  The motion of micro- and nanomechanical resonators can be coupled to
electromagnetic fields. Such optomechanical setups allow one to explore the
interaction of light and matter in a new regime at the boundary between quantum
and classical physics. We propose an approach to investigate non-equilibrium
photon dynamics driven by mechanical motion in a recently developed setup with
a membrane between two mirrors, where photons can be shuttled between the two
halves of the cavity. For modest driving strength we predict the possibility to
observe an Autler-Townes splitting indicative of Rabi dynamics. For large
drive, we show that this system displays Landau-Zener-Stueckelberg dynamics
originally known from atomic two-state systems.
