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We derive the displacement noise spectrum of a parametrically pumped resonator below the onset for self-excited oscillations. We extend the fluctuation-dissipation response of a thermomechanical-noise-driven resonator to the case of degenerate parametric pumping as a function of pump magnitude and frequency while properly accounting for the quadrature-dependence of the parametric thermal noise squeezing. We use measurements with a microelectromechanical cantilever to corroborate our model. 

Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors. Various methods have been utilized to tune the effective quality factor of MEM/NEM resonators, including external proportional feedback control, optical pumping, mechanical pumping, thermal-piezoresistive pumping, and parametric pumping. This work reviews these mechanisms and compares the effective Q tuning using a position-proportional and a velocity-proportional force expression. We further clarify the relationship between the mechanical Q, the effective Q, and the thermomechanical noise of a resonator. We finally show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques.