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 (L Δ k /2 ) function, where L is the length of the nonlinear medium and Δ k 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(−α x²) 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.