  We study theoretically nonlinear effects arising from the presence of a
Kerr-down conversion nonlinear crystal inside an optomechanical cavity. For
this system we investigate the influences of the two nonlinearities, i.e., the
Kerr nonlinearity and the parametric gain, on the dynamics of the oscillating
mirror, the intensity and the squeezing spectra of the transmitted field, and
the steady-state mirror-field entanglement. We show that in comparison with a
bare optomechanical cavity, the combination of the cavity energy shift due to
the Kerr nonlinearity and increase in the intracavity photon number due to the
gain medium can increase the normal mode splitting in the displacement spectrum
of the oscillating mirror and reduce its effective temperature. Our work
demonstrates that both the Kerr nonlinearity and down conversion process
increase the degree of squeezing in the transmitted field. Moreover, we find
that in the system under consideration the degree of entanglement between the
mechanical and optical modes decreases considerably because of the intracavity
photon number reduction in the presence of the Kerr medium.
