  We study the photon-blockade effect in a loop-coupled optomechanical system
consisting of two cavity modes and one mechanical mode. Here, the mechanical
mode is optomechanically coupled to the two cavity modes, which are coupled
with each other via a photon-hopping interaction. By treating the
photon-hopping interaction as a perturbation, we obtain the analytical results
of the eigenvalues and eigenstates of the system in the subspaces associated
with zero, one, and two photons. We find a phenomenon of optical
normal-mode-induced phonon-sideband splitting in the photon-blockade effect by
analytically and numerically calculating the second-order correlation functions
of the two cavity modes. This work not only presents a method to choose optimal
driving frequency for photon blockade by tuning the photon-hopping interaction,
but also provides a means to characterize the normal-mode splitting with cavity
photon statistics.
