Entanglement Control of Two-Level Atoms in Dissipative Cavities

: An open quantum bipartite system consisting of two independent two-level atoms interacting nonlinearly with a two-mode electromagnetic cavity ﬁeld is investigated by proposing a suitable non-Hermitian generalization of the Hamiltonian. The mathematical procedure of obtaining the corresponding wave function of the system is clearly given. Pancharatnam phase is studied to give a precise information about the required initial system state, which is related to artiﬁcial phase jumps, to control the degree of entanglement (DEM) and get the highest concurrence. We discuss the effect of time-variation coupling, and dissipation of both atoms and cavity. The effect of the time-variation function appears as frequency modulation (FM) effect in the radio waves. Concurrence rapidly reaches the disentangled state (death of entanglement) by increasing the effect of ﬁeld decay. On the contrary, the atomic decay has no effect.


Inroduction
Quantum systems promise enhanced capabilities in sensing, communications, and computing beyond what can be achieved with classical-based conventional technologies rather than quantum physics. Mathematical models are essential for analyzing these systems and building suitable quantum models from empirical data is an important research topic. In Dirac theory of radiation [1], he considered atoms and the radiation field with which they interact as a single system whose energy is represented by the frequency/energy of each atom solely, the frequency/energy of every mode of the applied laser field alone, and a small term is to the coupling energy between atoms and field modes. The interaction term is necessary if atoms and field modes are to affect each other. A simple model is that we consider a pendulum of resonant frequency ω 0 , which corresponds to an atom, and a vibrating string of resonant frequency ω 1 which corresponds to the radiation field. Jaynes-Cummings model (JCM) [2] is the first solvable analytical model to represent the atom-field interaction with experimental verification [3]. JCM has been subjected to intensive research in the last decades with many interesting phenomena explored [4][5][6][7]. The matter-field coupling term may be constant [8][9][10] or time-dependent [11,12], and that depends on the considered physical situation. Parametric coupling (QWIP), also packaged for high-speed operation and the resulting photocurrent was amplified and fed to a microwave spectrum analyzer, where the modulation amplitude was measured. In [47], the authors review the physical phenomena that arise when quantum mechanical energy levels are modulated in time. The dynamics resulting from changes in the transition frequency is a problem studied since the early days of quantum mechanics. It has been of constant interest both experimentally and theoretically, with the simple two-state model providing an inexhaustible source of novel concept. The choice of initial system parameters as we propose is related to the artificial phase jumps of Pancharatnam phase. Phase jumps are promising points such that they generate better entanglement degrees, and its successive repetition inside any system dynamics reflects a good sign of system capability to transfer information, as the geometric phases can be altered by changing the relative delay of the laser pulses [48].
Our work here is oriented around the interaction of an open quantum system of two independent two-level atoms with a quantization (non-classical) of electromagnetic field in a dissipative cavity in the multi-photon process. In Section 2, the considered physical scenario is introduced, the corresponding Hamiltonian is investigated, and the mathematical procedure for obtaining the solution of the wave equation is clearly given. In Section 3, we discuss the proposed technique to control the entanglement by properly choosing the initial values of the atomic state. Concurrence is also discussed to determine the effect of other parameters in the system. In Section 4, a brief conclusion and results are given.

