We consider an electromechanical system in which a microwave cavity is coupled to a mechanical resonator, with a mechanical frequency twice the microwave frequency. In this regime, the effective photon-phonon interaction is equivalent to that of a degenerate parametric amplifier, instead of the typical radiation pressure interaction. If the mechanical resonator is strongly driven, it undergoes a phase transition to a state in which the energy pumped into the mechanical mode is entirely converted to the photonic mode. Quantum fluctuations smear this phase transition. We describe these effects with a steady-state Fokker-Planck equation in the complex P representation and compute the photonic field intensity and quadrature variances, as well as the mechanical amplitude. This Fokker-Planck method performs better than the standard linearization results when compared to numerical simulations.

We analyze quantum effects occurring in optomechanical systems where the coupling between an optical mode and a mechanical mode is quadratic in displacement (membrane-in-the-middle geometry). We show that it is possible to observe quantum effects in these systems without achieving the single-photon strong-coupling regime. We find that zero-point energy causes a mechanical frequency shift, and we propose an experimental way to measure it. Further, we show that it is possible to determine the phonon statistics from the cavity transmission, and propose a way to infer the resonator temperature based on this feature. For completeness, we revisit the case of an isolated system and show that different types of mechanical quantum states can be created, depending on the initial cavity state. In this situation, mechanical motion undergoes collapse and revival, and we compute the collapse and revival times, as well as the degree of squeezing.