follows. In Sect. 2, we introduce SPA-TMTS and obtain its normal ordering form. In Sect. 3, we derive quasi-probability distribution such as Q function and Wigner function (WF) in phase space. By drawing three-dimensional graphics of Q function and WF (whose negativity indicate the nonclassicality), we analyze the nonclassicality of the SPA-TMTS. In Sect. 4, constructing a Clauser-Horrne (CH) Bell inequality from Q function and a Clauser-Horne-Shimony-Holt (CHSH) Bell inequality from Wigner function, we study the quantum nonlocality of the SPA-TMTS.

2 High-Order Superposition Photon Addition Two-Mode Thermal State (SPA-TMTS)

Here we introduce a new kind of quantum state, which can be generated by the superposition of high-order (N) photon addition to two-mode thermal state (SPA-TMTS), i.e. repeatedly operating a † and b † for N times on a two mode thermal state, respectively. Its density operator is given as

$$ \rho=C_{N} \bigl( a^{\dag N}+b^{\dag N} \bigr) \rho_{_{th1}}\rho_{_{th2}} \bigl( a^{N}+b^{N} \bigr), $$
(1)

where N⩾1 is the added photon number in each mode, and C N is the normalization coefficient to be determined by trρ=1. As special case, when N=0, C N =1/4, SPA-TMTS ρ will be two-mode thermal state \(\rho_{_{th1}}\rho_{_{th2}}\). Here \(\rho_{_{th1,2}}\) is a density operator of single-mode thermal state,

$$ \rho_{_{thj}}=\sum_{n=0}^{\infty} \frac{\bar{n}_{j}^{n}}{ ( \bar{n}_{j}+1 ) ^{n+1}}\vert n \rangle \langle n\vert , $$
(2)

where \(\bar{n}_{j}\) is the average photon number of thermal state \(\rho_{_{thj}}\) (j=1,2). For simplicity, we assume the average photon number of \(\rho_{_{th1}}\) and \(\rho_{_{th2}}\) to be identical, i.e. \(\bar{n}_{1}=\bar{n}_{2}=\bar{n}\). It is obvious to see that when \(\bar{n}=0\), SPA-TMTS ρ can be reduced to N00N states (a maximally path-entangled number state), which have important applications to quantum imaging, metrology, and sensing [39].

In addition the P-representation of density operator \(\rho_{_{thj}}\) can be expanded as [40]

$$ \rho_{_{thj}}=\frac{1}{\bar{n}}\int \frac{d^{2}\alpha_{j}}{\pi}e^{-\frac {1}{\bar{n}}\vert \alpha_{j}\vert ^{2}} \vert \alpha _{j} \rangle \langle \alpha_{j}\vert , $$
(3)

which is useful for later calculation and here |α j 〉 (j=1,2) are the coherent states. Therefore, we can re-write \(\rho_{_{th1}}\rho_{_{th2}}\) as

$$ \rho_{_{th1}}\rho_{_{th2}}=\int\!\!\! \int \frac{d^{2}\alpha_{1}d^{2}\alpha_{2}}{ \pi^{2}}P ( \alpha_{1},\alpha_{2} ) \vert \alpha_{1},\alpha_{2} \rangle \langle \alpha_{1},\alpha_{2}\vert , $$
(4)

with the P-representation of two mode thermal state (TMTS)

$$ P ( \alpha_{1},\alpha_{2} ) =\frac{1}{\bar{n}^{2}}\exp \biggl( -\frac{1}{\bar{n}}\vert \alpha_{1}\vert ^{2}- \frac{1}{\bar{n}}\vert \alpha_{2}\vert ^{2} \biggr) $$
(5)

and |α 1,α 2〉=|α 1〉⊗|α 2〉. Using the vacuum projector |0,0〉〈0,0|=:exp(−a † a−b † b): as well as the IWOP technique, we obtain the normal ordering form of \(\rho_{_{th1}}\rho_{_{th2}}\) as follow

$$ \rho_{_{th1}}\rho_{_{th2}}=\kappa^{2}\mathopen{:} \exp \bigl( - \kappa a^{\dag }a-\kappa b^{\dag}b \bigr) \mathclose{:}, $$
(6)

where we have set κ=1 \((\bar{n}+1)\) and used the following integral formula,

$$ \int \frac{d^{2}z}{\pi}\exp \bigl( \zeta \vert z\vert ^{2}+\xi z+ \eta z^{\ast} \bigr) =-\frac{1}{\zeta}\exp \biggl( -\frac{\xi \eta}{\zeta} \biggr) ,\quad \mathrm{Re} ( \zeta ) <0. $$
(7)

