plus measurement chain added noise) is simply given by the calibration factor for each measurement. This procedure yields a single-quadrature background noise $n_\mathrm{e} = 29.6~\mathrm{photons}$ at the microwave output port and $n_\mathrm{o} = 2.7~\mathrm{photons}$ at the optical output port. Inverting this calibration procedure, we can associate the number of output-referred photons on resonance in spectra like those in Fig. 3 of the main text with an effective mechanical bath temperature.

The microwave cavity coupling ratio $\kappa_\mathrm{ex,e}/\kappa_\mathrm{e}$ is easily measured using the HF2LI as a network analyzer (see Sec. \[microwaves\]). The optical cavity parameters are needed to predict the optical field output noise using optomechanical theory [@aspelmeyer_cavity_2014]. The mode matching between the cavity and the pump beam is obtained by directing a transmitted beam and the reflected pump onto the heterodyne detector, and measuring the visibility of the resulting interference pattern (see Sec. \[modematch\]). The optical cavity linewidth is measured by sweeping a probe with two sidebands at a known frequency through resonance. The optical cavity coupling ratio $\kappa_\mathrm{ex,o}/\kappa_\mathrm{o}$, which is used to determine $\kappa_\mathrm{ex,o}$ in Table \[tab:params\], is then fixed based on the measured converter efficiency. As a check, $\kappa_\mathrm{ex,o}/\kappa_\mathrm{o}$ is independently extracted from reflection and transmission measurements of both 