  We study photon, phonon statistics and the cross-correlation between photons
and phonons in a quadratically coupled optomechanical system. Photon blockade,
phonon blockade and strongly anticorrelated photons and phonons can be observed
in the same parameter regime with the effective nonlinear coupling between the
optical and mechanical modes, enhanced by a strong optical driving field.
Interestingly, an optimal value of the effective nonlinear coupling strength
for the photon blockade is not within the strong nonlinear coupling regime.
This abnormal phenomenon results from the destructive interference between
different paths for two-photon excitation in the optical mode with a moderate
effective nonlinear coupling strength. Further more, we show that phonon
(photon) pairs and correlated photons and phonons can be generated in the
strong nonlinear coupling regime with a proper detuning between the weak
mechanical driving field and mechanical mode. Our results open up a way to
generate anticorrelated and correlated photons and phonons, which may have
important applications in quantum information processing.
