  We identify signatures of the intrinsic nonlinear interaction between light
and mechanical motion in cavity optomechanical systems. These signatures are
observable even when the cavity linewidth exceeds the optomechanical coupling
rate. A strong laser drive red-detuned by twice the mechanical frequency from
the cavity resonance frequency makes two-phonon processes resonant, which leads
to a nonlinear version of optomechanically induced transparency. This effect
provides a new method of measuring the average phonon number of the mechanical
oscillator. Furthermore, we show that if the strong laser drive is detuned by
half the mechanical frequency, optomechanically induced transparency also
occurs due to resonant two-photon processes. The cavity response to a second
probe drive is in this case nonlinear in the probe power. These effects should
be observable with optomechanical coupling strengths that have already been
realized in experiments.
