  We present an atom-chip-based realization of quantum cavity optomechanics
with cold atoms localized within a Fabry-Perot cavity. Effective sub-wavelength
positioning of the atomic ensemble allows for tuning the linear and quadratic
optomechanical coupling parameters, varying the sensitivity to the displacement
and strain of a compressible gaseous cantilever. We observe effects of such
tuning on cavity optical nonlinearity and optomechanical frequency shifts,
providing their first characterization in the quadratic-coupling regime.
