  We propose a scheme to significantly increase quadratic optomechanical
couplings of optomechanical systems with the help of a nonlinear medium and two
driving lasers. The nonlinear medium is driven by one laser and the optical
cavity mode is driven by a strong laser. We derive an effective Hamiltonian
using squeezing transformation and rotating wave approximation. The effective
quadratic optomechanical coupling strength can be larger than the decay rate of
the cavity mode by adjusting the two optical driving fields. The thermal noise
of squeezed cavity mode can be suppressed totally with the help of a squeezed
vacuum field. Then, a driving field is applied to the mechanical mode. We
investigate the equal-time second order correlations and find there are photon,
phonon, and photon-phonon blockades even the original single-photon quadratic
coupling is much smaller than the decay rate of the optical mode. In addition,
the sub-Poissonian window of the two-time second order correlations can be
controlled by the mechanical driving field. Finally, we show the squeezing and
entanglement of the model could be tuned by the driving fields of the nonlinear
medium and mechanical mode.
