## 4 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 \(\lambda_{i}\) of the density matrix of the total system \(\hat{\rho}(t)\) in Equation (13): \[E(t)=\sum_{i=1}^{\infty}\lambda_{i}\ \ln(\lambda_{i}).\] (20)
2. The partial entropies are used to measure the coherence loss [23; 59] of some partitions. Based on the eigenvalues \(\pi_{n}^{i}\) of the reduced density matrices of the EMF \(\rho^{Ms}(t)\) in Equation (15) and of the TLAS system \(\rho^{AB}(t)\) in Equation (16), the partial entropies are calculated as: \[E_{i}(t)=\sum_{n=1}^{\infty}\pi_{n}^{i}\ \ln(\pi_{n}^{i}),\ i=AB,M\!s.\] (21)  
The entropy functions \(E(t)\), \(E_{AB}\) and \(E_{M\!s}\) satisfy the Araki-Lieb (A-LI) inequality as [60]:  
\[|E_{AB}(t)-E_{M\!s}(t)|\leq E(t)\leq E_{AB}(t)+E_{M\!s}(t). \tag{22}\]  
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 \(E_{AB}=E_{M\!s}\), and they are used to quantify the entanglement/mixedness. If the dissipation term is considered, then the TLAS-EMF state is a mixed; therefore, \(E(t)\
eq 0\) and \(E_{AB}\
eq E_{M\!s}\), 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 \(E_{M\!s}\) 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 \(J/\lambda\) and \(\gamma/\lambda\) are not equal to zero. Figure 3a shows the dynamics of the \(E(t)\), \(E_{AB}\), and \(E_{M\!s}\) 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 \(E_{M\!s}\). The functions of \(E_{AB}\) and \(E_{M\!s}\) 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 \(E_{M\text{s}}\) until they reach their time-independent stationary values, corresponding to the TLAS, EMF, and TLAS-EMF systems' mixed states. The \(E_{M\text{s}}\) of the EMF system and its maximum values are greater than those of the TLAS. The amplitudes of the oscillations of \(E_{M\text{s}}\) 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 \(\gamma/\lambda\) leads to growth of the entire entropy that is greater than growth for the partial entropies \(E_{M\text{s}}\) and \(E_{AB}\). The entropy functions rapidly reach their stationary values.