Tavis–Cummings Model with Moving Atoms

In this work, we examine a nonlinear version of the Tavis–Cummings model for two two-level atoms interacting with a single-mode field within a cavity in the context of power-law potentials. We consider the effect of the particle position that depends on the velocity and acceleration, and the coupling parameter is supposed to be time-dependent. We examine the effect of velocity and acceleration on the dynamical behavior of some quantumness measures, namely as von Neumann entropy, concurrence and Mandel parameter. We have found that the entanglement of subsystem states and the photon statistics are largely dependent on the choice of the qubit motion and power-law exponent. The obtained results present potential applications for quantum information and optics with optimal conditions.


Introduction
When the Jaynes-Cummings (JC) model was firstly proposed in 1963 [1,2], its practical significance was not clear, as it prescribes the ideal situation of the resonant interaction of a two-level atom with an electromagnetic field. A complete solvable quantum model of a qubit in a single-mode field was studied to examine the classical properties of spontaneous emission and to detect the presence of Rabi oscillations in the atomic excitation potentials of fields of sharply defined energy. In the 1980s, due to technical progress, the importance of this model increased significantly, since many of its predictions were confirmed experimentally [3][4][5]. It is worth noting that even though the JC model is simple and easy to implement, it still exhibits many physical effects, such as squeezing [6], Rabi oscillations [7,8], revivals and collapses [9][10][11][12], qubit-field entanglement [13,14], antibunching [15,16], and nonclassical states such as Fock states [17,18] and Schrödinger cat states [19]. The JC model was originally designed to describe the interaction of a single atom with a single-mode field, so it can be applied to various physical scenarios such as flux qubits [20], Josephson junctions [21,22] and Cooper-pair boxes [23]. This model can also be used in solid-state

