HIGHER-ORDER NONCLASSICAL AND ENTANGLEMENT PROPERTIES IN PHOTON-ADDED TRIO COHERENT STATE

This paper studies the higher-order nonclassical and entanglement properties in the photonadded trio coherent state (PATCS). We use the criterion of higher-order single-mode antibunching to evaluate the role of the photon addition operation. Furthermore, the general criteria for detection of higher-order three-mode sum squeezing and entanglement features in the PATCS are also investigated. The results show that the photon addition operation to a trio coherent state can enhance the degree of both the higher-order single-mode antibunching and the higher-order three-mode sum squeezing and enlarge the value of the higher-order three-mode entanglement factor in the photon-added trio coherent state. In addition, the manifestation of the single-mode antibunching and the entanglement properties are more obvious with increasing the higher values of orders.


Introduction
The nonclassical and entanglement properties of the nonclassical states have been applied to the quantum tasks in the quantum optics and quantum information, such as using antibunching for the generation of the single-photon sources [1], squeezing for the detection of the gravitational waves in the LIGO interferometer [2], and exploiting the entanglement for the implementation of protocols in quantum teleportation [3] and quantum secret sharing [4].
Therefore, the study of nonclassicality and entanglement of nonclassical states is an important work in the discovery of the quantum optics. It has been known that a classical state (e.g., coherent or thermal state) is transformed into a nonclassical one by adding photons on it [5,6]. As a further development, the addition of photons on two-mode states was studied and investigated, such as the photon-added pair coherent states [7], the photon-added displaced squeezed states [8], and the photon-added squeezed vacuum state [9].
Thanks to the photon addition operation, quantum features in these states, for example, the degree of the squeezing and the entanglement behaviours were enhanced [8,9]. This is meaningful in the processes of quantum information and computation, e.g., improving the quantum key distribution protocol [10]. Keeping this in mind, we study the addition of photon to three-mode states.
Obviously, the three-mode states play a central role in the network tasks of quantum information, including controlled teleportation [11] and joint remote state preparation [12]. Therefore, enhancing the nonclassical and entanglement properties of these states will raise the effectiveness where  = re i with r and  being real, |n, n + p, n + p + q = |n|n + p|n + p + q is denoted as the threemode Fock state, and Np,q is the normalized factor of the TCS given by The TCS is defined as the right eigenstate simultaneously of operators i.e., satisfying equations is the bosonic creation (annihilation) operator of mode x, x = {a, b, c}. Some nonclassical properties of the TCS in both usual and higher orders were investigated in [13,14].
Therein, the single-mode squeezing, the two-mode squeezing, as well as the three-mode sum squeezing do not exist in such a state. Besides, an experimental scheme for the generation of the TCS has been introduced [15]. Therefore, the addition of photons to the TCS may be feasible by using the protocol of Zavatta et al. [6].
Recently, a photon-added trio coherent state (PATCS) has been introduced [16]. The PATCS is written as follows: where Np,q;h,k,l is the normalized factor; h, k, and l are non-negative integers, which are referred to the number of photons added. In terms of the Fock states, the PATCS is given by It is easy to know that the PATCS is reduced to the TCS if h = k = l = 0. In the PATCS, the quantum average of operators n h n p k n p q l n h i n p k i n p q l i (8) where ia, ib, and ic are non-negative integers, and X Some usual nonclassical and entanglement properties in the PATCS, such as the Wigner distribution function, the three-mode sum squeezing, and the three-mode entanglement, have been studied in detail in [16]. In this paper, we focus on the study of the higher-order nonclassical, as well as entanglement properties in the PATCS.
We investigate the higher-order single-mode antibunching property in Section 2. Section 3 presents the higher-order three-mode sum squeezing behaviours. Section 4 clarifies the higher-order entanglement characteristic. Finally, we briefly summarize the main results of the paper in the conclusions. Lee [18], then further extended by others [14].
According to An [14], the factor to determine the antibunching degree of mode x in higher-order i is given by  denotes the quantum average. A certain state exists with the higher-order single-mode antibunching (HOSMA) when Ax;i < 0; the more negative the Ax;i is, the larger the degree of HOSMA will be. Let us consider the PATCS, from Eq. (8), the factor measuring the degree of HOSMA in mode a is given as Similarly, with respect to mode b, we obtain For mode c, it is determined as We use the analytical expressions in Eqs.  3

Higher-order three-mode sum squeezing
Squeezing property was applied in numerous quantum tasks [19]. Various criteria for the detection of squeezing were introduced and investigated, such as sum squeezing, difference squeezing, single-mode squeezing, multimode squeezing, usual squeezing, and higher-order squeezing [8,14]. In this section, we define a generalized criterion for the detection of higherorder three-mode sum squeezing. Let us consider two orthogonal Hermitian operatorŝˆˆ, where ja, jb, and jc are non-negative integers. The above operators obey the commutative relation in which ˆˆˆ.
A state exists with higher-order three-mode Factor SX or SY also manifests the degree of higher-order three-mode sum squeezing. The more negative these factors are, the higher the squeezing degree will become. Note that in case jb = jc = 0 or ja = jb = jc, the above criteria correspond to the higherorder single-mode squeezing or the higher-order three-mode sum squeezing (HOTMSS) [14].
However, when ja = jb and jc = 0, they become the higher-order two-mode sum squeezing criteria [8].
In the PATCS, the inequalities in Eq. (16) are written as . Y Therefore, we expect that it will be revealed in . X We use the analytic expression in Eq. (19) to clarify the property of the HOTMSS in the PATCS (Figure 3). It is shown that the PATCS exists with the HOTMSS in any orders. In addition, the negativity of SX;j becomes more obvious when order j decreases or/and parameter r increases. It is not difficult to see that the HOTMSS disappears in the small region of r. The numerical investigation indicates that the larger the photon-added number is, the higher the degree of HOTMSS will become. For example, when p = q = 0, r = 8 and j = 2, the degree of HOTMSS approaches 7, 11, 12, and 13% corresponding to h = k = l = 1, 2, 3, and 4, respectively.
Note that if h + k + l is fixed, the degree of HOTMSS is the highest when h = k = l. For example, when h + k + l = 6, p = q = 0, r = 8 and j = 2, the degree of HOTMSS approaches 11, 10, and 9% corresponding to (h, k, l) = (2, 2, 2), (4, 1, 1), and (6, 0, 0), respectively. property has been studied for quantum tasks such as quantum teleportation, quantum cryptography, quantum dense code, and quantum error correction [20]. Quantum entanglement only exists in multimode states and is detected by some criteria, for example, the Hillery-Zubairy criterion [21], the Shchukin-Vogel criterion [22]. In addition, there are several criteria for the detection of the entanglement degree, such as the von Neumann entropy [23], the linear entropy [24], and the concurrence [25]. In the three-mode case, the Duc et al. criterion [26] in the form of the inequality is given as We define the factor of higher-order threemode entanglement (HOTME) as follows: We use the analytical expression in Eq. (24) to evaluate the property of the HOTME in the PATCS. In Figure 4

Conclusions
In this paper, we investigated the higher-order nonclassical and entanglement properties in the PATCS, including the higher-order single-mode antibunching, the higher-order three-mode sum squeezing, and the higher-order three-mode entanglement. If the order is fixed, the role of photon addition operation in the PATCS is clearly Funding statement