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&=-\epsilon_{12}(nm)\delta_{sn}\delta_{tm} \nonumber  \end{align}  \end{equation}  where $\bar{n}$ is the negation of $n$, $H_1=\hbar\omega_0 D^{\dagger}(\hat{\gamma})\adag a D(\hat{\gamma})$, $H_2=\hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z$, $H_3=- 2\hbar\frac{\Gamma_1\Gamma_2}{\omega_0}\sigma^{(1)}_x\otimes\sigma^{(2)}_x$, 2\hbar\Gamma_1\Gamma_2\sigma^{(1)}_x\otimes\sigma^{(2)}_x$,  $\epsilon_N=\hbar\omega_0 N$, $\epsilon_1=\hbar \frac{\omega_1}{2}$, $\epsilon_2= \hbar \frac{\omega_2}{2}$ and $\epsilon_{12}=2\hbar\Gamma_1\Gamma_2$. One can easily identify two diagonal terms and an off-diagonal one. The overlap between two displaced number states $|N_{nm}>$ is  \begin{equation}\label{eqn:Lag}