Based on a scheme proposed to experience the dynamical Casimir effect in optomechanical systems, we show how to squeeze mechanical motion and entangle the optical field with mechanical
motion in an optomechanical system containing a parametric amplification. The scheme is based
on optical bistability which emerges in the system for a strong enough driving field. By considering
the steady state’s lower branch of the bistability, the system shows weak entanglement and almost
no mechanical squeezing. When the steady state is on the upper branch of the bistable shape,
both squeezing and entanglement are greatly enhanced. Specifically, the entanglement shows three
degrees of magnitude enhancement. However, this giant entanglement is fragile against decoherence
and thermal fluctuation. Regarding the mechanical squeezing, it reaches the standard quantum
limit (SQL) in the upper branch of the bistability. Our proposal provides a way to improve quantum effects in optomechanical systems by taking advantage of nonlinearities. This scheme can be
realized in similar systems such as superconducting microwave, and hybrid optomechanical systems.