## 5 The TLAS-EMF System's Negativity Entanglement  
Entanglement between the EMF and TLAS is quantified via the negativity of the total density matrix \(\rho(t)\)[61]. It is calculated using the matrix \(\rho(t)^{T_{AB}}\) of the partial transposition of the density matrix \(\rho(t)\) in Equation (13) with respect to the TLAS states' space \(\{|\omega_{i}\rangle\}\). The matrix elements \(\rho(t)^{T_{AB}}\) are expressed as:  
\[\langle\omega_{i},j|(\rho(t))^{T_{AB}}|\omega_{m},n\rangle=\langle\omega_{m},j |\rho(t)|\omega_{i},n\rangle. \tag{23}\]  
The negativity's close form is expressed as:  
\[N(t)=\sum_{j=1}^{4}|\lambda_{j}|, \tag{24}\]  
where \(\lambda_{j}\) represents the negative eigenvalues of the matrix \(\rho(t)^{T_{AB}}\). 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 \(\rho(t)^{T_{AB}}\).  
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 \(\gamma\in[0.0,\,0.03\lambda]\). At \(\gamma/\lambda=0\), \(N(t)\) has regular oscillatory behavior with a \(\pi\)-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 \(\frac{\lambda t}{\pi}=n\) (where \(n\varepsilon Z\)) and reaches maximum values before and after the center of the revival  
Figure 4: As Figure 1 (**a**,**b**) but for \(J/\lambda=30\).  
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 \(\gamma/\lambda\) 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 \(J/\lambda=30\). For small values of \(\gamma/\lambda\), \(N(t)\) 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 \(J/\lambda\) is more pronounced due to the lesser development of the entanglement compared with the previous case \(J/\lambda=0\). 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 \((k,\alpha)=(1,5)\). The states of the atomic system and the two-mode coherent field cavity are disentangled periodically at the times \(\lambda t=n\pi(n=1,2,3,...)\). 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.  
Figure 6: The \(N(t)\) for \(k=\frac{1}{4}\) and \(\alpha=5\) with different decoherence values of \(\gamma\): \(\gamma/\lambda=0\) (green solid curves), \(\gamma/\lambda=0.001\) (blue dashed curves), and \(\gamma/\lambda=0.03\) (red dashed dot curves). With \(J/\lambda=0\) in (**a**) and \(J/\lambda=30\) in (**b**).