Nonlinear Dynamics of a Cavity Containing a Two-Mode Coherent Field Interacting with Two-Level Atomic Systems

This study analytically explored two coupled two-level atomic systems (TLAS) as two qubits interacting with two modes of an electromagnetic field (EMF) cavity via two-photon transitions in the presence of dipole–dipole interactions between the atoms and intrinsic damping. Using special unitary su(1, 1) Lie algebra, the general solution of an intrinsic noise model is obtained when an EMF is initially in a generalized coherent state. We investigated the population inversion of two TLAS and the generated quantum coherence of some partitions (including the EMF, two TLAS, and TLAS–EMF). It is possible to generate quantum coherence (mixedness and entanglement) from the initial pure state. The robustness of the quantum coherence produced and the sudden appearance and disappearance of coherence depended not only on dipole–dipole coupling but also on the intrinsic noise rate. The growth of mixedness and entanglement may be enhanced by increasing dipole–dipole coupling, leading to more robustness against intrinsic noise.

Matter-cavity interactions are used to produce entangled states [26][27][28] and are important examples of the physical realization of quantum information processing (QIP) [29]. The Jaynes-Cummings model (JCM) was introduced [30] to describe a two-level atomic system (TLAS) coupled to a single-mode radiation field. This model has attracted considerable attention for multi-photon transitions and interactions between TLAS and multi-mode EMFs [21]. Increasing the number of atoms interacting with an electromagnetic field (EMF) is another promising direction. Two TLAS with different extensions interacting with one-mode or two-mode fields have been investigated [22,31].
Many generalizations depending on the number of atomic levels, such as 2, 3, and 4-level atoms interacting with different types of cavity fields, have been investigated [22]. The quantum correlation dynamics between their subsystems (atom-atom, atom-field, and field-field) have also been studied [32]. The Kerr medium effects on the dynamics of nonlocal correlations between the two atoms that interact with a single mode field have also been studied [33]. Two TLAS interacting with a single mode vacuum field in the presence of atom-field and atom-atom entanglements were investigated using quantum relative entropy [22].
In real physical models, effective dissipation degrades the purity and entanglement [34][35][36][37][38][39][40]. One cause of dissipation is intrinsic decoherence in which the quantum effects deteriorate as the quantum system evolves. Intrinsic decoherence was studied in several previous articles [41][42][43][44][45][46][47]. Thus, this work introduces a physical model wherein the su(1, 1)-system is initially assumed to have a generalized coherent state, and an analytical description is used to investigate the entire system's quantum coherence and its different partitions.
In this paper, the physical model is introduced in Section 2. The effects of the decay terms and dipole-dipole interactions on the atomic inversion of two atoms are studied in Section 3. The influences of the intrinsic dissipation and coupling rates between the two atoms on the purity are discussed in Section 4. In Sections 5 and 6, QE in different partitions are investigated by using negativity measurements. Section 7 concludes this paper.

