  We experimentally surpass the 3dB limit to steady state parametric squeezing
of a mechanical oscillator. The localization of a AFM cantilever, achieved by
optimal estimation, is enhanced by up to 6.2 dB in one position quadrature when
a detuned parametric drive is used. This squeezing is, in principle, limited
only by the oscillator Q-factor. Used on low temperature, high frequency
oscillators, this technique provides a pathway to achieve robust quantum
squeezing below the zero-point motion. Broadly, our results demonstrate that
control systems engineering can overcome well established limits in
applications of nonlinear processes. Conversely, by localizing the mechanical
position to better than the measurement precision of our apparatus, they
demonstrate the usefulness of nonlinearities in control applications.

Mesoscale and Nanoscale Physics;Statistical Mechanics;Optics;Quantum Physics