A unified description of sum frequency generation, parametric down conversion and two-photon fluorescence

A superoperator non-equilibrium Green's function formalism is presented for computing nonlinear optical processes involving any combination of classical and quantum optical modes. Closed correlation-function expressions based on superoperator time-ordering are derived for the combined effects of causal response and non-causal spontaneous fluctuations. Coherent three wave mixing (sum frequency generation and parametric down conversion) involving one and two quantum optical modes, respectively, are compared with their incoherent counterparts: two-photon-induced fluorescence and two-photon-emitted fluorescence.