  We analyze quantum effects occurring in optomechanical systems where the
coupling between an optical mode and a mechanical mode is quadratic in
displacement (membrane-in-the-middle geometry). We show that it is possible to
observe quantum effects in these systems without achieving the single-photon
strong coupling regime. We find that zero-point energy causes a mechanical
frequency shift, and we propose an experimental way to measure it. Further, we
show that it is possible to determine the phonon statistics from the cavity
transmission, and propose a way to infer the resonator's temperature based on
this feature. For completeness, we revisit the case of an isolated system and
show that different types of mechanical quantum states can be created,
depending on the initial cavity state. In this situation, mechanical motion
undergoes collapse and revivals, and we compute the collapse and revival times,
as well as the degree of squeezing.
