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

Authors: Neveen Sayed-Ahmed; M. M. Amein; Taghreed M. Jawa; Tahani A. Aloafi; F. S. Bayones; Azhari A. Elhag; J. Bouslimi
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

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 atomicQ 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.


---

## Body

## 1. 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 was studied [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 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 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: Section2 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.


## 2. 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.(1)H^ℏ=Ωc∧†c^+ω02σ^z+λc∧†σ^−+c^σ^+,where σ^+,σ^−, and σ^z are su(2) representation, and c^ and c∧† are the annihilation and creation operator, respectively. The frequency ω0 is the energy difference between the upper and the lower levels of the atom. The σ^± is connected with the inversion operator σ^z by the commutation relations
(2)σ^+,σ^−=σ^z,σ^z,σ^±=±2σ^z.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 operatorY^ is given by
(3)ddtY^=1iℏY^,H^+∂Y^∂t.Hence, the equation of motion forσ^z and c∧†c^ can be given as follows:
(4)dσ^zdT=2iλc^1†σ^−−c^σ^+,(5)dc∧†c^dT=−iλc∧†σ^−−c^σ^+,Thus, we deduce that the operator2N^=σ^z+2c∧†c^ is a constant of motion. Therefore, the Hamiltonian (1) becomes
(6)H^ℏ=ωN^−Δσ^z+λ1c^σ^++c∧†σ^−,whereΔ=1/2ω−ω0.The time evolution operator is given by(7)U^t=exp−iH^t=F^1n^,texp−iωn^t−iexp−iωn^tλE^1n^,tc^−iexp−iωn^−1tλE^2n^,tc∧†exp−iωn^−1tF¯^2n^,t,(8)F^jn^,t=cosμ^jn^t−iΔμ^jn^sinμ^jn^t,E^jn^,t=sinμ^jn^tμjn^,j=1,2,(9)μj2n^=Δ2+νjn^,j=1,2,(10)ν^1n^=λ2n^+1,ν^2=ν^1n^−1.Assume that the atom is initially in the excited state∣+>, and the field in a Binomial state η,M [18], where
(11)η,M=∑n=0MBη,M,nn,(12)Bη,M,n=Mnηn1−ηM−n,whereη 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,(13)ψAFt=a++b−,where
(14)a=∑n=0MBη,M,nF1n,tn,(15)b=−i∑n=0MBη,M,nexp−iωntν1nE1n,tn+1.


## 3. 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
(16)ρAt=TrFψAFtψAFt=a∣ab∣aa∣bb∣b,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,(17)SAT=−TrAρAtlnρAt=−∑j=12ejlnej,where e1,e2 are the eigenvalues of the atomic reduced density matrix, which are defined as follows,
(18)e1t=1+1−4a∣ab∣b−a∣b22,(19)e2t=1−1−4a∣ab∣b−a∣b22.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,
(20)QAt=12πθ,ϕρAtθ,ϕ,where θ,ϕ are the atomic phase space parameters and θ,ϕ is the atomic coherent state, which is defined as
(21)θ,ϕ=cosθ2++sinθ2exp−iϕ−.The expansion in terms ofθ and ϕ can be deciphered as follows:
(22)QAt=12π1+a∣a−b∣bcosθ+2Rea∣bsinθcosϕ+2Ima∣bsinθsinϕ,(23)0≤θ≤π,0≤ϕ≤2π.Another measure of classical correlation for quantum systems in phase space is the Wigner function [23, 24], which is given by
(24)Wx,y=1π∑k=0∞∑n,m=0∞−1kAntAm∗tGk,nΓGm,kΓ+BntBm∗tGk,n+1ΓGm+1,kΓ,(25)Ant=Bη,M,nFn,t,Bnt=−iBη,M,nexp−iωntν1nE1n,t,(26)Gk,nΓ=exp−Γ22∑j=0mink,n−Γk−jΓ∗n−jk!n!k−j!n−j!j!,(27)Γ=x+iy.


