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 ma
ster 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.
We theoretically investigate two quantum modes interacting via local couplings to a dissipative field. Our model considers two mechanical modes with distinct frequencies coupled optomechanically to the same cavity mode. The dissipative cavity field m
ediates the interaction between the mechanical modes but also leads to decoherence of the mechanical oscillators. Depending on the ratio between effective interaction strength and dissipation rate, which can be chosen via the pump detuning, the interaction assumes a quantum mechanical or classical character. For any cavity decay, there is a regime where the two mechanical modes interact in a non-classical way, which leads us to conclude that optomechanical systems can serve as a model to experimentally study the effects of long-range interactions mediated by classical or quantum-mechanical fields.