Physical Model and System Dynamics
The general expression of a one-dimensional PLP is introduced as [68] V(x, k) where V o and a describe the dimensions of energy and length, respectively. k is a positive real number known as the power-law exponent. These PLPs can be utilized to introduce a large class of quantum systems by a proper choice of the exponent k.
The Hamiltonian associated to PLPs has the following expression the corresponding eigenvalue equations are given bŷ H(k)|n = E n,k |n , n ≥ 0.
The Fock states |n are the eigenstates and E n,k are the corresponding eigenenergies. Substituting Equation (2) into Equation (3), we obtain where The eigenenergy spectrum E n,k can be obtained within the Wentzel-Kramers-Brillouin (WKB) approximation, such that where ±x o are the classical turning points. Here, g is the Maslov index, which accounts for the boundary effects at the classical turning points, E = V(x); we then have Using Equation (1) and Equation (5), Equation (6) can be written as This integral can be solved using the substitution, u = ( x a ) k with dx = a k u 1 k −1 du, and we have 2a where E Vo Therefore, the eigenenergy spectrum is given by where is the effective frequency.
The parameter k determines the type of potential. To gain insight into the structure of the energy spectrum given by Equation (11), we take into account the energy difference between levels Equation (12) shows that for k = 2, E (k) n is independent on n, so the energy spectrum is equally spaced. For the exponent k = 2, the level spacing varies with n. For k > 2, the energy difference increases with n (tightly binding potentials), whereas for k < 2, the energy between adjacent levels decreases with n (loosely binding potentials).
The CSs associated to PLPs are defined by [65,66] |n (13) where Annihilation and creation operators, A and A † , associated to PLPs act on the Fock state |n asÂ|n = N n−1 |n − 1 ,Â † |n = N n |n + 1 with N n = √ E n+1 − E 0 . Now, we investigate the time dependence of the two qubits that are coupled with a single-mode cavity field. Hence, the system Hamiltonian can be written as follows: The operators |e j (|g j ) (j = 1, 2) represents the excited (ground) state for the qubit. Whileâ † andâ denote the creation and annihilation operators, ω and Ω j , j = 1, 2 denote the frequencies of the cavity mode and the qubits, respectively, while ζ j (T) is the timedependent coupling between the field and the qubits. Some previous studies confirmed that the coupling parameter between the field and the qubits depends on the function of the wave number and the direction of the propagation (cos λp or sin λp), where λ denotes the wave number and p denotes the direction of propagator [69]. For a moving qubit, the direction of the propagation depends on the velocity and acceleration of the qubits as p = ϑT 2 + ϕT + c, where ϑ, ϕ and c are the acceleration, the velocity and phase coefficients, respectively [70]. Suppose that the coupling function takes the follows form, where ε (T), χ(T) and η j (T) are arbitrary functions that we will define later [71]. Suppose we introduce the scaled time operators, After we apply the scaled time, the above Hamiltonian (15) becomes, (e (i(χ(T)−ηj(T))) + e (−i(χ(T)−ηj(T))) ) where ω(T) = ω + ∂χ ∂T and Ω j (T) = Ω j + ∂η j ∂T . The interaction terms in (18) are separated into four quantities; the first two terms are |0 j 1 j | e (2i(χ(T)−ηj(T))) and |1 j 0 j | e (−2i(χ(T)−ηj(T))) , while the last two terms are the fast-varying terms, so it can be neglected without loss of generality. When we take χ(T) = η 1 (T) = η 2 (T) the Hamiltonian (18) becomes as followŝ In the exact resonance case, the above Hamiltonian can be rewritten as, where (t) = sin(ϑT 2 + ϕT + c) is the function that describes the time-dependent interaction. We consider that the two qubits to be initially in the Bell state and the field takes associated with the state The value r = 0 corresponds to the case of CSs for PLPs and r = 1 to for even cat states. The wave function of the proposed system at T = εt > 0 can be formulated as with the initial condition where The two qubits density matrix can be determined by calculating the trace over the field basis asρ where the diagonal elements of the two atoms density matrix are given by while the off-diagonal elements satisfy ρ ij = ρ * ji and have the form According to Equation (26), we are able to study the time evolution of the qubits-field entanglement and qubit-qubit entanglement. Moreover, we use the Mandel's parameter to discuss the quantum statistics of the field.

