Theory of high-gain twin-beam generation in waveguides: from Maxwell's equations to efficient simulation

We provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM), and also include the effects of both self-phase modulation (SPM) of the pump, and of cross-phase modulation(XPM) of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labelled by a frequency. We provide a proof that these fields satisfy bonafide commutation relations, and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin-beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.