Quantum-theoretical treatments of three-photon processes

We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive-P function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.