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

TL;DR

This paper investigates the Gaussian approximation of the sinc function in SPDC processes, proposing an optimal parameter choice and exploring super-Gaussian and cosine-Gaussian alternatives to improve theoretical predictions of biphoton states.

Contribution

It introduces an optimal value for the Gaussian approximation parameter and compares alternative approximations to enhance modeling accuracy in SPDC.

Findings

Optimal Gaussian parameter $eta$ maximizes approximation validity.

Super-Gaussian and cosine-Gaussian models offer improved experimental predictions.

Provides guidelines for choosing approximation functions in quantum optics simulations.

Abstract

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 $sinc(L\Delta k/2)$-function, where $L$ is the length of the nonlinear medium and $\Delta k$ 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 $\exp{(-\alpha x^2)}$ 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 \textit{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 \textit{super}-Gaussian and \textit{cosine}-Gaussian approximations as practical alternatives with improved predictive power for experiments.