Modulated electromechanics: large enhancements of nonlinearities

It is well-known that the nonlinear coupling between a mechanical oscillator and a superconducting resonator or optical cavity can be used to generate a Kerr-nonlinearity for the cavity mode. We show that the strength of this Kerr-nonlinearity, as well as the effect of the photon-pressure force can be enormously increased by modulating the strength of the nonlinear coupling. We describe an electromechanical circuit where this enhancement could be readily realized.