Unified space–time description of pulsed twin beams

This work provides a mathematical derivation of a quasi-stationary (QS) model for multimode parametric down-conversion (PDC), which was presented in Gatti <i>et al</i>. (Gatti <i>et al</i>., <i>Sci. Rep</i>. <b>13</b>, 16786) with heuristic arguments. Here, the model is derived from the 3D + 1 propagation equation of the quantum fields in a nonlinear crystal, and its approximations are discussed thoroughly. Thanks to its relative simplicity, and to the fact that it is valid in any gain regime, both at a quantum and classical level, it allows a unified description of disparate experimental observations conducted over the last 20 years, often described in the past by means of limited ad hoc models.This article is part of the theme issue 'The quantum theory of light'.