A general theory is presented to describe optomechanical interactions of acoustic phonons, having extremely long lifetimes in superfluid $^4$He, with optical photons in the medium placed in a suitable electromagnetic cavity. The acoustic nonlinearity
in the fluid motion is included to consider processes beyond the usual linear process involving absorption or emission of one phonon at a time. We first apply our formulation to the simplest one-phonon process involving the usual resonant anti-Stokes upconversion of an incident optical mode. However, when the allowed optical cavity modes are such that there is no single-phonon mode in the superfluid which can give rise to a resonant allowed anti-Stokes mode, we must consider the possibility of two phonon upconversion. For such a case, we show that the two step two phonon process could be dominant. We present arguments for large two step process and negligible single step two phonon contribution. The two step process also shows interesting quantum interference among different transition pathways.
We discuss the possible cooling of different phonon modes via three wave mixing interactions of vibrational and optical modes. Since phonon modes exhibit a variety of dispersion relations or frequency spectra with diverse spatial structures, dependin
g on the shape and size of the sample, we formulate our theory in terms of relevant spatial mode functions for the interacting fields in any given geometry. We discuss the possibility of Dicke like collective effects in phonon cooling and present explicit results for simultaneous cooling of two phonon modes via the anti-Stokes up
