  Single atoms coupled to a cavity offer unique opportunities as quantum
optomechanical devices because of their small mass and strong interaction with
light. A particular regime of interest in optomechanics is that of
"single-photon strong coupling," where motional displacements on the order of
the zero-point uncertainty are sufficient to shift the cavity resonance
frequency by more than its linewidth. In many cavity QED platforms, however,
this is unfeasible due to the large cavity linewidth. Here, we propose an
alternative route in such systems, which instead relies on the coupling of
atomic motion to the much narrower cavity-dressed atomic resonance frequency.
We discuss and optimize the conditions in which the scattering properties of
single photons from the atom-cavity system become highly entangled with the
atomic motional wave function. We also analyze the prominent observable
features of this optomechanical strong coupling, which include a per-photon
motional heating that is significantly larger than the single-photon recoil
energy, as well as mechanically-induced oscillations in time of the
second-order correlation function of the emitted light. This physics should be
realizable in current experimental setups, such as trapped atoms coupled to
photonic crystal cavities, and more broadly opens the door to realizing
qualitatively different phenomena beyond what has been observed in
optomechanical systems thus far.
