  We experimentally investigate a mechanical squeezed state realized in a
parametrically-modulated membrane resonator embedded in an optical cavity. We
demonstrate that a quantum characteristic of the squeezed dynamics can be
revealed and quantified even in a moderately warm oscillator, through the
analysis of motional sidebands. We provide a theoretical framework for
quantitatively interpreting the observations and present an extended comparison
with the experiment. A notable result is that the spectral shape of each
motional sideband provides a clear signature of a quantum mechanical squeezed
state without the necessity of absolute calibrations, in particular in the
regime where residual fluctuations in the squeezed quadrature are reduced below
the zero-point level.
