  We analyze the photon correlations in an optomechanical system containing two
nonlinear optical modes and one mechanical mode which are coupled via a
three-mode mixing. Under a weak driving condition, we determine the optimal
conditions for photon antibunching in the weak Kerr-nonlinear regime and we
find that the analytical calculations are consistent with the numerical
results. The photon blockade effect is attributed to destructive quantum
interference in the two-photon excitation pathways created as a result of the
three-mode interaction.
