Maximizing the validity of the Gaussian approximation for the biphoton state from parametric down-conversion

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 is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the sinc behavior of the phase mismatch has often been approximated by a Gaussian function exp(−αx2) 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 sinc and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of α provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of α which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.