  Nonlinear optomechanical coupling is the basis for many potential future
experiments in quantum optomechanics (e.g., quantum non-demolition
measurements, preparation of non-classical states), which to date have been
difficult to realize due to small non-linearity in typical optomechanical
devices. Here we introduce an optomechanical system combining strong nonlinear
optomechanical coupling, low mass and large optical mode spacing. This
nanoscale "paddle nanocavity" supports mechanical resonances with hundreds of
fg mass which couple nonlinearly to optical modes with a quadratic
optomechanical coupling coefficient $g^{(2)} > 2\pi\times400$ MHz/nm$^2$, and a
two phonon to single photon optomechanical coupling rate $\Delta \omega_0 >
2\pi\times 16$ Hz. This coupling relies on strong phonon-photon interactions in
a structure whose optical mode spectrum is highly non--degenerate. Nonlinear
optomechanical readout of thermally driven motion in these devices should be
observable for T $> 50 $ mK, and measurement of phonon shot noise is
achievable. This shows that strong nonlinear effects can be realized without
relying on coupling between nearly degenerate optical modes, thus avoiding
parasitic linear coupling present in two mode systems.
