Statistical Study of the Entanglement for Qubit Interacting with an Electromagnetic Field

A statistical method is applied to predict the behaviour of a quantum model consisting of a qubit interacting with a single-mode cavity field. The qubit is prepared in excited state while the field starts from the binomial distribution state. The wave function of the proposed model is obtained. A von Neumann entropy is used to investigate the behaviour of the entanglement between the field and the qubits. Moreover, the atomic Q and Wigner functions are used to identify the behaviour of the distribution in a phase space. The simulation method is used to estimate the parameters of the proposed model to reach the best results. A numerical study is performed to estimate the specific dependency of the binomial distribution state. The results of entanglement were compared with the atomic Q and Wigner functions. The results showed that there are many maximum values of entanglement periodically. The results also confirmed a correlation between von Neumann entropy, the atomic Q, and Wigner functions.


Introduction
The problem of field interaction with an atom or atoms has gained the attention of many researchers in the field of optics and quantum information. Through the previous literature, it becomes clear that the interaction model of a qubit with two levels with a single-mode of the optical field is the most easiest model for estimating properties that have physical applications [1]. Therefore, researchers have been interested in studying this model on a large scale in the direction of information and quantum optics [2]. Moreover, many generalizations of this model were appeared, especially in field formulas; therefore, the combinations of the field modes such as bimode as either a converter or amplifier were discussed [3,4]. Continuing the development of studies in the interaction of the field with the qubit, the effect of the interaction of two fields together on the interaction of the atom with the field was studied. The results showed nonclassical features in the phenomenon of collapse, revival, and entropy of squeezing [3,4]. On the other hand, a new type of generalization emerged from the Jaynes-Cummings model, which is the dependence of the coupling between the qubit and the field on time, which led to the results of increasing the correlation between the parts of the quantum system [5]. The theoretical efforts have been dealt with extensively by the experimental cavity of QED. Progress has been made in the quantum computation model, in which the quantum system is studied as a density operator or a matrix [6]. The study of the interaction of qubits with the single-mode field has received great interest from researchers. For example, the effect of stark displacement on the entanglement between quantum systems was investigated [7]. The analytical solution of the fractional Schrödinger equation was also used to describe the interaction between an atom with two levels and the placement of a single electromagnetic field inside a cavity [8]. Moreover, the effect of the interaction between the time-dependent field and a two-level atom with a single electromagnetic field     [9]. The effect of the decay on the interaction of a three-level atom with a multiphoton field in the presence of Kerr like medium was also studied [10,11]. In fact, the initial values of the parameters describing the quantum system are taken at random, while in this work, a statistical method is used to estimate the initial values that improve the entanglement between the parts of the system.
Undoubtedly, the entanglement of parts of quantum systems is an important measure for discovering the strength of the interaction between these systems. Therefore, attention has been given to many types of these measures to measure the degree of quantum entanglement between two systems. For example, the von Neumann entropy or linear entropy was used in the case of closed systems that start the

Advances in Mathematical Physics
interaction from a separate state [12]. Moreover, the qubit and the electromagnetic field were prepared in a separate or superposition states. They were followed by a partially entangled state during the interaction of the quantum system. Therefore, von Neumann entropy or linear entropy is the two main measures that mean the local classical entanglement process.
The process of controlling the degree of entanglement between two atoms with two thermal fields through the amount of decomposition between the atomic transition and the gaps is one of the most important problems in quantum optics [13]. Open system correlation cannot be measured with a standard scale such as entropy or von Neumann. Therefore, various attempts have been made to discover a new quantum estimator of correlation between parts of quantum systems [14,15]. It is worth noting that the initial state is strongly influencing the amount of entanglement between parts of a quantum system. The effect of the nondeformed pair coherent states on a quantum system was investigated [16]. The effect of the nondeformed pair coherent state on the field cavity was studied, where strong entanglement was established in most of the interaction periods between quantum systems, while the deformed state generated weak entanglement [17].
The main goal of this work is to predict the degree of entanglement between quantum systems containing a qubit     Advances in Mathematical Physics interaction with a single-mode cavity field. Statistical simulation is used to obtain the regions of entanglement strength and to obtain nonclassical behaviour by studying the Q and Wigner functions. The paper is arranged as follows: Section 2 contains the solution of the differential equations to obtain the general solution of the quantum system. Section 3 is devoted to discover the regions of entanglement between the parts of the system. Moreover, von Neumann entropy, the atomic Q -function, and W-function are examined. Some statistical characteristics are presented in Section 4. Finally, conclusion is drawn in Section 5.

