  In a hybrid atom-optomechanical system, the optical coupling of a mechanical
mode of a nanomembrane in an optical cavity with a distant interacting atom gas
permits highly non-classical quantum many-body states. We show that the
mechanical mode can be squeezed by the back-action of internal excitations of
the atoms in the gas. A Bogoliubov approach reveals that these internal
excitations form a fluctuating environment of quasi-particle excitations for
the mechanical mode with a gaped spectral density. Nanomechanical squeezing
arises due to quasi-particle excitations in the interacting atom gas when the
mechanical frequency is close to resonance with the internal atomic
transitions. Interestingly, nanomechanical squeezing is enhanced by atom-atom
interactions.
