  We realize a nondegenerate parametric amplifier in an ultrahigh $Q$
mechanical membrane resonator and demonstrate two-mode thermomechanical noise
squeezing. Our measurements are accurately described by a two-mode model that
attributes this nonlinear mechanical interaction to a substrate-mediated
process which is dramatically enhanced by the quality factors of the individual
modes. This realization of strong multimode nonlinearities in a mechanical
platform compatible with quantum-limited optical detection and cooling makes
this a powerful system for nonlinear approaches to quantum metrology,
transduction between optical and phononic fields and the quantum manipulation
of phononic degrees of freedom.
