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 $sinc(L\mathrm{\Delta }k\mathrm{/}2)$-function, where $L$ is the length of the nonlinear medium and $\mathrm{\Delta }k$ the phase mismatch between the pump and down-converted fields. In theoretical investigations, the $\mathit{\text{sinc}}$ behavior of the phase mismatch has often been approximated by a Gaussian function $\exp (-\alpha x^2)$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\alpha$, for instance by comparing the widths of $\mathit{\text{sinc}}$ and Gaussian functions or the momentum of down-converted photons. As a consequence, different values for $\alpha$ 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 $\alpha$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called $\mathit{\text{super}}$-Gaussian and $\mathit{\text{cosine}}$-Gaussian approximations as practical alternatives with improved predictive power for experiments.