Thus the normally ordering form of the SPA-TMTS can also be given as

$$ \rho=C_{N}\kappa^{2}\mathopen{:} \bigl( a^{\dag N}+b^{\dag N} \bigr) \exp \bigl( -\kappa a^{\dag}a-\kappa b^{\dag}b \bigr) \bigl( a^{N}+b^{N} \bigr) \mathclose{:}, $$
(8)

which is important to calculate the normalization coefficient, Q function and Wigner function.

To fully describe a quantum state, we must normalize the SPA-TMTS. Inserting the completeness relation of coherent state \(\int \frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\vert z_{1},z_{2} \rangle \langle z_{1},z_{2}\vert =1\) into \(\operatorname{tr}\rho=1\) and using Eq. (8), we easily obtain the normalization coefficient

$$ C_{N}^{-1}=2N! ( \bar{n}+1 ) ^{N}=2N!/ \kappa^{N}, $$
(9)

3 Quasi-probability Distribution Function

In order to investigate the quantum nonlocality of the SPA-TMTS, Banaszek established a direct relationship between the quantum nonlocality and the positive phase-space Q function, as well as the nonpositive Wigner function [41]. In this section, we derive the analytical expressions of Q function and Wigner function for the SPA-TMTS.

3.1 Q Function

The Q function [42] for two-mode quantum state ρ is defined as

$$ Q_{ab}(\alpha;\beta)=\frac{1}{\pi^{2}} \langle \alpha,\beta \vert \rho \vert \alpha,\beta \rangle , $$
(10)

where |α,β〉=|α〉 a |β〉 b is the two-mode coherent and \(\alpha=\frac{x_{1}+iy_{1}}{\sqrt{2}},\beta=\frac{x_{2}+iy_{2}}{\sqrt{2}}\). Obviously the Q function is just the expectation value of the density operator in coherent state, so it is always positive. Substituting the density operator of SPA-TMTS state in Eq. (8) into Eq. (10), we have

$$ Q_{ab}(\alpha;\beta)=\frac{\kappa^{N+2}}{2\pi^{2}N!}e^{-\kappa \vert \alpha \vert ^{2}-\kappa \vert \beta \vert ^{2}}\bigl \vert \alpha^{N}+\beta^{N}\bigr \vert ^{2}, $$
(11)

This function is always positive in all space, but it lost the Gaussian charter and exhibit the characteristic of entanglement due to the existence of the intercross term β N α ∗N+α N β ∗N.

Using Eq. (11), the Q function Q ab (α;β) of the SPA-TMTS are depicted in Fig. 1 in phase space (x 1,0;x 2,0) for several different values N=1,2,3 (from top to bottom) and \(\bar{n}=0,0.2,0.6\) (from left to right). It is easy to see that the Q function exhibits two peaks in phase space for N=1, while N>2, the Q function exhibits four peaks in phase space. The Q function for N=2 have higher symmetrical distribution than other cases. In addition, for a given values N, the distribution of WF becomes wider as the average photon number \(\bar{n}\) increase. From Eq. (11), we also find that the phase space center α=β=0 always has the form Q(0;0)=0 for any \(N,\bar{n}\), which can be seen clearly from Fig. 1.

Fig. 1
figure 1

The Q function Q ab (α;β) in phase space (x 1,0;x 2,0) for several different SPA-TMTS with the following cases N=1,2,3 (from top to bottom) \(\bar{n}=0,0.2,0.6\) (from left to right) (Color figure online)

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3.2 Wigner Function

The Wigner function (WF) is a quasiprobability distribution, which fully describes the state of a quantum system in phase space. The partial negativity of the WF is indeed a good indication of the highly nonclassical character of the state [43]. Next we derive the analytical expression of Wigner function for the SPA-TMTS for further investigating the nonclassicality.

