Non-classical properties for SU(1, 1) pair coherent states

A new type of pair quantum states is introduced. Which can be considered as pair Barut-Girardello coherent states. It is an eigenstate of the operators K − ab = K − a K − b and K 3 a − K 3 b . We construct these eigenstates and generation scheme is proposed in terms of two mode state described in terms of su(1, 1) Lie algebra , We employ the second-order correlation function to discuss some non-classical properties,and violations of Cauchy-Schwarz inequalities. The phenomenon of squeezing is examined , squeezing is clear and Q-functions support that. Finally the phase distribution in the framework of an appropriate Pegg and Barnett formalism is considered and discussed.


Introduction
The SU(1, 1) Lie algebra has many applications in quantum optics because it can characterize many kinds of quantum optical systems [1][2][3][4]. In order to study many problems in this field, it has recently been used by many researchers to investigate the nonclassical properties of light in quantum optical systems [5][6][7]. In recent years there has been much interest in applications and generalizations of the Barut-Girardello coherent states(BG-CS) [8]. The BG-CS were introduced [9] as eigenstates of the lowering Weyl operator K − in the framework of SU(1, 1) Lie algebra symmetry. The BG-CS representation has been used for the explicit construction of squeezed states (SS) for the generators of the SU(1, 1) group which minimize the Schrödinger uncertainty relation for two observables [10]. Also, they are eigenstates of a general element of the complex field algebra [11]. The overcomplete families of eigenstates of elements of the Lie algebra were called algebraic CS [12] and algebra eigenstates [13]. In the present article, we suggest a new pair quantum state that depends on (BG-CS) idea. It is considered as a generalization of (BG-CS) and takes a different form from that considered earlier [14]. The organization of this paper is as follows. In the next section 2, we are going to find the state of two modes. A generation schema is considered in section 4. We introduce the probability distribution function in section 5. We devote section 6 to consider the Glauber second-order correlation function and normal squeezing. In section 7 we introduce and discus Q-function.We show the phase behavior by studying the phase space distribution function in section 8. The last section includes the conclusion 2. The SU(1, 1) pair coherent state Let us have two independent systems where operators are described by the generators of SU(1, 1) group. These generators are + - Let us define new operators which are given by the following } obey SU(1, 1) Lie algebra commutation relation [15] where ξ is an arbitrary complex number and q is a real number. The state can be decomposed over the orthonormal two mode state basis, The action of the operators -K ab and We assume that the eigenvalue q of the operator 3 is positive where = + -q m k n k 1 2 that is given through the condition of the state (2.2). The expansion of x ñ q k k , , , 1 2 | in the two-mode basis is composed of states of the form + + -ñ n q k k k n k , ; ,

|
Resolution of the identity(completeness) in terms of a certain set of states is very important because it allows the practical use of these states as bases in the Hilbert space. The problem here consists in finding a weight function σ (ξ) with x = q re i such that Then the weight function σ(ξ) is given by This completes requirements for the resolution of the identity

Generation scheme
It is to be mentioned that su (1, 1) Lie algebra can be realized in terms of boson annihilation and creation operators, where we can define  K i and K i 3 where i=a, b as follows Here, we are going to study a generation scheme of the state which is the eigenstate to = = --- ithin the framework of the motion of a trapped ion [18] in a two dimensional harmonic potential. Consider a single ion of mass M trapped in a two dimensional harmonic potential with frequencies ω 1 (in the x-direction), ω 2 (in the y-direction). In the rotating wave approximation the Hamiltonian of the system is written as The Hamiltonian (4.1) describes a two-level ion confined within a two dimensional trap that is approximated as harmonic oscillators of frequencies ω 1 and ω 2 The frequency ω 0 is the energy difference between the two levels of the atom. The σ + s -( ) and σ z are the raising (lowering) and phase operator , and represent the Pauli operator of the electronic two-level ion. The parameter μ is the dipole matrix element and k i , ( ¢ k i ) is the wavevector of ith driving the laser field of amplitude E 1 and E 2 . The position of the center-of-mass of the trapped ion is given by (x , yˆ) quantized by the operators â , â † and b , b † which the annihilation and creation operators of the vibrational motion of the center-of-mass of the ion. The quantized centre-of-mass position x and yˆcan be written as We may use a vibrational rotating wave approximation and neglect the terms with fast oscillations [19].
| |are the Rabi frequencies to the laser fields and η i is the Lamb-Dicke parameter, 3 is a constant of motion for the Hamiltonian (4.2). For small Lamb-Dicke parameters h = i 1, 1, 2 i  , one may consider lowest terms with m 1 =0=m 2 in the summation. Hence the Hamiltonian (4.2) can be approximated to the term between parentheses can be written as given by (2.2) and (2.4)

Probability distribution
A probability distribution P n for any quantum state yñ | is defined as For the state (2.4) x = á + + -ñ = ¼ P n n q k k n k k q k k n , ; , , , , , 0, 1, 2, .
( )is given in (2.4). To study the effect of both ξ , q, k 1 and k 2 on probability distribution function we plot P(n) against n , At figure 1(a) q=1,5,10 , ξ=20 , k 1 =1 and k 2 =2. We find that when q increases the maximum value of the probability distribution curve moves towards lower value of n.We not that at q=0 the distributional behavior like a Gaussin distribution. In figure 1(b) ξ=5,10,20 and k 1 =1 , k 2 =2 , q=5 we find that when ξ increases the maximum value of the probability distribution curve moves towards higher values of n. It is observed that at x = 5 the distributional behavior like a thermal distribution.

