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.
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