  We propose to use cavity optomechanical systems in the regime of optical
bistability for the detection of weak harmonic forces. Due to the
optomechanical coupling an external force on the mechanical oscillator
modulates the resonance frequency of the cavity and consequently the switching
rates between the two bistable branches. A large difference in the cavity
output fields then leads to a strongly amplified homodyne signal. We determine
the switching rates as a function of the cavity detuning from extensive
numerical simulations of the stochastic master equation as appropriate for
continuous homodyne detection. We develop a two-state rate equation model that
quantitatively describes the slow switching dynamics. This model is solved
analytically in the presence of a weak harmonic force to obtain approximate
expressions for the power gain and signal-to-noise ratio that we then compare
to force detection with an optomechanical system in the linear regime.
