# Entanglement Control of Two-Level Atoms in Dissipative Cavities
## 2 Physical Scenario  
The theoretical model as illustrated in Figure 1 can be written as a non-Hermitian Hamiltonian  
\[ \begin{split}\hat{H}=\sum_{j=1}^{2}\sum_{\ell=1}^{2}& \frac{(-1)^{j+1}}{2}(\Omega_{j}^{\ell}-i\Gamma_{j}^{\ell})\vartheta_{ij}^{ \ell}+\sum_{j=1}^{2}(\omega_{j}-i\gamma_{j})\hat{a}_{j}^{\dagger}\hat{a}_{j}\\ &+\sum_{\ell=1}^{2}\bar{\lambda}_{\ell}(t)\big{(}\hat{a}_{\ell}^ {\kappa}\hat{\sigma}_{12}^{\ell}+\hat{a}_{\ell}^{\kappa}\hat{\sigma}_{21}^{ \ell}\big{)}.\end{split}  \tag{1}\]  
where \(\Omega_{j}^{\ell}\) is the associated frequency/energy of level \(j\) of the corresponding atom \(\ell\), with \(\Gamma_{j}^{\ell}\) is the atomic corresponding decay rate, and \(\hat{\sigma}_{ij}^{\ell}=|i\rangle\,\langle j|\), (\(i,j=1,2\)) are the atomic-flip operators for \(|j\rangle\rightarrow|i\rangle\), they satisfy the commutation relation \([\hat{\sigma}_{ij},\hat{\sigma}_{\kappa\beta}]=\hat{\sigma}_{i\beta}\hat{ \sigma}_{aj}-\hat{\sigma}_{\kappa j}\hat{\sigma}_{i\beta}\). \(\omega_{j}\) is the energy/frequency of the quantized electromagnetic cavity field mode \(j\) with a corresponding decay rate \(\gamma_{j}\) and \(\hat{a}_{j}\) (\(\hat{a}_{j}^{\dagger}\)) is the annihilation (creation) operator for the field mode \(j\), and they obey the commutation relation \([\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}\). Here, we consider that \(\Omega_{j}^{\ell}>>\Gamma_{j}^{\ell}\), and \(\omega_{j}^{\ell}>>\gamma_{j}^{\ell}\)[49]. \(\bar{\lambda}_{\ell}(t)\) is the time-dependent coupling torque of the matter-field interaction. It is more realistic to consider that the interaction intensity is not uniform, and in the following  
\[\alpha_{1} =\frac{1}{2}(\Omega_{1}^{1}+\Omega_{1}^{2})+n_{1}\omega_{1}+n_{2} \omega_{2}, \tag{3}\] \[\alpha_{2} =\frac{1}{2}(\Omega_{1}^{1}-\Omega_{2}^{2})+n_{1}\omega_{1}+(n_{2} +\kappa)\omega_{2},\] \[\alpha_{3} =\frac{1}{2}(\Omega_{1}^{2}-\Omega_{2}^{1})+(n_{1}+\kappa)\omega_ {1}+n_{2}\omega_{2},\] \[\alpha_{4} =\frac{-1}{2}(\Omega_{2}^{1}+\Omega_{2}^{2})+(n_{1}+\kappa)\omega _{1}+(n_{2}+\kappa)\omega_{2}.\]  
By applying the time-dependent Schrodinger equation to the system, we get the following coupled differential equations. The trigonometric function in \(\tilde{\lambda}_{\ell}(t)\) can be reformulated in an exponential form. There exist exponential terms with two different powers in the differential equations, \(e^{\pm i(\Delta+\omega)t}\) and \(e^{\pm i(\Delta-\omega)t}\). Approximately, we can ignore the counter oscillating terms \(e^{\pm i(\Delta+\omega)t}\). This approximation is similar to the RWA and has been accepted physically for numerous models [29; 50]. Therefore, the differential equations can be recomposed to be  
\[i\left[ \begin{array}{c}\dot{A}_{1}^{(n_{1},n_{2})}(t)\\ \dot{A}_{2}^{(n_{1},n_{2})}(t)\\ \dot{A}_{3}^{(n_{1},n_{2})}(t)\\ \dot{A}_{4}^{(n_{1},n_{2})}(t)\end{array} \right] =\left[ \begin{array}{cccc}k_{1}&g_{2}e^{i(\Delta-\omega)t}&g_{1}e ^{i(\Delta-\omega)t}&0\\ g_{2}e^{-i(\Delta-\omega)t}&k_{2}&0&g_{1}e^{i(\Delta-\omega)t}\\ g_{1}e^{-i(\Delta-\omega)t}&0&k_{3}&g_{2}e^{i(\Delta-\omega)t}\\ 0&g_{2}e^{-i(\Delta-\omega)t}&g_{1}e^{-i(\Delta-\omega)t}&k_{4}\end{array}  \right]\times \tag{4}\] \[\times\left[ \begin{array}{cccc}A_{1}^{(n_{1},n_{2})}(t)&A_{2}^{ (n_{1},n_{2})}(t)&A_{3}^{(n_{1},n_{2})}(t)&A_{4}^{(n_{1},n_{2})}(t)\end{array}  \right]^{T}\]  
After using the method in [29], we get  
\[i\left[ \begin{array}{c}\dot{B}_{1}^{(n_{1},n_{2})}(t)\\ \dot{B}_{2}^{(n_{1},n_{2})}(t)\\ \dot{B}_{3}^{(n_{1},n_{2})}(t)\\ \dot{B}_{4}^{(n_{1},n_{2})}(t)\end{array} \right] =\left[ \begin{array}{cccc}k_{1}&g_{2}&g_{1}&0\\ g_{2}&k_{2}&0&g_{1}\\ g_{1}&0&k_{3}&g_{2}\\ 0&g_{2}&g_{1}&k_{4}\end{array} \right]\left[ \begin{array}{c}B_{1}^{(n_{1},n_{ 2})}(t)\\ B_{2}^{(n_{1},n_{2})}(t)\\ B_{3}^{(n_{1},n_{2})}(t)\\ B_{4}^{(n_{1},n_{2})}(t)\end{array} \right] \tag{5}\]  
Figure 1: Energy level diagram of two two-level atoms coupled to two-mode field in a dissipative cavity.