  Coupling electromagnetic waves in a cavity and mechanical vibrations via the
radiation pressure of the photons [1,2] is a promising platform for
investigations of quantum mechanical properties of motion of macroscopic bodies
and thereby the limits of quantum mechanics [3,4]. A drawback is that the
effect of one photon tends to be tiny, and hence one of the pressing challenges
is to substantially increase the interaction strength towards the scale of the
cavity damping rate. A novel scenario is to introduce into the setup a quantum
two-level system (qubit), which, besides strengthening the coupling, allows for
rich physics via strongly enhanced nonlinearities [5-8]. Addressing these
issues, here we present a design of cavity optomechanics in the microwave
frequency regime involving a Josephson junction qubit. We demonstrate boosting
of the radiation pressure interaction energy by six orders of magnitude,
allowing to approach the strong coupling regime, where a single quantum of
vibrations shifts the cavity frequency by more than its linewidth. We observe
nonlinear phenomena at single-photon energies, such as an enhanced damping due
to the two-level system. This work opens up nonlinear cavity optomechanics as a
plausible tool for the study of quantum properties of motion.
