  We study a parametrically-driven nanomechanical resonator capacitively
coupled to a microwave cavity. If the nanoresonator can be cooled to near its
quantum ground state then quantum squeezing of a quadrature of the
nanoresonator motion becomes feasible. We consider the adiabatic limit in which
the cavity mode is slaved to the nanoresonator mode. By driving the cavity on
its red-detuned sideband, the squeezing can be coupled into the microwave field
at the cavity resonance. The red-detuned sideband drive is also compatible with
the goal of ground state cooling. Squeezing of the output microwave field may
be inferred using a technique similar to that used to infer squeezing of the
field produced by a Josephson parametric amplifier, and subsequently, squeezing
of the nanoresonator motion may be inferred. We have calculated the output
field microwave squeezing spectra and related this to squeezing of the
nanoresonator motion, both at zero and finite temperature. Driving the cavity
on the blue-detuned sideband, and on both the blue and red sidebands, have also
been considered within the same formalism.
