Abstract
The generation of large-amplitude coherent states of a massive mechanical resonator, and their quantum-limited detection represent useful tools for quantum sensing and for testing fundamental physics theories. In fact, any weak perturbation may affect the coherent quantum evolution of the prepared state, providing a sensitive probe for such a perturbation. Here we consider a cavity optomechanical setup and the case of the detection of a weak mechanical nonlinearity. We consider different strategies, first focusing on the stationary dynamics in the presence of multiple tones driving the system, and then focusing on non-equilibrium dynamical strategies. These methods can be successfully applied for measuring Duffing-like material nonlinearities, or effective nonlinear corrections associated with quantum gravity theories.