deletions | additions       diff --git a/Energies.tex b/Energies.tex  index e621fc2..fc3452e 100644  --- a/Energies.tex  +++ b/Energies.tex 
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\end{equation}  Calculate now the matrix's elements in the basis of the eigenstates (\ref{eqn:basis}) in order to find later the eigenenergies of the system  %\begin{equation} \begin{equation}  \begin{align}  &=\epsilon_N\delta_{sn}\delta_{tm}\\  &=\epsilon_1(1-\delta_{sn})\delta_{tm}+\epsilon_2\delta_{sn}(1-\delta_{tm})\\  &=-\epsilon_{12}(nm)\delta_{sn}\delta_{tm}  \end{align}  % \label{eqn:Matrix_Elements}  %\end{equation} \end{equation}  where $\bar{n}$ is the negation of $n$, $H_1=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma})\adag a D(\omega^{-1}_0 \hat{\gamma})$, $H_2=\hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z$, $H_3=- 2\hbar\frac{\Gamma_1\Gamma_2}{\omega^2}\sigma^{(1)}_x\otimes\sigma^{(2)}_x$, $\epsilon_N=\hbar\omega_0 N$, $\epsilon_1=\hbar \frac{\omega_1}{2}$, $\epsilon_2= \hbar \frac{\omega_2}{2}$ and $\epsilon_{12}=2\hbar\frac{\Gamma_1\Gamma_2}{\omega_0^2}$. One can easily identify two diagonal terms and an off-diagonal one.  The overlap between two displaced number states $|N_{nm}>$ is