  We describe an opto-mechanical system in which the coupling between optical
and mechanical degrees of freedom takes the form of a fully quantised
third-order parametric interaction. Two physical realisations are proposed: a
harmonically trapped atom in a standing wave and the `membrane in the middle'
model. The dominant resonant interaction corresponds to a stimulated Raman
process in which two phonons are converted into a single cavity photon. We show
that this system can exhibit a stable limit cycle in which energy is
periodically exchanged between optical and mechanical degrees of freedom. This
is equivalently described as a parametric self-pulsing.