Physical Scenario
The theoretical model as illustrated in Figure 1 can be written as a non-Hermitian Hamiltonian where Ω j is the associated frequency/energy of level j of the corresponding atom , with Γ j is the atomic corresponding decay rate, andσ ij = |i j|, (i, j = 1, 2) are the atomic-flip operators for |j → |i , they satisfy the commutation relation [σ ij ,σ αβ ] =σ iβ δ αj −σ αj δ iβ . ω j is the energy/frequency of the quantized electromagnetic cavity field mode j with a corresponding decay rate γ j andâ j (â † j ) is the annihilation (creation) operator for the field mode j, and they obey the commutation relation [â i ,â † j ] = δ ij . Here, we consider that Ω j >> Γ j , and ω j >> γ j [49].λ (t) is the time-dependent coupling torque of the matter-field interaction. It is more realistic to consider that the interaction intensity is not uniform, and in the following calculations we consider thatλ (t) = λ cos( t). κ refers to photon number process. To study the dynamics and properties of this model, we need to get the corresponding wave function |ψ(t) , which can be formulated in the following form, where A (n 1 ,n 2 ) m (t) (m = 1, 2, 3, 4) are functions of time and field modes, called the probability amplitudes. α m are field-dependent functions, and can be defined as follows, (3) Figure 1. Energy level diagram of two two-level atoms coupled to two-mode field in a dissipative cavity.
By applying the time-dependent Schrödinger equation to the system, we get the following coupled differential equations. The trigonometric function inλ (t) can be reformulated in an exponential form. There exist exponential terms with two different powers in the differential equations, e ±i(∆+ )t and e ±i(∆− )t . Approximately, we can ignore the counter oscillating terms e ±i(∆+ )t . This approximation is similar to the RWA and has been accepted physically for numerous models [29,50]. Therefore, the differential equations can be recomposed to be After using the method in [29], we get The solution of the coupled system in Equation (7) depends on the initial states of the system. We pay an attention for determining the formulas of probability amplitudes by preparing the atoms initially to be made in superposition of the states |ψ(0) atom = cos(θ) |e 1 , e 2 + e −iφ sin(θ) |g 1 , g 2 , where φ is the corresponding phase of the two states. If we take θ = 0, both atoms are in excited states. Whereas if we take θ = 1 2 π, the ground states are considered. The solution of this system is given by F (n 1 , n 2 , t) = exp − iM(n 1 , n 2 )t F (n 1 , n 2 , 0).
where M(n 1 , n 2 ) and F (n 1 , n 2 , t) take the following forms, This coupled system of differential equations can be solved numerically or analytically be using a method to get the exponential of the 4 × 4 matrix in Equation (5) as illustrated in [51] and applied in [52][53][54]. By applying Newton interpolation method [51] for getting the matrix exponential, which states that for a matrix A with eigenvalues λ j , (j = 1, 2, .., n), n is the dimension of the matrix, where I is the unitary matrix and the divided differences λ 1 , ..., λ j depend on t and defined recursively by Thus, by using the above method to e −iMt , where A = −iM and the eigenvalues E m (t) of A are defined in Equation (5), then with s = 1 2 where the divided differences are formulated as follows, After the derivation of the matrix exponential, we can calculate the formulas of the probability amplitudes of the wave function of the sytem. The atoms are initially in superposition of states, and the initial field is oriented in the coherent states. Then, the final form of the probability amplitudes be formulated to be  For simplicity, in the next calculations we consider λ 1 = λ 2 = λ, 1 = 2 = , Γ 1 = Γ 2 = Γ, and γ 1 = γ 2 = γ.

