  Phonon nonlinearities play an important role in hybrid quantum networks and
on-chip quantum devices. We investigate the phonon statistics of a mechanical
oscillator in hybrid systems composed of an atom and one or two standard
optomechanical cavities. An efficiently enhanced atom-phonon interaction can be
derived via a tripartite atom-photon-phonon interaction, where the atom-photon
coupling depends on the mechanical displacement without practically changing a
cavity frequency. This novel mechanism of optomechanical interactions, as
predicted recently by Cotrufo et al. [Phys. Rev. Lett. 118, 133603 (2017)], is
fundamentally different from standard ones. In the enhanced atom-phonon
coupling, the strong phonon nonlinearity at a single-excitation level is
obtained in the originally weak-coupling regime, which leads to the appearance
of phonon blockade. Moreover, the optimal parameter regimes are presented both
for the cases of one- and two- cavities. We compared phonon-number correlation
functions of different orders for mechanical steady states generated in the
one-cavity hybrid system, revealing the occurrence of phonon-induced tunneling
and different types of phonon blockade. Our approach offers an alternative
method to generate and control a single phonon in the quantum regime, and has
potential applications in single-phonon quantum technologies.
