Critical phenomena and quantum statistics in the four-wave interaction

We develop a theory of intracavity four-wave interaction. As a result of such interaction, two photons of the pump fields produce a pair of photons of the cavity's signal mode with the same frequency. In addition to taking into account the parametric process of photon transformation and dissipation processes, we allow for the effects of phase self-modulation of the signal mode and its cross modulation due to the pump fields. We find an exact steady-state solution of the Fokker-Planck equation for the distribution function of the quasiprobability of the amplitude in the generalized P-representation of the density matrix. This solution is then used to calculate the moments of the signal-mode operators, the distribution function for the number of photons, and the Wigner function. The results are applicable to all oscillating modes, including the threshold and above-threshold regions and the bistable mode. We study critical phenomena and the quantum statistical effects of photon correlation, the squeezing of fluctuations of the quadrature component, and sub-Poisson statistics. We find, in particular, that the pair correlation of photons increases significantly in the threshold region with the bistable oscillating mode present. We also show that squeezing in the threshold region is manifested only in the absence of bistability. Finally, we analyze the threshold characteristics of the nonlinear system and the characteristic features of the transition to nonthreshold behavior at high levels of quantum noise and large nonlinearities. O 1996 American Institute of Physics. [S 1063-776 1 (96)004 1 1 -81