The WF of a two-mode system ρ in the coherent state representation is given by

$$ W(\alpha;\beta)=\frac{4e^{2\vert \alpha \vert ^{2}+2\vert \beta \vert ^{2}}}{\pi^{2}}\int \frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}} \langle -z_{1},-z_{2}\vert \rho \vert z_{1},z_{2} \rangle e^{-2 ( z_{1}\alpha^{\ast}-z_{1}^{\ast}\alpha ) }e^{-2 ( z_{2}\beta^{\ast}-z_{2}^{\ast}\beta ) }, $$
(12)

where |z 1,z 2〉=|z 1〉 a |z 2〉 b is the two-mode coherent state and \(\alpha=\frac{x_{1}+iy_{1}}{\sqrt{2}},\beta=\frac{x_{2}+iy_{2}}{\sqrt{2}}\). Substituting the density operator in Eq. (8) into Eq. (12), we obtain the WF of SPA-TMTS as follow

(13)

where we have set \(\varkappa=1/ ( 2\bar{n}+1 )\), \(\varsigma= ( \bar{n}+1 ) / ( 2\bar{n}+1 ) \) and L N (x) is the N-order Laguerre polynomial. Equation (13) is just the analytical expression of the WF for the SPA-TMTS. It is obvious that the WF lost its Gaussian property in phase space due to the presence of a function of two Laguerre polynomials and an interference term.

Next we would like to discuss the changes in the WF W(α;β) of the SPA-TMTS as we vary the parameters N and \(\bar{n}\). Using Eq. (13), the WFs of the SPA-TMTS are depicted in Fig. 2 in phase space (x 1,0;x 2,0) for several different values N=1,2,3 (from top to bottom) and \(\bar{n}=0,0.2,0.6\) (from left to right). It is easy to see that the WF of the SPA-TMTS exhibits the nonclassicality due to the presence of partial negativity in phase space, which is evidently different from Fig. 1 (in Fig. 1 the Q function has no negative region). For a given values N, the distribution of WF becomes wider as the average photon number \(\bar{n}\) increase; while for the same values \(\bar{n}\), there are more wave valleys and peaks as N increasing. We also find from Eq. (13) that

$$ W(0;0)=\frac{4 ( -1 ) ^{N}}{\pi^{2} ( 2\bar{n}+1 ) ^{N+2}}=\frac{4 ( -1 ) ^{N}\varkappa^{N+2}}{\pi^{2}}, $$
(14)

at the phase space center, which can be seen clearly from Fig. 3. In particular, for the case N=1, L 1(x)=1−x, Eq. (13) reduces to

$$ W_{N=1}(\alpha;\beta)=\frac{4\varkappa^{3}}{\pi^{2}}\exp \bigl( -2\varkappa \vert \alpha \vert ^{2}-2\varkappa \vert \beta \vert ^{2} \bigr) \bigl( 2\varsigma \vert \alpha+\beta \vert ^{2}-1 \bigr) , $$
(15)

which implies that the WF always has its negative region at the phase space region \(\vert \alpha+\beta \vert ^{2}< ( 2\bar{n}+1 ) / ( 2\bar{n}+2 )\) (see Fig. 2(a)–(c)).

Fig. 2
figure 2

The Wigner function W(α;β) in phase space (x 1,0;x 2,0) for several different SPA-TMTS with the following cases N=1,2,3, (from top to bottom) \(\bar{n}=0,0.2,0.6\) (from left to right) (Color figure online)

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Fig. 3
figure 3

Three-dimensional region \((J,\varphi,\bar{n})\) for CH<−1 with (a) N=1. (b) N=2 (c) N=3 (d) N=4 (Color figure online)

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4 Violation of Bell Inequalities for the SPA-TMTS

In this section, we turn our attention to the nonlocal properties [44] of the SPA-TMTS in terms of the Bell’s inequality and demonstrate how nonlocality of the SPA-TMTS is revealed by the Q function and the Wigner function. Here we construct Clauser-Horne (CH) inequality from Q function and CHSH inequality from Wigner function.

4.1 CH Inequality

The Q function contains direct information on nonlocal quantum correlations. In order to link the quantum nonlocality with the phase space quasidistribution, we redefine \(\bar{Q}_{ab}(\alpha;\beta)=\pi^{2}Q_{ab}(\alpha;\beta)\) for two mode and \(\bar{Q}_{a}(\alpha)=\pi Q_{a}(\alpha)\) (\(\bar{Q}_{b}(\beta)=\pi Q_{b}(\beta)\)) for one mode. The one-mode Q functions in two-mode quantum state are given by

$$ Q_{a}\bigl(\alpha,\alpha^{\ast}\bigr)=\frac{1}{\pi} \langle \alpha \vert \operatorname{tr}_{b}\rho \vert \alpha \rangle ,\qquad Q_{b}\bigl(\beta ,\beta^{\ast}\bigr)= \frac{1}{\pi} \langle \beta \vert \operatorname{tr}_{a}\rho \vert \beta \rangle . $$
(16)