## 4. Statistical Study

### 4.1. Statistical Study and Discussions

It is worth to investigate the relationship betweenS,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 [25–30]. 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 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 parameterη of the binomial distribution is a random variable follows the standard uniform distribution in the form fη=1,0≤η≤1(2)
The parameterτ is a random variable follows a uniform distribution in the form fτ=1/50,1<τ<50,0≤τ≤50(3)
X has a standard normal distribution in the form fx=1/2πe−0.5x2,−∞<x<∞(4)
Y has a standard normal distribution in the form fy=1/2πe−0.5y2,−∞<y<∞(5)
The number of values which simulated was 5000 runs andM=5,20,50 and 100. Then, finding the maximum and minimum for S,Q, W, τd,ηd,θ,ϕ,x, and y for each value of M, where d=S,Q, and W. For instance, max ηS=0.0203785 when M=5 means that the value of η=0.0203785 when S was maximum and similarly for Q and W. Some graphs are obtained to see the behaviour of S,Q, and W graphically at the maximum and minimum points which are given at Tables 1–4Table 1
(M=5).
MaxMinMaxMinS0.6930130.0000226021ϕ136.362106.99τS7.0232531.4164W0.00553529-0.00140655ηS0.02037850.620389τW2.8796111.0223Q0.00277485.2415×10–6ηW0.9858840.99747τQ0.010009331.3941x0.28476-0.0930154ηQ0.2175740.00494274y0.79185-0.738397θ113.035128.731Table 2
(M=20).
MaxMinMaxMinS0.6929331.49224×10–7ϕ349.46440.5101τS9.0207747.1239W0.00552633-0.0006944ηS0.8116480.664846τW19.375632.0177Q0.00275695.47404×10–6ηW0.9994270.996774τQ37.688234.555x-0.2998-1.01915ηQ0.3525510.184801y0.8534620.785669θ75.449765.8915Table 3
(M=50).
MaxMinMaxMinS0.6931250.0000714152ϕ197.61564.7959τS37.963934.5578W0.005506932.45124×10–6ηS0.7459450.548057τW24.025434.5578Q0.002763720.0000900447ηW3.41675×10–60.548057τQ47.12656.23383x0.104398-1.05545ηQ0.1560820.0424479y0.1058260.345778θ131.806159.928Table 4
(M=100).
MaxMinMaxMinS0.6930610.0000714152ϕ205.4964.7959τS28.08234.5578W0.0052686–5.89211×10–8ηS0.3602090.548057τW40.78969.4861Q0.002766320.0000900447ηW0.2955080.646471τQ12.61246.23383x0.1133950.426285ηQ0.001991630.0424479y-0.3745660.37842θ113.02159.928Here, the results were obtained from the simulation method which are used to discuss the entanglement in the case ofM=50. For the data that estimated the maximum values of SAT, the entanglement between the cavity field and the qubit returns between maximum and minimum values 0,ln2; moreover, it repeats periodically. The function SAT reaches zero (pure state) only in two points, as evident from Figure 1. Whereas in the case of the minimum function SAT, the entanglement between the field and the qubits reaches a state of separation periodically, as seen in Figure 2.Figure 1
Maximum points for theS-function, M=50, 0≤τ≤50.Figure 2
Minimum points for theS-function, M=50, 0≤τ≤50.For the data that estimated the maximum values ofQAt, 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 QAt is improved and oscillations appeared regularly as observed in Figure 3. For the data that estimated the minimum values of QAt, 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.Figure 3
Maximum points for theQ-function, M=50, 0≤τ≤50.Figure 4
Minimum points for theQ-function, M=50, 0≤τ≤50.For the data that estimated the maximum values ofWx,y, in the case M=50, for the maximum data, the quasidistribution Wx,y has one-peak centered in the center (0,0). The nonclassical behaviour appears around the peak. The maximum values of the function Wx,y 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.Figure 5
Minimum points for thew-function, M=50.


### 4.2. Concluding Remarks

The aim of this study is to discover the behaviour of the three main functionsS,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.Figure 6
Maximum points for thew-function, M=50.


### 4.3. Remarks

(1)
TheS function always positive and less than ln2 and may take 0 by rounding, while the W function may take negative values(2)
The behaviour of the functionsS,Q, and W does not depend on the values of τ and η. A direct or an indirect relation between the specific functions and their parameters cannot be discovered(3)
A similar behaviour of theS,Q, and W does not observe. There is no remark recorded that the three functions have a maximum or minimum at the same time and at the same values(4)
The impact of increasingM is too low on S,Q, and W(5)
The upper limit ofQ function is ln1.002779≃0.0027748 and does not less than 0(6)
The upper limit ofW function is ln1.0055≃0.00553 and is not less than -0.00096(7)
Depending on simulation study, there is no relation betweenθ and ϕ and the behaviour of Q(8)
The signs ofx and y may be the same or different in many cases, and there are no relation between x and y and the behaviour of W(9)
In all cases ofM, the relation W<Q<S was verified


