  We propose a scheme to squeeze mechanical motion and to entangle 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.
