  We theoretically investigate interactions between non-degenerate mechanical
oscillators mediated by a time-dependent cavity field. We obtain a reduced
master equation valid for all optomechanical systems operating in the weak
coupling regime. This master equation includes all forms of decoherence and
back-action due to the dissipation of the field mediating the interaction. We
apply the master equation to study two resonant coupling schemes within a
rotating-wave approximation: the beam splitter Hamiltonian and the two-mode
parametric amplifier. In both cases, the effective unitary interaction can be
made arbitrarily strong compared to the decoherence due to dissipation of the
mediating field by choosing appropriate detunings.
