  Squeezing the quadrature noise of a harmonic oscillator used as a sensor can
enhance its sensitivity in certain measurment schemes. The canonical approach,
based on parametric modulation of the oscillation frequency, is usually limited
to a squeezing of at most 3 dB. However, this can be overcome by additional
stabilization of the anti-squeezed quadrature. Here, we apply this approach to
highly-stressed silicon nitride membrane resonators, with effective masses of
the order few nanograms and quality factors routinely exceeding 108, which hold
promise for sensing applications in both the classical and quantum regimes. We
benchmark their performance using either piezo or capacitive parametric
modulation. We observe maximum thermomechanical squeezing by record-high 17 dB
and 21 dB, respectively, and we argue that even larger values can be attained
with minimal changes to the device design. Finally, we provide a full quantum
theory of a combination of this approach with quantum-limited motion
measurement and conclude that quantum squeezing is attainable at moderate
cryogenic temperatures.