The Physical Model and Its Density Matrices
Much experimental work has been done on quantum entanglement using photons. It is worth noting that the individual polarization states of the photons can be easily controlled and their quantum coherence can be maintained over several kilometers in optical fibers [48]. Therefore, photons cannot be held for long time, and many difficulties arise in controlling collective entanglement states even if the photons are confined to the same cavity. It is interesting to note that the parametric amplifier type for an interaction between two fields can be realized in a three-wave mixing parametric amplifier device [49], and in quantum signals with Josephson circuits [50]. Therefore in this manuscript, the physical system of two two-level atomic systems, A and B, interacting with two nondegenerate modes of an EMF in an ideal cavity via four-photon transitions in the presence of dipole-dipole interactions is considered. The interaction that realizes nondegenerate parametric amplifier is an important nonlinear phenomenon [51,52]. The Hamiltonian that describes the parametric amplification interactions is expressed asĤ whereâ † (â) andb † (b) represent the field mode operators with frequencies of ω j (j = 1, 2) and µ is the linear coupling parameter between the waveguide. The model described by Equation (1) does not include any interactions with the two qubits. We posit that the two nondegenerate modes of the EMF in Equation (1) interact with the two coupled TLAS via dipole-dipole interactions. After applying rotating wave approximations, the model's Hamiltonian becomes: where ω represents the TLAS frequency andσ ± andσ z are the Pauli matrices. λ is the coupling of the TLAS-EMF interactions and J is the coupling of the dipole-dipole interactions. If the two modes are identical, that is,â =b, the degenerate two-photon transition of one field mode occurs in the Hamiltonian system in Equation (2) in the degenerate case. We set ω 1 = ω 2 = ω 2 and introduce the su(1, 1) generatorsK ± andK 0 as follows: which satisfies the relationship [K 0 ,K ± ] = ±K ± and [K − ,K + ] = 2K 0 . Therefore, Î is the Bargmann number. The Hamiltonian of Equation (2) is governed by su(1, 1) and a su(2) Lie algebra as: Thus, the su(1, 1) operators in the number state representation induce the following operators: Of note S k,n = n(n + 2k − 1), the connection between the Bargmann number and the field modes' photon number is k = 1 2 (I +â †â −b †b ). In this study, we consider the intrinsic decoherence [41]. This decoherence is described by the Milburn equation, which is expressed as: where γ is the intrinsic decoherence parameter. This equation is used to develop the density matrix approach to the dissipative systems. The solution of the previous differential Equation (6) through the Milburn equation can expressed as [41,[53][54][55]: where M k (t) = H k e −iHt e − γt 2 H 2 , and ρ(0) is the initial density system. Based on the completeness relation, ∑ n |ψ n ψ n | =Î, of the eigenvectors of the Hamiltonian system in Equation (4) into Equation (7); the density matrix is given by where |ψ l are the eigenvectors of the Hamiltonian (4) and E l are their corresponding eigenvalues. This density matrix is used to study the dynamical character of the quantum phenomena for different initial states where the coupling system is initially in the pure state ρ(0) = |ϕ(0) ϕ(0)|. In the space states { |1 = |1 A 1 B , k, n , |2 = |1 A 0 B , k, n + 1 , |3 = |0 A 1 B , k, n + 1 , |4 = |0 A 0 B , k, n + 2 }, the eigenstates |Ψ l (i = 1, 2, 3, 4) are expressed as: Appl. Sci. 2020, 10, 7150

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The coefficients χ lm satisfyĤ|Ψ n l = E l |Ψ l , where E l is the eigenvalue ofĤ E l = J(1 − l) + ω(n + k + 1), f or l = 1, 2, It is worth mentioning that the eigenvectors of the Hamiltonian system in Equation (4) can be written by another symmetric combination of |2 and |3 [56] that will also be interesting for examining the behavior of the system when starting in an asymmetric correlated state.
We assume that the two su(2) systems are initially in the upper state,ρ AB (0) = |1 A 1 B 1 A 1 B |, while the field system in the Barut-Girardello (BG) coherent states is [57], with where I ν (x) is the modified Bessel function with The su(1, 1)-system's density matrix ρ Ms (t) obtained by tracing over the TLAS A and B as: where The reduced density matrix of the TLAS ρ AB (t) is obtained by tracing the cavity field degrees that have space states {|n, k }(n = 0, 1, 2, ..., ∞) as: Appl. Sci. 2020, 10, 7150 The density matrices in Equations (17) and (18) describe the time evolution of some quantum phenomena, such as population inversion, coherence loss, and entanglement.