Quantum System
Now, we assume that a two-level atom is injected into a cavity containing a medium described by Hamiltonian containing single-mode of cavity field and the two-level atom.
where b σ + ,b σ − , and b σ z are su (2) representation, andĉ and c∧ † are the annihilation and creation operator, respectively. The frequency ω 0 is the energy difference between the upper and the lower levels of the atom. The b σ ± is connected with the inversion operator b σ z by the commutation relations The Heisenberg equation of motion is used to obtain the constants of motion. Through it, the general solution for the proposed quantum system is obtained. The equation of motion for any dynamical operatorŶ is given by Hence, the equation of motion for b σ z and c∧ †ĉ can be given as follows: Thus, we deduce that the operator 2N = b σ z + 2c∧ †ĉ is a constant of motion. Therefore, the Hamiltonian (1) becomeŝ where Δ = 1/2ðω − ω 0 Þ: The time evolution operator is given by Assume that the atom is initially in the excited state |+ > , and the field in a Binomial state jη, Mi [18], where where η is the characteristic probability of having each photon occurring, and M is the maximum photon number present in the field. Therefore, the time-dependent wave function describing the proposed quantum system is given from the following relation, where

Nonclassical Properties
In this part, the values of the parameters of this system are estimated. This leads to the improvement of the values of the entanglement between the proposed system. There is no doubt that the use of a statistical method in estimating the parameters leads to better results than the use of random values for these parameters. In quantum information, systems of qubits that interact remarkably with photons (quantum electrodynamics) are a suitable environment for studying open quantum systems [19]. Aside from being a major candidate for quantum information manipulation, such quantum optical systems are of fundamental theoretical importance [20]. In particular, the experimental and theoretical study of quantum optical systems may yield insight into the links between mixedness and entanglement. The mixedness which is associated with the impurity of the quantum state is always measured by atomic entropy [21]. The reduced density operator is given by Here, the von Neumann formula is used to measure the entanglement between parts of a quantum system. Therefore, the von Neumann entropy is given by the following relation, where e 1 , e 2 are the eigenvalues of the atomic reduced density matrix, which are defined as follows, Recently, some measures have emerged that lead to measuring the entanglement between parts of quantum systems. One of the most well-known of these measures is Wehrl entropy [22], which is formulated on atomic Q-function. Therefore, the atomic Q-function is defined as follows, where θ, ϕ are the atomic phase space parameters and jθ, ϕi is the atomic coherent state, which is defined as The expansion in terms of θ and ϕ can be deciphered as follows: Another measure of classical correlation for quantum systems in phase space is the Wigner function [23,24], which is given by

