Photon-phonon parametric oscillation induced by the quadratic coupling in an optomechanical resonator

A direct photon-phonon parametric effect of the quadratic coupling on the mean-field dynamics of an optomechanical resonator in the large-scale-movement regime is found and investigated. Under a weak pumping power, the mechanical resonator damps to steady state with a nonlinear static response sensitively modified by the quadratic coupling. When the driving powerincreases beyond the static energy balance, the steady states lose their stabilities via Hopf bifurcations and the resonator produces stable self-sustained oscillation(limit-circle behavior) of discrete energies with step-like amplitudes due to the parametric effect of the quadratic coupling, which can be understood roughly by the power balance between gain and loss on the resonator. A further increase of the pumping power can induce chaotic dynamic of the resonator via a typical routine of period-doubling bifurcation but which can be stabilized by the parametric effect through an inversion bifurcation process back to limit-circle states. The bifurcation-to-inverse-bifurcation transitions are numerically verified by the maximal Lyapunov exponents of the dynamics and which indicate an efficient way to suppress the chaotic behavior of the optomechanical resonator by the quadratic coupling. Furthermore, the parametric effect of the quadratic coupling on the dynamic transitions of an optomechanical resonator can be conveniently detected or traced by the output power spectrum of the cavity field.