Qubits-Field Entanglement and Qubit-Qubit Entanglement
To quantify the entanglement between two qubits with the field, we use the von Neumann entropy, which is given by: This quantity can be evaluated through the eigenvalues of the density operatorρ AB .
To evaluate the amount of the qubit-qubit entanglement, we use the concurrence defined as in which Υ j are the eigenvalues given in decreasing order of R = ρ AB (σ y ⊗ σ y )ρ * AB (σ y ⊗ σ y ), where ρ * AB denotes the conjugate of ρ AB and σ y is the Pauli operator. When the two qubits are in separable state then C AB = 0. While C AB = 1 indicates that two-qubit is in a maximally entangled state.
In Figure 1, we display and analyze the entanglement between the two qubits and the field through the Equation (26), by setting the parameter ξ = √ 5 (see references [72,73]), and the other parameters (ϑ, ϕ, c) = (0, 0, π/2). For first case, the harmonic well potential (k = 2, g = 2) and a coherent state (r = 0), the entanglement function S N (T) grows gradually and reaches its peak on the revival periods, while the S N (T) achieves its lowest value on the collapse periods [74,75]. When setting the field in the even coherent state (r = 1), the entanglement increases from the start of the interaction and the intensity of the oscillations increases. Therefore, a strong entanglement arises between the field and the two qubits, and the function does not approach the minimum value (zero) during the interaction period. For the triangular well (k = 1, g = 3) and a coherent state (r = 0), the intensity of the oscillations decreases and the entanglement decreases slightly compared to the previous case. When the field is placed in the even coherent state (r = 1), clear fluctuations are generated between the maximum and minimum values with an increase in the amplitude of the oscillations. For the infinite barrier (k → ∞, g = 4) and a coherent state (r = 0), strong entanglement appears immediately after the beginning of the interaction with random oscillations. Moreover, the amplitude of the oscillations decreases and the S N (T) function does not approach minimum value (zero) during the interaction. When the even coherent state (r = 1) is considered, the entanglement is weakened slightly and the S N (T) function regularly reaches its minimal values. In Figure 2, we study the effect of time dependence on the entanglement between the cavity field and the qubits with the same conditions mentioned in the first Figure. In general, when taking the velosity and the acceleration into account (ϑ = ϕ = 1, c = 0), a strong entanglement arises and the S N (T) function never reaches a minimum value (zero) during the interaction period. When there is only an effect of the atomic speed (ϑ = c = 0, ϕ = 1), the behavior of the S N (T) is completely affected. For the first case, harmonic well potential (k = 2) and coherent state (r = 0), strong entanglement is generated at regular intervals during the interaction time and the S N (T) function reaching minimum values every π. When the SCSs for PLPs (r = 1) are considered, the entanglement is significantly enhanced. For the triangular well (k = 1) with the field state in a coherent state (r = 0), more entanglement increases periodically and the S N (T) function does not reach a minimum value (zero). For the field in SCSs for PLPs (r = 1), S N (T) function tends to attain zero value. For the infinite barrier (k → ∞) and the field state with r = 0, the amplitude of the oscillations of S N (T) decreases and the entanglement between the field and the two qubits increases. In the case of the field state with r = 1, the entanglement decreases slightly with increase in the amplitude of the oscillations at the smallest values of the function S N (T). In Figure 3, we display the plots the concurrence with the same previous conditions as in Figure 1, the concurrence is considered to study the entanglement between the two qubits. For the first case, the harmonic well potential (k = 2) with a coherent state (r = 0), the entanglement starts from the maximaly entangled Bell state, followed by partial entanglement until the function C AB (T) reaches the minimum values. The phenomena of sudden death and sudden birth are achieved in multiple inverals during the interaction time. The entanglement can be enhanced after preparing the field in the even coherent state (r = 1) and the periods of sudden death and sudden birth decrease. For the triangular well (k = 1) with a coherent state (r = 0), the intensity of the oscillations decreases and the entanglement between the qubits decreases slightly compared to the previous case. For the infinite barrier (k → ∞) with a coherent state r = 0, the entanglements are more powerful than the previous two cases, while the periods of sudden death and sudden birth are reduced. Moreover, the amount of entanglement between the two qubits increases and the intensity of oscillations decreases after setting the field in the even coherent state (r = 1).
In Figure 4, we study the effect of time dependence on the entanglement the qubits with the same conditions mentioned in the first figure. For the first case, k = 2 with r = 0 and ϑ = c = 0, ϕ = 1, the concurrence function C AB (T) fluctuates smoothly between the maximum and minimum values. Therefore, periods of sudden death and sudden birth are realized regularly. Note that the periods of sudden death and sudden birth increase after the inclusion of the even coherent state (r = 1). In the second case (the triangular well), the entanglement function behavior the saming as the first case (the harmonic well potential), with an increase in the intensity of fluctuations. For the infinite barrier, the entanglement increases significantly to the point where the phenomenon of sudden death and sudden birth disappear. Moreover, the entanglement function reaches maximum values for every time interval 2π. The phenomena of sudden death and sudden birth returns after preparing the field in the even coherent state (r = 1). For the second case of time dependence (ϑ = ϕ = 1 and c = 0), the entanglement can be enhanced for the fields with r = 0, especially in the harmonic well potential case. The entanglement decreases when preparing the field in the even coherent state (r = 1).

