a peak of width $\Gamma_\mathrm{T}/2\pi=200~\mathrm{Hz}$ around $\omega/2\pi=f_\mathrm{m}$ (Fig. \[correlations\]a). The peak height relative to background, $\langle (X^{\mathrm{(th)}}_\mathrm{e})^2\rangle = 69.2~\mathrm{photons}$, is obtained from a Lorentzian fit to the data and attributed to thermally driven mechanical motion at a bath temperature $T = 87 \pm 4 ~\mathrm{mK}$ (see supplement), which exceeds the cryostat base temperature of $35~\mathrm{mK}$. The elevated membrane temperature is consistent with independently measured optical heating (see supplement). The background noise level, 31.8 photons, corresponds to $n_\mathrm{e}=29.6~\mathrm{photons}$ from vacuum noise and the added noise of the microwave measurement chain, with the remaining 2.2 photons due primarily to parameter noise in the LC circuit (independently calibrated, see supplement).

The optical real spectral density, $\langle X_\mathrm{o}(\omega) X_\mathrm{o}(\omega) \rangle$, shows a similar peak (Fig. \[correlations\]b). The peak height above noise, $\langle (X^{\mathrm{(th)}}_\mathrm{o})^2\rangle = 33.1~\mathrm{photons}$, provides a second measure of the bath temperature, $T = 80 \pm 4 ~\mathrm{mK}$, consistent with Fig. \[correlations\]a. The background noise, $n_\mathrm{o}=2.7$ photons, corresponds to vacuum noise plus the effect of loss in the optical measurement chain. Note that $n_\mathrm{o} < n_\mathrm{e}$, indicating that the optical measurement apparatus is closer to ideal. The real cross-spectral density, $\langle X_\mathrm{e}(\omega) X_\mathrm{o}(\omega)\rangle$, has a $47.7$ photon peak at $\omega/2\pi=f_\mathrm{m}$, indicating that the 