## 2 The Physical Model and Its Density Matrices  
Much experimental work has been done on quantum entanglement using photons. It is worth noting that the individual polarization states of the photons can be easily controlled and their quantum coherence can be maintained over several kilometers in optical fibers [48]. Therefore, photons cannot be held for long time, and many difficulties arise in controlling collective entanglement states even if the photons are confined to the same cavity. It is interesting to note that the parametric amplifier type for an interaction between two fields can be realized in a three-wave mixing parametric amplifier device [49], and in quantum signals with Josephson circuits [50]. Therefore in this manuscript, the physical system of two two-level atomic systems, \(A\) and \(B\), interacting with two nondegenerate modes of an EMF in an ideal cavity via four-photon transitions in the presence of dipole-dipole interactions is considered. The interaction that realizes nondegenerate parametric amplifier is an important nonlinear phenomenon [51; 52]. The Hamiltonian that describes the parametric amplification interactions is expressed as  
\[\hat{H}_{Modes}=\omega_{1}(\hat{a}^{\dagger}\hat{a}+\frac{1}{2})+\omega_{2}( \hat{b}^{\dagger}\hat{b}+\frac{1}{2})+\mu(\hat{a}\hat{b}+\hat{a}^{\dagger}\hat {b}^{\dagger}). \tag{1}\]  
where \(\hat{a}^{\dagger}(\hat{a})\) and \(\hat{b}^{\dagger}(\hat{b})\) represent the field mode operators with frequencies of \(\omega_{j}\) (\(j=1,2\)) and \(\mu\) is the linear coupling parameter between the waveguide. The model described by Equation (1) does not include any interactions with the two qubits. We posit that the two nondegenerate modes of the EMF in Equation (1) interact with the two coupled TLAS via dipole-dipole interactions. After applying rotating wave approximations, the model's Hamiltonian becomes:  
\[\hat{H} = \frac{1}{2}\sum_{i=1,2}(\omega_{i}+\omega\hat{v}_{i}^{z})+\omega _{1}\hat{a}^{\dagger}\hat{a}+\omega_{2}\hat{b}^{\dagger}\hat{b}+J(\sigma_{A}^ {+}\sigma_{B}^{-}+\sigma_{A}^{-}\sigma_{B}^{+}) \tag{2}\] \[\qquad\qquad+\sum_{\ell=A,B}\lambda(\hat{a}\hat{b}\sigma_{\ell}^ {+}+\hat{a}^{\dagger}\hat{b}^{\dagger}\sigma_{\ell}^{-}).\]  
where \(\omega\) represents the TLAS frequency and \(\hat{\sigma}_{\ell}^{\pm}\) and \(\hat{\sigma}_{\ell}^{z}\) are the Pauli matrices. \(\lambda\) is the coupling of the TLAS-EMF interactions and \(J\) is the coupling of the dipole-dipole interactions. If the two modesare identical, that is, \(\hat{a}=\hat{b}\), the degenerate two-photon transition of one field mode occurs in the Hamiltonian system in Equation (2) in the degenerate case. We set \(\omega_{1}=\omega_{2}=\frac{\omega}{2}\) and introduce the \(\mathrm{su}(1,1)\) generators \(\hat{K}_{\pm}\) and \(\hat{K}_{0}\) as follows:  
\[\hat{K}_{-}=\hat{a}\hat{b}=\hat{K}^{\dagger}{}_{+},\quad\hat{K}_{0}=\frac{1}{2 }(1+\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}), \tag{3}\]  
which satisfies the relationship \([\hat{K}_{0},\hat{K}_{\pm}]=\pm\hat{K}_{\pm}\) and \([\hat{K}_{-},\hat{K}_{+}]=2\hat{K}_{0}\). Therefore, \(\hat{K}_{+}\hat{K}_{-}=\hat{K}_{0}^{2}-(\hat{K}_{0}+\hat{K}^{2})\) is the Casimir operator and \(\hat{K}^{2}=\hat{K}_{0}^{2}-\frac{1}{2}(\hat{K}_{+}\hat{K}_{-}+\hat{K}_{-} \hat{K}_{+})=k(k-1)\hat{I}\) is the Bargmann number. The Hamiltonian of Equation (2) is governed by \(\mathrm{su}(1,1)\) and a \(\mathrm{su}(2)\) Lie algebra as:  
\[\hat{H}=\omega K_{o}+\sum_{\ell=A,B}\{\frac{\omega}{2}\hat{\sigma}_{\ell}^{z} +\lambda(K_{-}\sigma_{\ell}^{+}+K_{+}\sigma_{\ell}^{-})\}+J(\sigma_{A}^{+} \sigma_{B}^{-}+\sigma_{A}^{-}\sigma_{B}^{+}). \tag{4}\]  
Thus, the \(\mathrm{su}(1,1)\) operators in the number state representation induce the following operators:  
\[ \begin{array}{rcl}\hat{K}_{0}|n,k\rangle&=&(n+k)|n,k\rangle,\hat{K}^{2}|n,k \rangle=k(k-1)|n,k\rangle,\\ \hat{K}_{-}|n,k\rangle&=&S_{k,n}|n-1,k\rangle,\hat{K}_{+}|n,k\rangle=S_{k,n+1 }|n+1,k\rangle,\end{array}