  We consider the physics of an optomechanical cavity subject to coherent
two-phonon driving, i.e. degenerate parametric amplification of the mechanical
mode. We show that in such a system, the cavity mode can effectively "inherit"
parametric driving from the mechanics, yielding phase-sensitive amplification
and squeezing of optical signals reflected from the cavity. We also demonstrate
how such a system can be used to perform single-quadrature detection of a
near-resonant narrow-band force applied to the mechanics with extremely low
added noise from the optics. The system also exhibits strong differences from a
conventional degenerate parametric amplifier: in particular, the cavity
spectral function can become negative, indicating a negative effective photon
temperature.
