deletions | additions       diff --git a/Energies.tex b/Energies.tex  index 3ac0d89..8d22ec5 100644  --- a/Energies.tex  +++ b/Energies.tex 
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and neglecting the constant terms, the Hamiltonian becomes  \begin{equation}  H_{2q}=\hbar\omega_0 b^{\dagger}b - 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  \end{equation}  the eigenstates of this Hamiltonian are:  \begin{equation}\label{eqn:basis}  |N_{nm}nm>=\ddag{\omega_0^{-1}\gamma_{nm}}|N>|n>|m> |N_{nm}nm>=\ddag{\gamma_{nm}}|N>|n>|m>  \end{equation}  with $N\in\mathbb{N}$, $n,m=\{+,-\}$ and $\gamma_{nm}=n\Gamma_1+m\Gamma_2$ $\in\mathbb{R}$ are the eigenvalues of the operator $\hat{\gamma}$ on the 2-qubits basis $\{|nm>\}=\{|++>,|+->,|-+>,|-->\}$.  Negletting the constant terms the Hamiltonian (\ref{eqn:DHD}) becomes  \begin{equation}  H_{2q}=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma})\adag D^{\dagger}(\hat{\gamma})\adag  a D(\omega^{-1}_0 \hat{\gamma}) D(\hat{\gamma})  + \hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z - 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  \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 
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&=-\epsilon_{12}(nm)\delta_{sn}\delta_{tm} \nonumber  \end{align}  \end{equation}  where $\bar{n}$ is the negation of $n$, $H_1=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma})\adag a D(\omega^{-1}_0 \hat{\gamma})$, $H_2=\hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z$, $H_3=- 2\hbar\frac{\Gamma_1\Gamma_2}{\omega_0}\sigma^{(1)}_x\otimes\sigma^{(2)}_x$, $\epsilon_N=\hbar\omega_0 N$, $\epsilon_1=\hbar \frac{\omega_1}{2}$, $\epsilon_2= \hbar \frac{\omega_2}{2}$ and $\epsilon_{12}=2\hbar\frac{\Gamma_1\Gamma_2}{\omega_0}$. $\epsilon_{12}=2\hbar\Gamma_1\Gamma_2$.  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}