Quantum statistics of parametric oscillation

We present a quantum-statistical analysis of the steady-state light fields in driven parametric oscillation in a cavity. Using the solution of a Fokker-Planck equation for the generalized $P$ representation of the signal and idler modes, we calculate the mean photon numbers, second-order correlation functions, and intermode correlation function as functions of the driving field. The generalized $P$ distribution describes the signal and idler modes' statistics over the whole range of driving-field strengths except in the region far below the oscillation threshold. The second-order correlation functions are found to violate the Cauchy-Schwartz inequality, a violation allowed because the $P$ distribution is complex. Squeezing is also found in a linear combination of the signal and idler fields.