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We consider an electromechanical system where a microwave cavity is coupledto a mechanical resonator, with a mechanical frequency twice the microwavefrequency. In this regime, the effective photon-phonon interaction isequivalent to that of a degenerate parametric amplifier, instead of the typicalradiation pressure interaction. If the mechanical resonator is strongly driven,it undergoes a phase transition to a state where the energy pumped into themechanical mode is entirely converted to the photonic mode. Quantumfluctuations smear this phase transition. We describe these effects with asteady-state Fokker-Planck equation in the complex P-representation, andcompute the photonic field intensity and quadrature variances, as well as themechanical amplitude. This Fokker-Planck method performs better than thestandard linearization results, when compared to numerical simulations.

Realization of the degenerate parametric oscillator in electromechanical systems