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We explore the physics of optomechanical systems in which an optical cavitymode is coupled parametrically to the square of the position of a mechanicaloscillator. We derive an effective master equation describing two-phononcooling of the mechanical oscillator. We show that for high temperatures andweak coupling, the steady-state phonon number distribution is non-thermal(Gaussian) and that even for strong cooling the mean phonon number remainsfinite. Moreover, we demonstrate how to achieve mechanical squeezing by drivingthe cavity with two beams. Finally, we calculate the optical output andsqueezing spectra. Implications for optomechanics experiments with themembrane-in-the-middle geometry or ultracold atoms in optical resonators arediscussed.

Cooling and squeezing via quadratic optomechanical coupling