Four-wave mixing in all degrees of freedom

A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon-number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.