## 2 The Physical Model and Its Density Matrices
 \tag{5}\]  
Of note \(S_{k,n}=\sqrt{n(n+2k-1)}\), the connection between the Bargmann number and the field modes' photon number is \(k=\frac{1}{2}(I+\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b})\). In this study, we consider the intrinsic decoherence [41]. This decoherence is described by the Milburn equation, which is expressed as:  
\[\frac{d}{dt}\rho(t)=-i[\rho,H]-\frac{\gamma}{2}[H,[H,\rho]], \tag{6}\]  
where \(\gamma\) 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]:  
\[\rho(t) = \sum_{k=0}^{\infty}\frac{(\gamma t)^{k}}{k!}M^{k}(t)\rho(0)M^{ \mathrm{t}k}(t), \tag{7}\]  
where \(M^{k}(t)=H^{k}e^{-iHt}e^{-\frac{\gamma t}{2}H^{2}}\), and \(\rho(0)\) is the initial density system. Based on the completeness relation, \(\sum_{n}|\psi_{n}\langle\psi_{n}|=\hat{I  
The coefficients \(\chi_{lm}\) satisfy \(\hat{H}|\Psi_{l}^{n}\rangle=E_{l}|\Psi_{l}\rangle\), where \(E_{l}\) is the eigenvalue of \(\hat{H}\)  
\[E_{l} = J(1-l)+\omega(n+k+1),\quad for\ l=1,2,\] \[E_{3} = \tfrac{1}{2}\{J+2\omega(n+k+1)\] \[+ (-1)^{l}\sqrt{J^{2}+8\lambda^{2}(S_{k,n+1}^{2}+S_{k,n+2}^{2})}\},\quad for\ l=3,4.\]  
It is worth mentioning that the eigenvectors of the Hamiltonian system in Equation (4) can be written by another symmetric combination of \(|2\rangle\) and \(|3\rangle\)[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 \(\mathrm{su}(2)\) systems are initially in the upper state, \(\hat{\rho}^{AB}(0)=|1_{A}1_{B}\rangle\langle 1_{A}1_{B}|\), while the field system in the Barut-Girardello (BG) coherent states is [57],  
\[\hat{\rho}^{f}(0)=\sum_{n,m=0}^{\infty}\beta_{m}\beta_{n}^{*}\ |m,k\rangle \langle n,k|, \tag{11}\]  
with  
\[\beta_{m}=\frac{1}{\sqrt{n!\Gamma(2k+m)}}\sqrt{\frac{\alpha^{2m}|\alpha|^{2k- 1}}{L_{2k-1}(2|\alpha|)}}, \tag{12}\]  
where \(I_{v}(x)\) is the modified Bessel function  
\[\hat{\rho}(t)=\sum_{m,n=0}\sum_{s=1,3,4}\beta_{m}^{*}\beta_{n}\left\{\sum_{ \ell=1}^{4}(1-\delta_{\ell 2})\alpha^{\varepsilon\ell}|\Psi_{s}^{m}\rangle \langle\Psi_{\ell}^{n}|\right\} \tag{13}\]  
with  
\[\alpha^{mn}=A_{m1}^{m}A_{n1}^{n}e^{-i\lambda(E_{i}^{m,k}-E_{j}^{n,k})t-\gamma (E_{i}^{m,k}-E_{j}^{n,k})^{2}t}, \tag{14}\]  
The \(\mathrm{su}(1,1)\)-system's density matrix \(\rho^{Ms}(t)\) obtained by tracing over the TLAS \(A\) and \(B\) as:  
\[\rho^{Ms}(t)=\mathrm{tr}_{AB}\{\rho(t)\}=\sum_{i=1}^{4}\langle\varpi_{i}|\rho (t)|\varpi_{i}\rangle. \tag{15}\]  
where \(\{|\varpi_{1}\rangle=|1_{A}1_{B}\rangle,|\varpi_{2}\rangle=|1_{A}0_{B}\rangle, |\varpi_{3}\rangle=|0_{A}1_{B}\rangle,|\varpi_{4}\rangle=|0_{A}0_{B}\rangle\}\) is the TLAS space states. The reduced density matrix of the TLAS \(\rho^{AB}(t)\) is obtained by tracing the cavity field degrees that have space states \(\{|n,k\rangle\}(n=0,1,2,...,\infty)\) as:  
\[\rho^{AB}(t)=\mathrm{tr}_{Ms}\{\rho(t)\}=\left( \begin{array}{cccc}\rho_{11} ^{AB}&\rho_{12}^{AB}&\rho_{13}^{AB}&\rho_{14}^{AB}\\ \rho_{21}^{AB}&\rho_{22}^{AB}&\rho_{23}^{AB}&\rho_{24}^{AB}\\ \rho_{31}^{AB}&\rho_{32}^{AB}&\rho_{33}^{AB}&\rho_{34}^{AB}\\ \rho_{41}^{AB}&\rho_{42}^{AB}&\rho_{43}^{AB}&\rho_{44}^{AB}\end{array} \right), \tag{16}\]  
Figure 1b shows the influence of the dipole-dipole rate \(J/\lambda\) 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 \(J/\lambda\) and \(\gamma/\lambda\) 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.