Maximizing the validity of the Gaussian approximation for the biphoton state from parametric down-conversion
Baghdasar Baghdasaryan, Fabian Steinlechner, S. Fritzsche · Physical Review A · 2022
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 $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where $L$ is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the $\mathrm{sinc}$ behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of $\mathrm{sinc}$ and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$ 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.