  Cavity optomechanics offers powerful methods for controlling optical fields
and mechanical motion. A number of proposals have predicted that this control
can be extended considerably in devices where multiple cavity modes couple to
each other via the motion of a single mechanical oscillator. Here we study the
dynamical properties of such a multimode optomechanical device, in which the
coupling between cavity modes results from mechanically-induced avoided
crossings in the cavity's spectrum. Near the avoided crossings we find that the
optical spring shows distinct features that arise from the interaction between
cavity modes. Precisely at an avoided crossing, we show that the particular
form of the optical spring provides a classical analog of a
quantum-nondemolition measurement of the intracavity photon number. The
mechanical oscillator's Brownian motion, an important source of noise in these
measurements, is minimized by operating the device at cryogenic temperature
(500 mK).
