Time ordering in spontaneous parametric down-conversion

Abstract

We consider the multi‐mode parametric down‐conversion Hamiltonian that governs the evolution of the fields inside a nonlinear crystal. If the Hamiltonian does not commute with itself at all times—as is the case for the multimode down‐conversion Hamiltonian—then the expansion of the evolution operator must take the form of the time‐ordered Dyson series, as opposed to the simplified Taylor series. By expanding the evolution operator to third order, the conditions under which the Taylor series is a valid approximation are revealed. In addition, some new and interesting behaviour is predicted