$q(t)$ and $p(t)$, are used to compute, $$a(t) = ( q(t) + i p(t) ) e^{i 2 \pi f_\mathrm{d} t},$$ which is the complex-equivalent signal of $s(t)$, $$a(t) = s(t) + i H[ s( t ) ],$$ where $H$ denotes the Hilbert transform. The Fourier transform of $a(t)$, $a( \omega ) = \int_{-\infty}^{+\infty} a(t) e^{-i \omega t}\, \mathrm{d}t$ is then computed approximately using DFT. $a( \omega )$ is a complex-valued function, $$a(\omega) = X( \omega ) + i Y( \omega ),$$ with real component $X(\omega)$ and imaginary component $Y(\omega)$, and is zero when $\omega < 0$. In the limit of narrowband demodulation the quadrature variances satisfy e.g., $\mathrm{Var}(q_\mathrm{o}) =  \langle X_\mathrm{o}( \omega_\mathrm{m} ) X_\mathrm{o}( \omega_\mathrm{m} ) \rangle$. A demodulation bandwidth of $2 \pi \times 50~\mathrm{Hz}$ is used for all quadrature data. Note that away from mechanical resonance, $X(\omega)$ is not just the Fourier transform of $q(t)$, but rather contains contributions from both $q(t)$ and $p(t)$. In the main text, spectra are displayed shifted so that $f_\mathrm{c}$ appears at zero frequency.

Microwave and optical calibrations {#sec:omcal}
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A temperature sweep of the base plate of the dilution refrigerator is used to calibrate output-referred noise spectra like those shown in Fig. 3 