  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.