Statistical Study and Discussions.
It is worth to investigate the relationship between S, Q, and W (i.e., when S is a maximum, is that would lead to Q to be maximum or minimum? Then what is the effect on the related parameters η, τ, and else). Some questions about the behaviour of equations (17), (23), and (24) and the related parameters in many cases have not a precise answer. Therefore, statistical studies are given in this subsection to give suitable answers in this point. Statistical studies are used frequently to address and solve many of research questions. Simulation studies are creating data with numerical techniques like Monte-Carlo simulation by pseudorandom sampling based on computer packages for conducting experiments on the computer [25][26][27][28][29][30]. A Monte-Carlo simulation is defined as using random sampling from probability distributions in the computer experiments. Many researchers have used the simulation studies in their research, among them [16,17]. The aim of simulation studies of this research is the ability to understand the behaviour of certain functions. This allows to consider some properties of the required equations (17), (23), and (24). The Monte-Carlo simulations are employed to investigate the behaviour of S, Q, and W functions and the related parameters by assuming that the parameters of 6 Advances in Mathematical Physics the S, Q, and W are random variables. Mathematica 10 program with 4 GB RAM and processor Core i7 is employed to generate samples for τ, η, θ, ϕ, x, y, and M and then finding the corresponding dependent variables S, Q, and W.
Depending on the interval of each variable, we assumed that (1) The parameter η of the binomial distribution is a random variable follows the standard uniform distribution in the form f ðηÞ = 1, 0 ≤ η ≤ 1 (2) The parameter τ is a random variable follows a uniform distribution in the form f ðτÞ = 1/50, 1 < τ < 50 , 0 ≤ τ ≤ 50  Tables 1-4 Here, the results were obtained from the simulation method which are used to discuss the entanglement in the case of M = 50. For the data that estimated the maximum values of S A ðTÞ, the entanglement between the cavity field and the qubit returns between maximum and minimum values ð0, ln ð2ÞÞ; moreover, it repeats periodically. The function S A ðTÞ reaches zero (pure state) only in two points, as evident from Figure 1. Whereas in the case of the minimum function S A ðTÞ, the entanglement between the field and the qubits reaches a state of separation periodically, as seen in Figure 2.
For the data that estimated the maximum values of Q A ðtÞ, in the case of M = 50, the atomic Q-function fluctuates chaotically, and the negative values are not shown of completely, see Figure 3. Moreover, the function Q A ðtÞ is improved and oscillations appeared regularly as observed in Figure 3. For the data that estimated the minimum values of Q A ðtÞ, in the case of M = 50, the oscillations have the chaotic behaviour. The intensity of the oscillations increases when M = 50 is taken into account, as shown in Figure 4.
For the data that estimated the maximum values of Wð x, yÞ, in the case M = 50, for the maximum data, the quasidistribution Wðx, yÞ has one-peak centered in the center (0,0). The nonclassical behaviour appears around the peak. The maximum values of the function Wðx, yÞ which decrease after the minimum values data are included. The previous peak was divided into a crater with multiple peaks at the edges. In addition, the negative values appear significantly as shown in Figure 5.

Concluding
Remarks. The aim of this study is to discover the behaviour of the three main functions S, Q, and W between each others and studying the impact of the related variables. A standard uniform distribution has used to generate values for η, τ, θ, ϕ, and the standard normal distribution has used to generate values for x and y. A random samples of 0 < τ < 50, 0 < η ≤ 1, 0 < θ < π, 0 < ϕ < 2π, −∞<x <∞ and −∞ < y < ∞ are generated and then S, Q, and W are estimated by using Mathematica 10 program to make a numerical study. Depending on the statistical study presented on the previous Section 4, the values of S and Q are always positive but W may take negative values. There is no affection of τ, η, θ, ϕ, x, y, and M on the behaviour of S, Q, and W functions. In addition, all cases of M, W < Q < S as seen in Tables 1-4 and Figures 1-6, the remarks on the Tables can be written as follows.

Conclusion
A statistical method is used to predict the behaviour of a quantum model consisting of a qubit interacting with a single-mode cavity field is addressed. The qubit is prepared in the excited state while the field is in the binomial distribution. The wave function of the proposed model is obtained and studied numerically. A von Neumann entropy is used to investigate the behaviour of the entanglement between the field and the qubits. Moreover, the atomic Q and Wigner functions are used to identify the behaviour of the distribution in a phase space. A numerical study is performed to 7 Advances in Mathematical Physics estimate the specific dependency of the binomial distribution state. The results of entanglement were compared with the atomic Q and Wigner functions (see Tables 1-4 and Figures 1-6). Finally, different values of the parameters τ, η , θ, ϕ, x, and y are simulated and then the maximum and minimum values of S, Q, and W are obtained. In brief, depending on the statistical study which was presented on Section 4, the values of S and Q are always positive but W may take negative values. Furthermore, there is no affection of τ, η, θ, ϕ, x, y, and M on the behaviour of S, Q, and W functions.

Data Availability
The data used are generated from the mathematical models in the article.

Conflicts of Interest
The authors have no conflicts of interest regarding the publication of the paper.