is the thermal lifetime quantifying the rate of decoherence of the ground-state-cooled nanomechanical resonator due to bath phonons, and $S_{w_{o}}~=~\hbar\omega_{o}\kappa^{2}/16P_{i}$ is the shot-noise-limited sensitivity of an ideal Pound-Drever-Hall detector for cavity loss rate $\kappa$ (cooling the device to the ground state will be further discussed below). With $Q_m$ = $10^8$ and $L$ = 500 $\mu$m, $\Sigma^{(0)}$ &gt; 1, however this can also be achieved with lower $Q_m$ and longer $L$. We note that calculating the signal-to-noise ratio as in Ref. [@paraiso2015position] gives similar results.

For measurements of phonon shot noise (the granularity of large phonon number coherent states) [@ClerkPRL2010] with signal-to-noise ratio $S=8\bar{n}_{d}n_{th}\Sigma^{(0)}$ &gt; 1, the required drive amplitude $x_d$ and its correspondence in units of phonon number $\bar{n}_d$, assuming a thermal phonon number $n_{th}=1/4$, is shown in Fig. \[figureSix\](b). At $L$ = 500 $\mu$m, the number of required coherent drive phonons is &lt; 400, a relatively small value that is more than three orders of magnitude less than what is required to drive the Si device previously discussed [@Kaviani:15]. This leaves significant available headroom in drive strength to compensate for non-optimal devices parameters and reduced $\Sigma^{(0)}$.

However, the above figures of merit do not consider the effect of parasitic 