  In a recent publication [K. Hammerer et al., Phys. Rev. Lett. 103, 063005
(2009)] we have shown the possibility to achieve strong coupling of the
quantized motion of a micron-sized mechanical system to the motion of a single
trapped atom. In the proposed setup the coherent coupling between a SiN
membrane and a single atom is mediated by the field of a high finesse cavity,
and can be much larger than the relevant decoherence rates. This makes the
well-developed tools of CQED (cavity quantum electrodynamics) with single atoms
available in the realm of cavity optomechanics. In this paper we elaborate on
this scheme and provide detailed derivations and technical comments. Moreover,
we give numerical as well as analytical results for a number of possible
applications for transfer of squeezed or Fock states from atom to membrane as
well as entanglement generation, taking full account of dissipation. In the
limit of strong-coupling the preparation and verification of non-classical
states of a mesoscopic mechanical system is within reach.