Second order correlation
To study the quantum statistical properties of any quantum state, we must pay attention to the nonclassical behavior such as (sub-Poissonian behavior [20]). So that we introduce the second-order correlation function [21,22], which leads to better understanding of the nonclassical behavior of quantum states [4,23], A state for which < g 1 . Therefore, we devote the present section to discussing this correlation function. This can be introduced for the SU(1, 1) group generated as follows In order to discuss the behavior of the correlation function , we calculate the expectation values of the quantities ( ) ( ) , + -K K i i at i=a for the first mode , i=b for the second mode. For the first mode = i a,the second order correlation function is . Figure 2(a) for first, figure 2(b) for second mode. We find that the state has nonclassical behavior at all values of q and ξ, as it may be expected from the form of the C n coefficient and their dependence on n

Cauchy-Schwarz inequality
We now consider violation of the Cauchy-Schwarz inequality between the single mode and cross-correlation second-order coherence functions. In the classical theory, this inequality can be expressed as In order to measure the deviation from the classical inequality , we define the quantity [4,24] x x As we can observe in figure 3 this function is always negative, which means that the inter-mode correlation is larger than the correlation between the same mode The strongest violations of the Cauchy-Schwarz inequality occurs at lower q for a fixed values of k 1 =1, k 2 =2.

Squeezing effect
Squeezing fluctuations are important in quantum measurement and communication theories [25]. In the SU(1, 1) Lie group [7],one can define two hermitian operators X and P as follows The uncertainty relation for these operators takes the form fluctuations in the X (or P) component are squeezed if the following condition is satisfied To measure the degree of squeezing, we define the following squeezing parameters, The squeezing condition can be expressed as S X <0 or S P <0, In figure 4 we note that state (2.4) achieves squeezing phenomenon in S P . It is to be observed that squeezing increases when q decreases at the fixed values k 1 =1, k 2 =2 as shown in figure 4

Q-function
It is well known that the quasiprobability distribution functions are important tools to give insight on the statistical description of quantum dynamics [26]. Therefore, we devote the present section to concentrate on one of these functions, that is the Husimi Q-function [27]. In fact the Q-function is not only a convenient tool to calculate expectation values of anti-normally ordered products of the operators, but also interpreted as a true phase space probability distribution. For the state (2.4) x ñ q k k , , , 1 2 | , we present the following definition for the Q-function of two modes as [28] ( ) |́́́| | Figure 3. I 0 as a function of ξ for different values of q, To study Q-function, we assume the subspace a b = + = x iy and by using some numerical computations we plot some figures. Details of the behavior can be seen when we plot the function against x and y for some different values of ξ and q where k 1 =1, k 2 =2 are fixed. This is shown in figure 5. We can observe in figure 5(a) for ξ=5 and q=5 that the graph of the Q-function shows two almost merged peaks of a squeezed Gaussian shape centered on the line y=0, By increasing of ξ=10 , q=5 the shape is divided into two peaks connected as shown in figure 5(b). More increase of ξ to 20 with q=5 two-peaks figure is shown clearly in figure 5(c). When we take q=0, 10,15 and fix ξ=20 ,the effect happens vice versa , at q=0, the Q-function shows two separated peaks , when q increases to q=10 two peak join near the bases while at q=15 the two peaks are almost merged as shown in figures 5(d), (e), (f) respectively. Therefore the Q-function is sensitive to the changes in the values of q and ξ for fixed values of k 1 , k 2

Phase distribution
To study the phase distribution of the state (2.4) we use the definition for the SU(1, 1) phase state [29]  where θ=θ 1 +θ 2 and C n (q, k 1 , k 2 ) is defined in (2.4). To show behavior of P(θ) we plot it for −π<θ<π and different values of ξ and q for fixed k 1 =1, k 2 =2. In figure 6(a) we take q=5 and plot P(θ) against θ, ξ. We note that phase distribution function appears as a peak centered at θ=0. No information for ξ=0 but as ξ increases P(θ) increase and information about phase starts to build up around θ=0. At figure 6(b) we take ξ=20 and plot P(θ) against θ, q we note that phase distribution function appears as a peak centered at θ=0 when q increases the pack of P(θ) decreases in height and information about phase decreases for the fixed values of ξ.

Conclusions
In this article we have introduced and examined some statistical properties of a new pair coherent quantum state of the SU(1, 1) algebra. A suggested generation scheme is presented based on the vibrational motion of the center of mass of a trapped ion in two-dimensional harmonic potential. The present scheme could be realized experimentally.We calculate and plot probability distribution function.Quantum statistical properties of these states have been studied in some detail. We have found interesting nonclassical features of these states. The sub-Poissonian distribution, Cauchy-Schwarz inequality valuation and squeezing phenomenon were displayed for these particular states for fixed parameter values.We studied the Q-function and showed its behavior for different parameters. Finally, we introduced phase distribution function. We note that this state has non-classical properties for squeezing ,phase distribution and these properties are sensitive to change in the parameters of the state ORCID iDs S Sanad https:/ /orcid.org/0000-0003-4385-5351 Figure 6. Phase distribution at = = k k 1, 2 1 2 .