deletions | additions       diff --git a/Energies.tex b/Energies.tex  index 9e02441..68628c8 100644  --- a/Energies.tex  +++ b/Energies.tex 
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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  \begin{equation} \begin{equation}\label{eqn:Lag}  ==e^{-2\alpha_{stnm}^2}(2\alpha_{stnm})^{(M-N)}\sqrt{\frac{N!}{M!}}L^{(M-N)}_N(4\alpha_{stnm}^2)\;\;\;\;\;(M\geq N)  \end{equation}  where $\alpha_{stnm}=\omega_0(\gamma_{st}-\gamma_{nm})$. Therefore, one can easily see that for $s=n$ and $t=m$ the matrix element becomes $=\delta_{MN}$ (since $D(0)=\id$).  As \textit{Rebolini} done in his thesis work \cite{A_Rebolini} I perform the \textit{adiabatic approximation} that consist in discarding the far-from diagonal matrix elements, i.e. those therms for which $M\neq N$.  \begin{comment} Adiabatic approximation consists in discarding the far-from diagonal matrix  elements (NM for N 6= M). This approximation can be legitimized  mathematically by inspecting the structure of Laguerre polynomials (2.2.3),  that fall o as N M grows. But from a physical point of view it can be  interpreted in this way: in the limit !0 !, the qubit causes a much weaker  boost on the system dynamics than that caused by the oscillator; therefore,  only transitions between states with the same N are energetically favoured.\end{comment}  \begin{equation}  \begin{bmatrix}