  There has been much interest recently in the analysis of optomechanical
systems incorporating dielectric nano- or microspheres inside a cavity field.
We analyse here the situation when one of the mirrors of the cavity itself is
also allowed to move. We reveal that the interplay between the two oscillators
yields a cross-coupling that results in, e.g., appreciable cooling and
squeezing of the motion of the sphere, despite its nominal quadratic coupling.
We also discuss a simple modification that would allow this cross-coupling to
be removed at will, thereby yielding a purely quadratic coupling for the
sphere.