For the SPA-TMTS, using Eq. (8), we obtain

(17)
(18)

From the resulting three different functions we construct the CH combination

$$ \mathrm{CH}=\bar{Q}_{ab}(0;0)+\bar{Q}_{ab}(\alpha;0)+ \bar{Q}_{ab}(0;\beta)-\bar{Q}_{ab}(\alpha;\beta)- \bar{Q}_{a}(0)-\bar{Q}_{b}(0), $$
(19)

which for local theories satisfies the inequality −1≤CH≤0. If the CH combination given in Eq. (19) violate the inequality −1≤CH≤0, this immediately certifies the nonlocal properties of the quantum state. Substituting Eqs. (11), (17) and (18) into (19), we have the analytical expression of CH inequality for the SPA-TMTS as follow

$$ \mathrm{CH}=\frac{\kappa^{N+2}}{2N!}\vert \alpha \vert ^{2N}e^{-\kappa \vert \alpha \vert ^{2}}+ \frac{\kappa^{N+2}}{2N!}\vert \beta \vert ^{2N}e^{-\kappa \vert \beta \vert ^{2}}- \frac {\kappa^{N+2}}{2N!}\bigl \vert \alpha^{N}+\beta^{N}\bigr \vert ^{2}e^{-\kappa \vert \alpha \vert ^{2}-\kappa \vert \beta \vert ^{2}}-\kappa. $$
(20)

Thus we can say that the SPA-TMTS is quantum mechanically nonlocal as CH<−1, and the nonlocality is stronger with the increase of |CH|. Here we will take equal magnitudes of the coherent displacement |α|2=|β|2=J and a certain phase difference between them β=e 2iφ α, then

$$ \mathrm{CH}=\frac{\kappa^{N+2}}{N!}J^{N}e^{-\kappa J}-\frac{\kappa^{N+2}}{N!}J^{N}e^{-2\kappa J}\cos^{2}N\varphi-\kappa, $$
(21)

From Eq. (21) one can see that the degree of nonlocality not only depends on the coherent amplitude J, on the phases Ω, but also on the parameter \(N,\bar{n}\). In Fig. 3 we make a plot showing the three-dimensional region \((J,\varphi,\bar{n})\) in which CH<−1 is true for different N=1,2,3,4. It is found that the region of violation decrease as N increasing. Moreover, only for small \(\bar{n}\) and some N, there is the violation. In Table 1, we enumerate the minimum of CH value at the position \((J,\varphi,\bar{n})\) for different N. When N>5, the minimum limit to −1 showing no violation.

Table 1 The minimum CH values for several different SPA-TMTS
Full size table

In particular, when \(\bar{n}=0\), Eq. (21) just reduces to

$$ \mathrm{CH}=\frac{e^{-J}}{N!}J^{N}-\frac{J^{N}e^{-2J}}{N!}\cos^{2}N \varphi -1. $$
(22)

which correspond to the CH inequality of N00N state. As a special case, when \(\bar{n}=0\) and N=1, Eq. (22) reduces to

$$ \mathrm{CH}_{N=1}=Je^{-J}-2Je^{-2J}\cos^{2} \varphi-1. $$
(23)

In order to show the violation, we plot CH as the function of J and φ for different N00N states in Fig. 4. It is easy to see that the cases N=1 have some region under the lower bound CH<−1. As depicted in Fig. 4(a), we find that the strongest violation is obtained for φ=0, i.e. when the coherent displacement have same phases. When φ=0, there is some region imposed CH<−1 in the interval J∈[0,ln2] for N=1,2. This result violates the lower bound imposed by local theories (see Fig. 4(a)) when φ=π/2, the weak violation only happen for N=2 (see Fig. 4(b)).

Fig. 4
figure 4

The plot of the CH inequality defined in Eq. (22) as a function of the intensity of coherent displacements J=|α|2=|β|2, for different N00N state. The CH=−1 indicates the lower bound imposed by local theories (Color figure online)

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4.2 CHSH Inequality

Using correlations between parity measurements, we consider the combination by building the CHSH inequality defined by

$$ B=\frac{\pi^{2}}{4} \bigl( W(0;0)+W(\alpha;0)+W(0;\beta)-W(\alpha ;\beta) \bigr) , $$
(24)

for which local theories impose the bound −2≤B≤2. If the CHSH combination given in Eq. (24) violate the inequality −2≤B≤2, this immediately certifies the nonlocal properties of the quantum state. Next we shall determine how the CHSH inequality of the SPA-TMTS is violated as a function of \(N,\bar{n}\).