## 4.1. Statistical Study and Discussions

It is worth to investigate the relationship betweenS,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 [25–30]. 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 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 parameterη of the binomial distribution is a random variable follows the standard uniform distribution in the form fη=1,0≤η≤1(2)
The parameterτ is a random variable follows a uniform distribution in the form fτ=1/50,1<τ<50,0≤τ≤50(3)
X has a standard normal distribution in the form fx=1/2πe−0.5x2,−∞<x<∞(4)
Y has a standard normal distribution in the form fy=1/2πe−0.5y2,−∞<y<∞(5)
The number of values which simulated was 5000 runs andM=5,20,50 and 100. Then, finding the maximum and minimum for S,Q, W, τd,ηd,θ,ϕ,x, and y for each value of M, where d=S,Q, and W. For instance, max ηS=0.0203785 when M=5 means that the value of η=0.0203785 when S was maximum and similarly for Q and W. Some graphs are obtained to see the behaviour of S,Q, and W graphically at the maximum and minimum points which are given at Tables 1–4Table 1
(M=5).
MaxMinMaxMinS0.6930130.0000226021ϕ136.362106.99τS7.0232531.4164W0.00553529-0.00140655ηS0.02037850.620389τW2.8796111.0223Q0.00277485.2415×10–6ηW0.9858840.99747τQ0.010009331.3941x0.28476-0.0930154ηQ0.2175740.00494274y0.79185-0.738397θ113.035128.731Table 2
(M=20).
MaxMinMaxMinS0.6929331.49224×10–7ϕ349.46440.5101τS9.0207747.1239W0.00552633-0.0006944ηS0.8116480.664846τW19.375632.0177Q0.00275695.47404×10–6ηW0.9994270.996774τQ37.688234.555x-0.2998-1.01915ηQ0.3525510.184801y0.8534620.785669θ75.449765.8915Table 3
(M=50).
MaxMinMaxMinS0.6931250.0000714152ϕ197.61564.7959τS37.963934.5578W0.005506932.45124×10–6ηS0.7459450.548057τW24.025434.5578Q0.002763720.0000900447ηW3.41675×10–60.548057τQ47.12656.23383x0.104398-1.05545ηQ0.1560820.0424479y0.1058260.345778θ131.806159.928Table 4
(M=100).
MaxMinMaxMinS0.6930610.0000714152ϕ205.4964.7959τS28.08234.5578W0.0052686–5.89211×10–8ηS0.3602090.548057τW40.78969.4861Q0.002766320.0000900447ηW0.2955080.646471τQ12.61246.23383x0.1133950.426285ηQ0.001991630.0424479y-0.3745660.37842θ113.02159.928Here, the results were obtained from the simulation method which are used to discuss the entanglement in the case ofM=50. For the data that estimated the maximum values of SAT, the entanglement between the cavity field and the qubit returns between maximum and minimum values 0,ln2; moreover, it repeats periodically. The function SAT reaches zero (pure state) only in two points, as evident from Figure 1. Whereas in the case of the minimum function SAT, the entanglement between the field and the qubits reaches a state of separation periodically, as seen in Figure 2.Figure 1
Maximum points for theS-function, M=50, 0≤τ≤50.Figure 2
Minimum points for theS-function, M=50, 0≤τ≤50.For the data that estimated the maximum values ofQAt, 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 QAt is improved and oscillations appeared regularly as observed in Figure 3. For the data that estimated the minimum values of QAt, 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.Figure 3
Maximum points for theQ-function, M=50, 0≤τ≤50.Figure 4
Minimum points for theQ-function, M=50, 0≤τ≤50.For the data that estimated the maximum values ofWx,y, in the case M=50, for the maximum data, the quasidistribution Wx,y has one-peak centered in the center (0,0). The nonclassical behaviour appears around the peak. The maximum values of the function Wx,y 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.Figure 5
Minimum points for thew-function, M=50.


## 4.2. Concluding Remarks

The aim of this study is to discover the behaviour of the three main functionsS,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.Figure 6
Maximum points for thew-function, M=50.


## 4.3. Remarks

(1)
TheS function always positive and less than ln2 and may take 0 by rounding, while the W function may take negative values(2)
The behaviour of the functionsS,Q, and W does not depend on the values of τ and η. A direct or an indirect relation between the specific functions and their parameters cannot be discovered(3)
A similar behaviour of theS,Q, and W does not observe. There is no remark recorded that the three functions have a maximum or minimum at the same time and at the same values(4)
The impact of increasingM is too low on S,Q, and W(5)
The upper limit ofQ function is ln1.002779≃0.0027748 and does not less than 0(6)
The upper limit ofW function is ln1.0055≃0.00553 and is not less than -0.00096(7)
Depending on simulation study, there is no relation betweenθ and ϕ and the behaviour of Q(8)
The signs ofx and y may be the same or different in many cases, and there are no relation between x and y and the behaviour of W(9)
In all cases ofM, the relation W<Q<S was verified


## 5. 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 atomicQ and Wigner functions are used to identify the behaviour of the distribution in a phase space. 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 (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.



---

*Source: 5571796-2021-10-11.xml*