Nondegenerate four-wave mixing in a cavity: instabilities and quantum noise reduction

We consider the four-wave mixing interaction of three longitudinal cavity modes in a unidirectional ring cavity containing a system of two-level atoms. The set of equations which governs the dynamics of the system coincides with that of multimode optical bistability, and describes the behaviour of the system over all its parameter space. We identify an appropriate limit in which the model reduces exactly to a parametric interaction among the three modes. In this limit the model is fully quantum mechanical and it includes all the physical processes that are relevant in the limit (four-wave mixing, phase modulation, cross phase modulation). The stationary solutions of the parametric model are calculated analytically and display a complex scenario, which includes also the possibility of undamped oscillations of the modal amplitudes. Using the parametric model we calculate analytically also the spectrum of the fluctuations of the intensity difference between the signal fields, above the threshold of generation of the signals. It turns out that this coincides with the expression of the spectrum that is well known in the case of the nondegenerate parametric oscillator.