Four-wave mixing with self-phase matching due to collective atomic recoil

We describe a method for nondegenerate four-wave-mixing in a cold sample of four-level atoms. An integral part of the four-wave-mixing process is a collective instability which spontaneously generates a periodic density modulation in the cold atomic sample with a period equal to half of the wavelength of the generated high-frequency optical field. Because of the generation of this density modulation, phase matching between the pump and scattered fields is not a necessary initial condition for this wave-mixing process to occur; rather the density modulation acts to "self-phase match" the fields during the course of the wave-mixing process. We describe a one-dimensional model of this process, and suggest a proof-of-principle experiment which would involve pumping a sample of cold Cs atoms with three infrared pump fields to produce blue light.