  We describe a method for non-degenerate four-wave mixing in a cold sample of
  4-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. Due to 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 infra-red pump fields to
produce blue light.