Substituting Eq. (13) into Eq. (24), we have

(25)

Again we will take equal magnitudes of the coherent displacement |α|2=|β|2=J and a certain phase difference between them β=e 2iφ α. Thus the CHSH inequality takes the following form

$$ B=\kappa^{N}\varkappa^{2} ( -\varsigma ) ^{N} \biggl[ 1+e^{-2\varkappa J} \bigl( L_{N} ( 4\varsigma J ) +1 \bigr) -e^{-4\varkappa J} \biggl( L_{N} ( 4\varsigma J ) +\frac{ ( -4\varsigma ) ^{N}J^{N}}{N!} \cos2N\varphi \biggr) \biggr] . $$
(26)

Thus we can say that the SPA-TMTS is quantum mechanically nonlocal as |B|>2, and the nonlocality is stronger with the increase of |B|. From Eq. (26) one can see that the degree of nonlocality not only depends on the coherent amplitude J, on the phase φ, but also on the parameter \(N,\bar{n}\). In Fig. 5 we make a plot showing the small three-dimensional region \((J,\varphi,\bar{n})\) in which B<−2 is true for N=1. Moreover, we find the minimum value is −2.1759 at \(\{ \bar{n}=2.3016\times10^{-8},J=0.100148,\varphi=0.0008134\}\) for N=1; while for any N>1 and any \(\bar{n}\), there are no chance to find violation.

Fig. 5
figure 5

(a) Three-dimensional region \((J,\varphi,\bar{n})\) for B<−2 with N=1 and section of violation of (a). (b) φ=0.0008134, N=1, (c) J=0.100148, N=1, (d) \(\bar{n}=2.3016\times10^{-8}\), N=1 (Color figure online)

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In particular, when \(\bar{n}=0\), Eq. (26) just reduces to

$$ B= ( -1 ) ^{N}+e^{-2J} ( -1 ) ^{N} \bigl( L_{N} ( 4J ) +1 \bigr) -e^{-4J}\biggl[ ( -1 ) ^{N}L_{N} ( 4J ) +\frac{4^{N}J^{N}}{N!}\cos2N\varphi\biggr], $$
(27)

which correspond to the CHSH inequality of N00N state. In order to show the violation, we plot CHSH for different N00N states as the function of J for different N when φ=0,π/2 according to Eq. (27) in Fig. 6. As a special case, when \(\bar{n}=0\) and N=1, Eq. (27) reduces to

$$ B_{N=1}=-1+ ( 4J-2 ) e^{-2J}- \bigl( 8J \cos^{2}\varphi-1 \bigr) e^{-4J}. $$
(28)

It is easy to see that only the cases N=1 have some region under the lower bound B=−2, which indicates the violation of CHSH inequality for 1001 state. As before, the strongest violation is obtained for φ=0, i.e. when the coherent displacements have same phases.

Fig. 6
figure 6

The plot of the CHSH inequality defined in Eq. (27) as a function of the intensity of coherent displacements J=|α|2=|β|2, for different N00N state. The B=−2 indicates the lower bound imposed by local theories (Color figure online)

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5 Conclusion

In this paper, we have demonstrated that phase-space quasidistribution functions, the Q function and the WF, carry explicit information on nonlocality of the entangled SPA-TMTS. By using the P-representation of thermal state, we obtain the normal ordering form of the SPA-TMTS. Then we derive the normalization coefficient, the Q-function and the WF. It is shown that the WF of the SPA-TMTS with N=1 can possess negative region when the condition \(\vert \alpha+\beta \vert ^{2}< ( 2\bar{n}+1 ) / ( 2\bar{n}+2 ) \) is hold. In addition, the nonlocality of the SPA-TMTS is discussed from CH and CHSH inequalities. It is found that for a given small mean thermal photon number \(\bar{n}\), the CH inequality is violated for some small N and the violation decrease with the increasing N. Therefore the SPA-TMTSs show EPR correlation for any finite N. While the CHSH inequality is violated only for N=1, which show that the test of the different Bell-type inequalities leads to a different result.