membrane modulates $\omega_\mathrm{e}$ by $G_\mathrm{e} \approx 2\pi\times8~\text{MHz/nm}$ and $\omega_\mathrm{o}$ by $G_\mathrm{o} \approx 2\pi\times13~\text{MHz/nm}$. The entire assembly is housed in a cryostat with base temperature $T=35~\mathrm{mK}$ [@peterson_laser_2016], where both microwave and optical modes are close to their quantum ground state.

With a strong red-detuned pump beam incident on the optical cavity, the optomechanical interaction couples the mechanical oscillator to propagating optical fields at a rate $\Gamma_\mathrm{o}$ that exceeds the intrinsic mechanical damping rate $\gamma_\mathrm{m} = 2\pi\times11~\mathrm{Hz}$. At pump detuning $\Delta_\mathrm{o} = -\omega_\mathrm{m}$ and in the resolved sideband limit ($4\omega_\mathrm{m}/\kappa_\mathrm{o} \gg 1$), the optomechanical damping rate has the simple form[@andrews_bidirectional_2014] $$\Gamma_\mathrm{o} = \frac{4G_\mathrm{o}^2x_\mathrm{zp}^2\alpha^2_\mathrm{o}}{\kappa_\mathrm{o}},$$ where $\alpha_\mathrm{o}$ is the intracavity pump amplitude and $x_\mathrm{zp}$ is the zero-point amplitude of the membrane mode. The electromechancial damping in the presence of a strong red-detuned microwave pump tone has the same form: $$\Gamma_\mathrm{e} = \frac{4G_\mathrm{e}^2x_\mathrm{zp}^2\alpha^2_\mathrm{e}}{\kappa_\mathrm{e}}.$$ In practice, our converter is moderately sideband-resolved ($4\omega_\mathrm{m}/\kappa_\mathrm{o} \approx 4\omega_\mathrm{m}/\kappa_\mathrm{e} \approx 2.5$), with pump detunings $\Delta_\mathrm{e} / 2 \pi =-1.47~\mathrm{MHz}$ and $\Delta_\mathrm{o}/2 \pi = -1.11~\mathrm{MHz}$. These details change the precise form of the expressions for $\Gamma_\mathrm{o}$ and $\Gamma_\mathrm{e}$ but not the essential physics.

We set the optical pump detuning $\Delta_\mathrm{o}$ to maximize the optomechanical damping rate per incident photon. In the resolved 