Photon Statistics
In order to deeper our understanding on the underlying physics of this system, we analyze the nonclassicality of the field through studying the Poissonian distribution of the photons. These properties are identified by the Mandel's parameter [76] The field photon statistics is classified according to the value of the parameter P M as P M > 0 and P M < 0 for the case of super-Poissonian and sub-Poissonian distribution, respectively. The Poissonian distribution is obtained in the case P M has zero value.
In Figure 5, we study the Mandel parameter to define regions of the nonclassical distribution of photons. For the first case, (k = 2 with r = 0), the Mandel parameter P M indicates a super-Poissonian distribution. The Mandel parameter gradually increases with increasing time, whether the field is set in the coherent or even coherent states.
The nonclassical distribution appears in the second case, k = 1, with r = 0, it is more pronounced when setting the field in the even coherent state (r = 1). In the third case, k → ∞ with r = 0, the nonclassical distribution completely disappears and the super-Poisson distribution appears again. In Figure 6, we examine the influence of the time dependence on the Mandel parameter. The time dependence strongly affects the Mandel parameter. In the case of time dependence (ϑ = c = 0, ϕ = 1), we consider the first case (k = 2, r = 0). The Mendel parameter oscillates in the positive parts (super-Poisson distribution) and the distribution never reaches the negative regions (sub-Poisson distribution). In the second case (k = 1, r = 0) the nonclassical distribution (sub-Poisson distribution) appears uniformly during the interaction period. Moreover, the nonclassical distribution decreases after the inclusion of the even coherent sate of the interaction cavity. While the distribution super-Poisson appears again in the third case (k → ∞, r = 0), and the sub-Poisson distribution completely disappears, whether the field is set in the coherent or even coherent states. When considering the coupling dependence on time (ϑ = ϕ = 1, c = 0), we note that the nonclassical distribution appears in the first (k = 2) and second (k = 1) cases, while disappears in the last case (k → ∞).

Effect of Qubit-Qubit Interaction
In this section, we examine the effect of qubit-qubit interaction parameter D within the interaction Hamiltonian The solution of the above Hamiltonian (32) is calculated numerically. In Figure 7, we show the effect of the qubit-qubit interaction term on the entanglement and the Mandel parameter. We Note that the von Neumann entropy does not affected by adding the qubitqubit interaction to the interaction cavity. Therefore, consider the concurrence to test the effect of the qubit-qubit interaction on the entanglement. In the first case (k = 2, r = 0), the amount of the entanglement between the two qubits gradually reduced with increasing time. The phenomena of sudden death and sudden birth increase. The Mandel parameter is not affected by the inclusion of the two qubits interacting together, so the Mandel parameter P M indicates that the super-Poisson distribution and the function P M does not reach the nonclassical distribution. In the second case (k = 1, r = 0), the entanglement C AB (T) decreases with increasing interaction time. Whereas in the third case (k → ∞, r = 0), the improvement in entanglement is more pronounced. In general, the Mandel parameter states that the distribution of photons is classical for all parameters values and the nonclassical distribution appears in short intervals for the second case (k = 1, r = 0).

Conclusions
In this work, we have introduced a nonlinear version of the Tavis-Cummings model for two two-level atoms interacting with a single-mode field within a cavity in the context of power-law potentials. We have considered the effect of the particle position inside the cavity field that depends on the velocity and acceleration, and the coupling parameter is supposed to be time-dependent. We have examined the effect of velocity and acceleration on the dynamical behavior of some quantumness and nonlocality measures namely; von Neumann entropy, concurrence and Mandel parameter. We have considered that the two qubits are initially prepared in the Bell states and the field from the generalized coherent states for power-law potentials state. We have examined the time evolution of entanglement of the cavity field and the two qubits through the von Neumann entropy, the entanglement between two the qubits together through the concurrence as well as the distribution of the photons through the Mandel parameter, for different potentials. We have shown in detail the dynamical behavior of the different measures for the cases of harmonic, triangular and infinite barrier potentials with respect to the main physical parameters in the absence and presence of the qubit motion. The obtained results present potential applications for quantum information and optics with optimal conditions.