  We theoretically show that strong mechanical quantum squeezing in a linear
optomechanical system can be rapidly generated through the dynamical
instability reached in the far red-detuned and ultrastrong coupling regime. We
show that this mechanism, which harnesses unstable multimode quantum dynamics,
is particularly suited to levitated optomechanics, and we argue for its
feasibility for the case of a levitated nanoparticle coupled to a microcavity
via coherent scattering. We predict that for sub-millimeter-sized cavities the
particle motion, initially thermal and well above its ground state, becomes
mechanically squeezed by tens of decibels on a microsecond timescale. Our
results bring forth optical microcavities in the unresolved sideband regime as
powerful mechanical squeezers for levitated nanoparticles, and hence as key
tools for quantum-enhanced inertial and force sensing.
