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We study time ordering corrections to the description of spontaneousparametric down-conversion (SPDC), four wave mixing (SFWM) and frequencyconversion (FC) using the Magnus expansion. Analytic approximations to theevolution operator that are unitary are obtained. They are Gaussian preserving,and allow us to understand order-by-order the effects of time ordering. We showthat the corrections due to time ordering vanish exactly if the phase matchingfunction is sufficiently broad. The calculation of the effects of time orderingon the joint spectral amplitude of the photons generated in spontaneous SPDCand SFWM are reduced to quadrature.

Effects of time ordering in quantum nonlinear optics