NON-CLASSICAL PROPERTIES AND GENERATION SCHEMES OF SUPERPOSITION OF MULTIPLE-PHOTON-ADDED TWO-MODE SQUEEZED VACUUM STATE

In this paper, we study some non-classical properties and propose the generation schemes of the superposition of multiple-photon-added two-mode squeezed vacuum state (SMPA-TMSVS). Based on the Wigner function, we clarify that this state is a non-Gaussian state, while the original twomode squeezed vacuum state (TMSVS) is a Gaussian state. Besides, the SMPA-TMSVS is sum squeezing, as well as difference squeezing. In particular, the manifestation of the sum squeezing and the difference squeezing in the SMPA-TMSVS becomes more pronounced when increasing parameters r and . In addition, by exploiting the schemes of photon-added superposition in the usual order, we give some schemes that the SMPA-TMSVS can be generated with the higher-order photon-added superposition by using some optical devices.


Introduction
Non-classicality of non-classical states plays a central role in the quantum tasks such as detecting gravitational waves via the LIGO interferometer by using squeezing light [1], producing single-photon resources by exploiting the anti-bunching effect [2], and implementing protocols in the quantum information by applying entangled sources [3,4]. To improve the effectiveness of these tasks, researchers have studied the methods that can enhance the degree of non-classicality of the non-classical states [5][6][7][8].
We know that the local photon addition operation to multimode states changes classical states to non-classical states [16] and increases the manifestation of squeezing and entanglement in original non-classical states [6,8,12,15,17].
We pay attention to the two-mode squeezed vacuum state (TMSVS) in the two-mode continuous variables regime [21]. This state is a Gaussian state with a high degree of non-6 classicality when the squeezed amplitude is large.
Therefore, it is suitable to implement quantum tasks such as quantum teleportation [3], quantum cryptography, quantum dense code, and quantum error correction [22]. Interestingly, this state was generated in the laboratory to perform the quantum teleportation process [23]. Finally, we briefly summarize the main results of the paper in the conclusions.

Wigner function
In terms of the Fock states, the TMSVS state [21] is given by where |n, nab = |na|nb is a normal product of two Fock states corresponding to two modes a and b, and the squeezed parameter r is real. From the TMSVS given in Eq. (1), the SMPA-TMSVS was recently defined in the form as [24] |r; h, kab = Nh,k(r)(a +h + b +k )|rab, where the normalized coefficient Nh,k is determined by In terms of the two-mode Fock states, the   The Wigner function is a quasi-probability used to describe the quantum state in the phase space. The negativity of the Wigner function is a witness of non-classicality and non-Gaussian characteristics of a non-classical state [25]. In the two-mode continuous variables regime, the Wigner function of a state with a density operator  is given in terms of the coherent states as where αa and αb receive the complex values, and |γa, γbab = |γaa|γbb denotes a two-mode coherent state. We use Eq.
According to the calculation method shown in [12], the Wigner function of the SMPA-TMSVS is determined as where 2F0 denotes the hypergeometric function.
We use the analytical expression in Eq. (10) to investigate the non-Gaussian and non-classical characteristics in the SMPATMSVS. In Fig. 1 This proves that the SMPA-TMSVS is a non-Gaussian as well as a non-classical state.

Sum squeezing
The squeezing property plays an essential role in high-accuracy experiments, such as detecting gravitational waves [1] and cooling materials [26].
The sum squeezing property is associated with the creation of light with the sum frequency at the output. The sum squeezing criterion was first introduced by Hillery [27]. Then, it was applied to detect squeezing in several states [6,7]. To imagine this criterion, for two arbitrary modes a and b, we consider an operator in the form where  is real. A sum squeezing factor is defined as ( ) where (V) 2  = V 2  -V 2 and Nx = x + x, x = a, b.
A state is called a sum squeezing if it satisfies S < Using Eq. (7), we obtain ( ) We use the analytical expression in Eq. (14) to In this case, factor S becomes more and more negative when parameter r is getting bigger and bigger.

Difference squeezing
The difference squeezing criterion was also proposed by Hillery [27]. Accordingly, let us consider another two-mode operator as A state is called a difference squeezing if D < 0. This condition was verified in the two-mode photon-added displaced squeezed states [6]. In the SMPA-TMSVS, the difference squeezing factor becomes.   However, the negativity of factor D increases when r becomes bigger. In addition, we note that the degree of difference squeezing achieves a maximum when  = m with integer m (see Fig. 4).
It should be noted that the original TMSVS does not exist the difference squeezing. Therefore, the photon-added superposition operation on the TMSVS plays an important role to emerge the difference squeezing in the proposed state.

Generation schemes
In order to generate the SMPA-TMSVS with the superposition of photon addition in the usual order and second order as a + + b + and a 2+ + b 2+ , several experimental schemes have been proposed [18,19]. When h = k > 2, using the scheme to generate the superposition xa + + yb + of Fiurasek [18], we apply this operation on the Therefore, at the output, the action of the superposition is written as (x2a + + y2b + )(x1a + + y1b + )(xa + + yb + )|rab By setting  = yy1y2/xx1x2, the state in Eq. (20) returns to the SMPATMSVS when h = k = 3. The arrangement of these physical resources is depicted in Fig. 6.
where  is the coupling strength of the PDC,  << 1. On the other hand, the transformation of the TONC is described by where  is proportional to the third-order By setting  = -/ (/), the state in Eq. (26) or (27)

Conclusions
In this paper, we investigate the Wigner function, Combining this scheme with that of Fiurasek, Lee and Nha, it is possible to generate the SMPA-TMSVS with a higher-order photon-added superposition.