Pancharatnam Phase and Concurrence
We need to estimate a certain parameter for controlling the dynamics and entanglement of the system. A special attention is paid for the value of the initial latter phase parameter φ. To reach that goal, we investigate the evolution of Pancharatnam phase Φ(φ, t) = arg( ψ(φ, 0)|ψ(φ, t) ). To control the phase φ, we plot Φ(φ, t) vs. φ for three different values of the scaled time λt, as in Figure 2. The red, black, and blue curves are plotted for λt = π/3, π/4, and π/2, respectively. In the red curve, we note that there is a smooth evolution of the phase, whereas for the black and blue curves, they exhibit two artificial phase jumps for two different values of φ. The phase jump for the blue curve (λt = π/2) is repeated every period of π and in-between the jumps the evolution is semi-parabolic shaped and reflects a slow variation of the system. The two phase jumps of the black curve are repeated every ≈ 13 20 π and in-between the two jumps the variation is very slow, smooth and is separated by ≈ 7 20 π. Now, we can detect the dynamical behavior of the considered mutipartite system, by investigating the degree of entanglement (DEM) by using the concurrence measure, which was formulated as a convex measure to amount the DEM for two qubits in pure states by Wootters and Hills [55]. For two qubits in pure states, concurrence is C(t) = 2 (1 − Trˆ 2 ), whereˆ = |ψ(t) ψ(t)| is the reduced density operator. The definition of concurrence has been extended to include multiple qubits [56], and can be calculated generally by where ij are the elements of reduced density in matrix form. Figure 3 sketches the evolution of concurrence C(t) against the scaled time λt.
(a) = πλ, θ = 0. Figure 3. The evolution of concurrence C(t) vs. the scaled time, for the one-photon process (κ = 1), In Figure 3, we plot C(t) versus the scaled time by using the estimated initial value for the latter phase φ = π 2 , which is chosen due to the existence of the artificial phase jump at this value in the geometric phase in Figure 2. In Figure 3a, we set the atoms initially to be in excited (upper-most) states θ = 0, and = πλ, we note that DEM ≤ ln 2, which is less than the standard result in models initially prepared in superposition of states. In the next figures, we examine the results of considering superposition of atomic states. In Figure 3b, we set the coupling variation parameter = 0, and take three various values for the decay parameter γ of the field. We observe that, in the beginning of the interaction between the two atoms and the coherent field λt ≤ 3π, the effect of the decay parameter is not noticed and the concurrence curves are very similar, but at a drastic point of change, it differs dramatically, as we see that the black curve γ = 0, and then it fluctuates till reaching a stable case of concurrence to be ≥ 0.8; the red curve γ = (10 −4 λ) has a chaotic behavior, as in the beginning. It evolves to give a higher rate of concurrence compared with the absence of decay case (black curve), and after a sufficient time it decreases. The blue curve γ = (10 −3 λ) represents the system when concurrence rapidly reaches the disentangled state (death of entanglement). In Figure 3c, we set = πλ, and we note the effect of the oscillation in the matter-field coupling as proposed in the considered model. The effect of that function is clearly noted in the higher case of the decay rate (blue curve) as the interaction has become very weak and fluctuations affect the system evolution. The effect of the time variation function appears as the frequency modulation (FM) effect in the radio waves. FM is a method to encode information in a laser field by varying the instantaneous frequency of the coupling between matter and laser. Also, we note that the presence of or its absence, the system has reached a disentangled state in the same period of scaled time, but the evolution itself changes by the presence of . In Figure 4, by taking into consideration the effect of the decay in the atomic energy levels (Γ j ), the concurrence has not been affected.
In Figure 5, we display the evolution of Pancharatnam phase Φ(t) vs. the scaled time, for various values of the system decay parameters γ and Γ. Both curves approximately exhibit the same behavior and for λt ≤ 6π the phases exhibits a quick subsequent artificial phase jumps, then take a dominate saturation period till λt ≤ 16π, which is followed by a slow fluctuation that evolves to start another subsequent artificial phase jumps but less quick than the previous evolution.

Conclusions
The interaction between atoms with field of the system has been investigated by taking into consideration that cavity leaks energies of both atoms and field while the laser field couples the atom as a cosine wave function of time with a parameter in the multi-photon process. The RWA has been applied twice to approximate the interaction part of the system. By solving the coupled differential equations resulting by applying the time-dependent Schrödinger equation, we get the wave vector and the corresponding eigenenergies. To control the Degree of the Entanglement (DEM) of the system, we determine the initial latter phase by plotting the Pancharatnam phase for three different time points, and investigate the concurrence between the two atoms according to the best value of the latter phase. By increasing the effect of field decay parameter γ, the concurrence rapidly reaches to the disentangled state (death of entanglement). On the contrary, the atomic decay parameter Γ has no effect on the concurrence. The effect of the time-variation function appears as FM effect in the radio waves. FM is used to encode information between atom and field. The system reaches disentangled state in the same period of scaled time, but the evolution itself changes by the presence of . We note that for various values of the system decay parameters γ and Γ, the evolution of Pancharatnam phase in both curves approximately exhibits the same behavior.