signal-to-noise ratio $\Sigma^{(0)}$ for $Q_m$ = $10^8$ (top, green), $Q_m$ = $10^5$ (middle, orange), and $Q_m$ = $10^2$ (bottom, blue) at ultrahigh $Q_o$ = $10^6$. (b) The number of phonons $n_{th}$ in the device at $T_b$ = 10 mK (blue) and the drive amplitude $x_d$ (red, and in units of phonon number $n_d$) to achieve a signal-to-noise ratio $S$ &gt; 1 when measuring phonon shot noise.[]{data-label="figureSix"}](figure6v2.pdf){width="12.5cm"}

Although the longer supports make the quantized phononic energy fluctuations of the oscillator observable, there will be increased thermal decoherence associated with the lower $\omega_m$. There is also a trade-off in the increase in thermal phonon occupation number at achievable cryogenic operation temperature, requiring development of the ability to further cool the mechanical resonance from $T_b$ to an effective temperature corresponding to the quantum ground state. The number of phonons $n_{th}$ remaining in the device at $T_{b}$ and $\omega_m$ given by $L$ (according to Bose-Einstein statistics) is shown in Fig. \[figureSix\](b) (blue). Removing these phonons in order to demonstrate the quantum effects described above would require sideband-unresolved nonlinear optomechanical cooling [@Clark2017] to extract mechanical energy by scattering incident photons to higher energies (anti-Stokes scattering), as recently achieved in a low-frequency optomechanical system [@Clark2017]. By 