With an increasing coupling between light and mechanics, nonlinearities begin to play an important role in optomechanics. We solve the quantum dynamics of an optomechanical system in the multiphoton strong-coupling regime retaining nonlinear terms. This is achieved by performing a Schrieffer-Wolff transformation on the Hamiltonian including driving terms. The approach is valid away from the red-and blue-sideband drive. We show that the mechanical resonator displays self-sustained oscillations in regimes where the linear model predicts instabilities and that the amplitude of these oscillations is limited by the nonlinear terms. Related oscillations of the photon number are present due to frequency mixing of the shifted mechanical and cavity frequencies. This leads to additional peaks in the cavity's spectral density. Furthermore, we show that it is possible to create phonon states with sub-Poissonian statistics when the system is red detuned. This result is valid even with strong driving and with initial coherent states.
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