  Typical optomechanical systems involving optical cavities and mechanical
oscillators rely on a coupling that varies linearly with the oscillator
displacement. However, recently a coupling varying instead as the square of the
mechanical displacement has been realized, presenting new possibilities for
non-demolition measurements and mechanical squeezing. In this article we
present a quantum mechanical study of a generic quadratic-coupling
optomechanical Hamiltonian. First, neglecting dissipation, we provide
analytical results for the dressed states, spectrum, phonon statistics and
entanglement. Subsequently, accounting for dissipation, we supply a numerical
treatment using a master equation approach. We expect our results to be of use
to optomechanical spectroscopy, state transfer, wavefunction engineering, and
entanglement generation.
