  Optomechanical systems provide a unique platform for observing quantum
behavior of macroscopic objects. However, efforts towards realizing nonlinear
behavior at the single photon level have been inhibited by the small size of
the radiation pressure interaction. Here we show that it is not necessary to
reach the single-photon strong-coupling regime in order to realize significant
optomechanical nonlinearities. Instead, nonlinearities at the few quanta level
can be achieved, even with weak-coupling, in a two-mode optomechanical system
driven near instability. In this limit, we establish a new figure of merit for
realizing strong nonlinearity which scales with the single-photon
optomechanical coupling and the sideband resolution of the mechanical mode with
respect to the cavity linewidth. We find that current devices based on
optomechanical crystals, thought to be in the weak-coupling regime, can still
achieve strong quantum nonlinearity; enabling deterministic interactions
between single photons.
