  We present an approach for exponentially enhancing the single-photon coupling
strength in an optomechanical system using only additional linear resources. It
allows one to reach the quantum nonlinear regime of optomechanics, where
nonlinear effects are observed at the single photon level, even if the bare
coupling strength is much smaller than the mechanical frequency and cavity
damping rate. Our method is based on using a large amplitude, strongly detuned
mechanical parametric drive to amplify mechanical zero-point fluctuations and
hence enhance the radiation pressure interaction. It has the further benefit of
allowing time-dependent control, enabling pulsed schemes. For a two-cavity
optomechanical setup, we show that our scheme generates photon blockade for
experimentally accessible parameters, and even makes the production of photonic
states with negative Wigner functions possible. We discuss how our method is an
example of a more general strategy for enhancing boson-mediated two-particle
interactions and nonlinearities.
