  We theoretically analyze antibunching of the phonon field in an
optomechanical oscillator employ- ing the membrane-in-the-middle geometry. More
specifically, a single-mode mechanical oscillator is quadratically coupled to a
single-mode cavity field in the regime in which the cavity dissipation is a
dominant source of damping, and adiabatic elimination of the cavity field leads
to an effective cubic nonlinearity for the mechanics. We show analytically in
the weak coupling regime that the mechan- ics displays a chaotic phonon field
for small optomechanical cooperativity, whereas an antibunched single-phonon
field appears for large optomechanical cooperativity. This opens the door to
control of the second-order correlation function of a mechanical oscillator in
the weak coupling regime.
