Maximizing the Validity of the Gaussian Approximation for the biphoton State from Parametric Downconversion

Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $sinc(LΔk/2)$-function, where $L$ is the length of the nonlinear medium and $Δk$ the phase mismatch between the pump and down-converted fields. In theoretical investigations, the \textit{sinc} behavior of the phase mismatch has often been approximated by a Gaussian function $\exp{(-αx^2)}$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $α$, for instance by comparing the widths of \textit{sinc} and Gaussian functions or the momentum of down-converted photons. As a consequence, different values for $α$ provide different theoretical predictions for the same setup. Therefore, an informed and unique choice of this parameter is necessary. In this work, we present a choice of $α$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called \textit{super}-Gaussian and \textit{cosine}-Gaussian approximations as practical alternatives with improved predictive power for experiments.

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