  The study of the quantum to classical transition is of fundamental as well as
technological importance, and focusses on mesoscopic devices, with a size for
which either classical physics or quantum physics can be brought to dominate. A
particularly diverse selection of such devices is available in cavity
quantum-optomechanics. We show that these can be leveraged for the study of
dynamical-tunnelling in a quantum chaotic system. This effect probes the
quantum to classical transition deeply, since tunnelling rates sensitively
depend on the ability of the quantum system to resolve the underlying classical
phase space. We show that the effective Planck's constant, which determines
this phase space resolution, can be varied over orders of magnitude as a
function of tunable parameters in an opto-mechanical experiment. Specifically,
we consider a membrane-in-the-middle configuration of a mechanical oscillator
within an optical cavity, where the intracavity field is modulated periodically
by the external laser source. We demonstrate that a mixed regular and chaotic
phase space can be engineered in one spatial dimension, through a significant
quartic opto-mechanical interaction. For that case, we explore the expected
dynamical tunnelling rates using Floquet theory and map out values of the
effective Planck's constant that should be within practical reach.
