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The overlap between two displaced number states $|N_{nm}>$ is
\begin{equation}
==e^{-2\alpha_{stnm}^2}(2\alpha_{stnm})^{(M-N)}\sqrt{\frac{N!}{M!}}L^{(M-N)}_N(4\alpha_{stnm}^2)\:(M\geq ==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}