The Population Inversion of the Two TLAS
We now consider the atomic population inversion and discuss the behavior of the collapse and revival phenomena. The mathematical formula of atomic population inversion is the difference between the probabilities of finding the atom in excited and ground states. In this system, the two atoms are identical, so we can study the population of only one of the atoms in the su(2) subsystems-for example, A based on its reduced density matrix ρ A (t) =Tr B ρ AB (t) . Therefore, the population inversion W(t) is expressed as: As shown in Figure 1, the behavior of the function W(t) is illustrated with a fixed parameter α = 5, k = 1/4, and the other rates vary; specifically, γ/λ and J/λ, which are related to the dissipation terms and dipole-dipole interactions, respectively. For example, for γ = 0 and J = 0, W(t) exhibits quasi-regular fluctuations between 1 and −1 around the zero value. The population inversion demonstrates periods of revivals and collapses similar to the two-photon JCM [58]. When the  Figure 1b shows the influence of the dipole-dipole rate J/λ in the absence of the dissipation terms. The function W(t) has irregular oscillations, and the amplitudes of the oscillations of W(t) decrease gradually as the time increases. When the influences of the rates of J/λ and γ/λ are combined, the dipole-dipole interaction inhibits the fast deterioration of the damped oscillatory behavior of the atomic population inversion, which appears due to the dissipation term. Figure 2a shows that the transfer of the energy between the upper atomic states and the lower states depends on the Bargmann number k = 1, where, with the increase of the Bargmann number k, the atomic population inversion is regular oscillatory behavior with a π-period. The atomic system returns to its initial excited states periodically at the points λt = nπ(n = 1, 2, 3, ...). The influences of the initial coherent field intensity α on the atomic population inversion dynamics are shown in Figure 2b for the case (k, α) = ( 1 4 , 1). We note that the increase of the initial coherent field intensity reduces the transfer process of the energy between the atomic states, and reduces the effect of the decoherence term. For the case (k, α) = ( 1 4 , 1), the dynamics of W(t) are more robust against intrinsic noise than those of the case of (k, α) = ( 1 4 , 5) of Figure 1a.

Coherence Loss Measures
The quantum coherence in the present system is due to: (1) The unitary TLAS-EMF interaction, in which the coherence is the QE between the atomic system and the cavity subsystems or is the coherence-loss for one of them. (2) The intrinsic decoherence leads to coherence-loss; in this case the coherence differs from the QE. Further, the effects of both the rates of dipole-dipole interactions and the decoherence on the quantum coherence were studied via the entire system's von Neumann entropy, the EMF, and the TLAS.
(1) The TLAS-EMF entropy is numerically calculated using the eigenvalues λ i of the density matrix of the total systemρ(t) in Equation (13): (2) The partial entropies are used to measure the coherence loss [23,59] of some partitions. Based on the eigenvalues π i n of the reduced density matrices of the EMF ρ Ms (t) in Equation (15) and of the TLAS system ρ AB (t) in Equation (16), the partial entropies are calculated as: The entropy functions E(t), E AB and E Ms satisfy the Araki-Lieb (A-LI) inequality as [60]: If the dissipation term is neglected, Equation (11) is governed only by the unitary interactions of the TLAS-EMF system, so the TLAS-EMF system's state is pure, E(t) = 0. According to the A-LI inequality, Equation (22) becomes E AB = E Ms , and they are used to quantify the entanglement/mixedness. If the dissipation term is considered, then the TLAS-EMF state is a mixed; therefore, E(t) = 0 and E AB = E Ms , and the partial entropies are used to measure only the mixedness in the subsystems.
The numerical results of the entropy functions were obtained by numerically computing the eigenvalues of the density matrices of Equations (7), (10), and (11), and using the entropy Formulas (14) and (15). As shown in Figures 3-5, the quantum coherence's growth is analyzed by the functions E(t), E AB and E Ms using the same initial parameters as Figure 1. Since the atomic inversion's revival periods decrease in the presence of the rates of the dipole-dipole interactions and the decoherence term, the TLAS-EMF system never reaches an initial pure state when the rates J/λ and γ/λ are not equal to zero. Figure 3a shows the dynamics of the E(t), E AB , and E Ms in the absence of the dipole-dipole interactions and dissipation term. The TLAS, EMF, and TLAS-EMF systems' states start from the initial pure states and evolve into mixed states. The entire entropy E(t) has small values compared with the values of E AB and E Ms . The functions of E AB and E Ms have approximately the same dynamic behavior and reach their minimal values at the middle point of the revival intervals, and the curves never reach their initial pure state, as shown in Figure 2a. Of note, the entropies gradually increase to their maximum values before and after the revival periods and then decrease to minimum values in the middle of the collapse periods.  Figure 3b,c shows that the dissipation term leads to a gradual increase in the E(t), E AB , and E Ms until they reach their time-independent stationary values, corresponding to the TLAS, EMF, and TLAS-EMF systems' mixed states. The E Ms of the EMF system and its maximum values are greater than those of the TLAS. The amplitudes of the oscillations of E Ms and E(t) increase as the interaction time increases-that is, the system and its subsystems never come close to their initial pure states. The increase in the rate γ/λ leads to growth of the entire entropy that is greater than growth for the partial entropies E Ms and E AB . The entropy functions rapidly reach their stationary values.
As shown in Figure 4, the functions E(t), E AB , and E Ms are plotted at a dipole-dipole coupling J = 30λ and the other parameters are the same as in Figure 3a,b. The entropy functions E AB and E Ms show rapid fluctuations with the interference between the patterns most of the time. For the small decay rate value γ = 0.01λ, the growth of the entire entropy E(t) is less than that of the subentropies compared to J/λ = 0 in Figure 3b. We deduce that the dipole-dipole rate increases the chaotic oscillatory behavior of the entropy functions, and the generated mixedness of the TLAS and EMF states is more than that of the TLAS-EMF state. Figure 5a shows the dynamics of the mixedness of the system and its subsystems for the case (k, α) = (1, 5). By comparing this case with the case (k, α) = ( 1 4 , 5) of Figure 3a, we find that the mixedness of the entropies is generated regular with π-period, where the system and its subsystems are in initial pure states at the times λt = nπ(n = 1, 2, 3, ...). The effects of the initial coherent field intensity α on the dynamics of the mixedness are shown in Figure 5a for the case (k, α) = ( 1 4 , 1). The small coherent field intensity leads to enhancing the generated mixedness and weakening the effect of the decoherence term.

