## 6 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 \(\rho^{AB}(t)\) 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:  
\[(\rho^{AB}(t))^{T_{A}}=\left(\begin{array}{cccc}\rho_{11}^{AB}&\rho_{21}^{AB} &\rho_{13}^{AB}&\rho_{23}^{AB}\\ \rho_{12}^{AB}&\rho_{22}^{AB}&\rho_{14}^{AB}&\rho_{23}^{AB}\\ \rho_{31}^{AB}&\rho_{41}^{AB}&\rho_{33}^{AB}&\rho_{43}^{AB}\\ \rho_{32}^{AB}&\rho_{42}^{AB}&\rho_{34}^{AB}&\rho_{44}^{AB}\end{array}\right), \tag{25}\]  
where \(\rho_{ij}^{AB}(ij=1-4)\) are the elements of in Equation (16). Then the TLAS negativity is computed numerically using \(N_{Q}(t)=\sum_{i=1}^{4}|\lambda_{i}|\), where \(\lambda_{i}\) is the negative eigenvalue of the matrix \((\rho^{AB}(t))^{T_{A}}\).  
As shown in Figure 8, the effects of the rates of dipole-dipole interactions and dissipation term \(\gamma\in[0.0,\,0.03\lambda]\) 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 \(\lambda t\). 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 \(\gamma/\lambda\) 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 (\(J/\lambda=30\)). 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 \(J/\lambda=0\) in Figure 8a. The dipole-dipole rate \(J/\lambda\) 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 \((k,\alpha)=(1,5)\), the TLAS-EMF interactions lead to generate atomic entanglement periodically with \(\pi\)-period. Compared to the case of Figure 8a, the time intervals of the sudden appearance and disappearance of the entanglement are increased clearly. Figure 9 bshows 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].