sideband limit, this maximum damping occurs at $\Delta_\mathrm{o} = -\omega_\mathrm{m}$, but this is not generally the case with imperfect sideband resolution.

### Fabrication Details {#fabrication-details .unnumbered}

The mechanical oscillator is a $100~\mathrm{nm}$ thick, $500~\mathrm{\mu m}$ wide silicon nitride membrane suspended from a silicon chip. A $25~\mathrm{nm}$ thick niobium film which serves as one capacitor pad is fabricated on one quadrant of the membrane before it is released. The membrane chip is flipped over and affixed to a second silicon chip, on which a microfabricated niobium circuit comprising an inductor and a second capacitor pad was previously patterned using standard lithographic techniques. The full fabrication process has been described elsewhere [@andrews_2015]. The flip-chip assembly is constructed with a West Bond manual die bonder, and the chips are affixed using Stycast 2850. In the fully assembled flip-chip device, the two niobium pads form a parallel-plate capacitor with a plate spacing of $300~\mathrm{nm}$, and the $6.16~\mathrm{GHz}$ resonant frequency of the resulting LC circuit is modulated by vibrations of mode of interest. Propagating microwaves are wirelessly coupled to the LC circuit through a re-entrant microwave cavity, which also holds mirrors that form a Fabry-Perot optical cavity with a $281.8~\mathrm{THz}$ resonant frequency [@menke_reconfigurable_2017]. The optical 