The TLAS-EMF System's Negativity Entanglement
Entanglement between the EMF and TLAS is quantified via the negativity of the total density matrix ρ(t) [61]. It is calculated using the matrix ρ(t) T AB of the partial transposition of the density matrix ρ(t) in Equation (13) with respect to the TLAS states' space {| i }. The matrix elements ρ(t) T AB are expressed as: The negativity's close form is expressed as: where λ j represents the negative eigenvalues of the matrix ρ(t) T AB . If N(t) = 1, then the quantum state has maximal QE and it is zero for disentangled state; otherwise, the states have partial entanglement. The negativity will be calculated numerically by computing the negative eigenvalues of the matrix ρ(t) T AB . Figure 6, displays the effects of both the rates of intrinsic decoherence and the dipole-dipole interplay on the entanglement behavior between the EMF and TLAS with the same conditions using in the previous sections. In Figure 6a, N(t) is depicted when J = 0 has different dissipation rate values γ ∈ [0.0, 0.03λ]. At γ/λ = 0, N(t) has regular oscillatory behavior with a π-period. This behavior shows the ability of the unitary interactions between the EMF and TLAS to generate maximal and partial entanglement. The negativity N(t) gradually decreases to minimum values at the end of the period λt π = n (where n Z) and reaches maximum values before and after the center of the revival periods, as shown in Figures 1a and 6a. As the dissipation rate increases, the oscillations gradually decrease, and after a short time, the negativity damps the oscillatory behavior. The increase of γ/λ leads to a decrease in the entanglement until it reaches its time-independent stationary non-zero value, which is called the stationary entangled state. The EMF and TLAS rapidly reach their stationary entangled states by increasing the decoherence rate. Figure 6b displays the dynamic behavior of the TLAS-EMF entanglement with the rate of the dipole-dipole interactions J/λ = 30. For small values of γ/λ, N(t) exhibits more irregular oscillations. The maxima increased, whereas its minima shifted upward. This means that the dipole-dipole rate enhances the generated entanglement. We deduce that the effect of the rate J/λ is more pronounced due to the lesser development of the entanglement compared with the previous case J/λ = 0. In general, the dipole-dipole rate increases entanglement, delaying the appearance of stationary entanglement. Figure 7a illustrates that the regularity of the generated TLAS-EMF entanglement is enhanced under the effect of the Bargmann number for that case (k, α) = (1, 5). The states of the atomic system and the two-mode coherent field cavity are disentangled periodically at the times λt = nπ(n = 1, 2, 3, ...). By comparing Figures 6a and 7a, we find that the decoherence has the same effect on the dynamics of the generated TLAS-EMF entanglement. For the case where small initial coherent field intensity is considered, the generated TLAS-EMF entanglement can be enhanced compared to the case with large coherent field intensity.

Two-Atom System's Entanglement
In the following section, the negativity function N Q (t) is used to study the QE of the two TLAS via the state ρ AB (t) in Equation (16). To calculate the negativity of the two TLAS, we find the partial transposition of the matrix with respect to the subsystem that is expressed by the following matrix: where ρ AB ij (ij = 1 − 4) are the elements of in Equation (16). Then the TLAS negativity is computed numerically using N Q (t) = ∑ 4 i=1 |λ i |, where λ i is the negative eigenvalue of the matrix (ρ AB (t)) T A . As shown in Figure 8, the effects of the rates of dipole-dipole interactions and dissipation term γ ∈ [0.0, 0.03λ] on the TLAS entanglement are displayed with the same conditions used in the previous section. As demonstrated in Figure 8a, in the absence of the dissipation rate, the negativity oscillates between its extreme values with regular oscillatory behavior, which shows the unitary interaction's ability to generate TLAS entanglement. First, the negativity remains zero for a short period, until a particular time, when it suddenly increases from to its partial maximum value. After that, it decreases and suddenly disappears over a short period (the death period). The sudden appearance and disappearance of the entanglement [62][63][64] is repeated at further intervals of λt. The dissipative rate plays an important role in the negativity's dynamic behavior. At low dissipation rates, N Q (t) fluctuates with no death periods over the course of the interactions. Larger values γ/λ decrease the entanglement until it reaches its stationary state. Sudden appearance and disappearance does not occur.  Figure 8b, shows the behavior of the generated TLAS entanglement on the dipole-dipole rate when (J/λ = 30). At low dissipation rates, the negativity demonstrates chaotic oscillatory behavior and never reaches zero (the disentangled state), as shown in Figure 8b. After the damping term is taken into account, the negativity increases and its minimum values are greater than those of J/λ = 0 in Figure 8a. The dipole-dipole rate J/λ increases the amplitudes and the negativity's oscillations and also delays reaching stationary entanglement. Figure 9a confirms that the Bargmann number has the same the effect on the previous quantum phenomena. For the case (k, α) = (1, 5), the TLAS-EMF interactions lead to generate atomic entanglement periodically with π-period. Compared to the case of Figure 8a, the time intervals of the sudden appearance and disappearance of the entanglement are increased clearly. Figure 9b shows that the ability of the TLAS-EMF interactions to generate the entanglement is weakened with the small initial coherent field intensity. Our results have potential applications in quantum information, such as: Generation of mixedness and entanglement under the decoherence [29]. The maximally entangled state of the two-qubit system can be used to employ quantum teleportation [65,66] or quantum dense coding [67]. The generated stable, maximally mixed state can be used to realize quantum computation [68,69] and qubit-channel metrology [70].

Conclusions
In the present context, the coupled TLAS interacting with two nondegenerate modes of an electromagnetic field represented by su(1,1) Lie algebra were investigated. The intrinsic damping effect of an effective dissipation term was studied by solving the Milburn equation. We investigated the effects of the dissipation and dipole-dipole rates on some quantum effects, namely, the atomic population inversion, quantum coherence, mixedness, and entanglement of the entire EMF and TLAS states. The atomic inversion's revival and collapse periods depended on the dissipation and dipole-dipole rates. The dissipation and dipole-dipole rates may have controlled the behavior of some quantum effects that were measured using the negativity and von Neumann entropy. The oscillatory behavior of the different quantum effects and their stationary values were enhanced by the increase in the dipole-dipole rate. Through the results, we noticed clear effects of the dipole-dipole interaction and the decoherence on the phenomena of the collapses and revivals of the population inversion. Moreover, the oscillations decrease when increasing the dipole-dipole interaction and the function will fluctuate randomly. While increasing the decoherence, the amplitude of the oscillations decreases until it is erased after a short period of time. The entropies and negativity inspect the periodicity phenomenon and are generated mixedness and weaken the effect of the decoherence terms by adding the dipole-dipole interaction.
Author Contributions: The all authors contributed equally to the manuscript and typed, read, and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.