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

Mesoscale and Nanoscale Physics;Quantum Physics