Photon statistics in parametric interaction of three waves

The parametric approximation is used in a statistical analysis of the frequency conversion of an optical field by a noncentrosymmetric crystal which couples an input wave of frequency $0<\omega<\omega_L/2$ to output waves of frequency $\vert\omega+n\omega_L\vert$ ($n=0,\pm1,\pm2,\dots$, $\omega_L$ is the pump frequency). An expression is obtained for the characteristic function of the output field which is described in terms of a matrix of the coupling coefficients in the case of Gaussian (displaced) statistics of the input field with independent modes. The mutual statistics of the output waves with $n\ge0$ and $n'<0$ is not Gaussian but corresponds, in the lowest approximation with respect to the pump wave and in the absence of the input field, to a spontaneous (direct or cascade) decay of pump photons into photon pairs. The intensity of the radiation represented by the $n\ge0$ modes is equal to the intensity of the radiation emerging from a classical converter which receives one photon in each mode with $n'<0$. The special case of three interacting modes with $n=-1,0,1$ (upconverter or light modulator) is considered in detail. The noise intensities and correlations between the numbers of photons in two and three modes are found for this case, and the spatial-temporal correlation function of the fields of the Stokes ($n'= -1$) and anti-Stokes ($n=1$) frequencies is determined.