  Cavity optomechanical system involving an optical parametric amplifier (OPA)
can exhibit rich classical and quantum dynamical behaviors. By simply
modulating the frequency of the laser pumping the OPA, we find two interesting
parameter regimes, with one of them enabling to study quantum-classical
correspondence in system dynamics, while there exist no classical counterparts
of the quantum features for the other. For the former regime, as the parametric
gain of OPA increases to a critical value, the classical dynamics of the
optical or mechanical modes can experience a transition from the regular
periodic oscillation to period-doubling motion, in which cases the
light-mechanical entanglement can be well studied by the logarithm negativity
and can manifest the dynamical transition in the classical nonlinear dynamics.
Moreover, the optomechanical entanglement shows a second-order transition
characteristic at the critical parametric gain. For the latter regime, the kind
of normal mode splitting comes up in the laser detuning dependence of
optomechanical entanglement, which is induced by the squeezing of the optical
and mechanical hybrid modes and finds no classical correspondence. The OPA
assisted optomechanical systems therefore offer a simple way to study and
exploit quantum manifestations of classical nonlinear dynamics.
