$4.0~\times~10^4$, $Q_m~=~10^5$ ((a,b) top plots are with $L \sim 50~\mu$m long supports). The detector noise (lower, red dashed line) and shot noise (upper, black dashed line) are also shown. (c) The PSD at $L$ = 50 $\mu$m and $\omega/\omega_m$ = 2. (d) PSD for high $Q_o$, low $Q_m$ as given in (a,b).[]{data-label="figureFive"}](figure5v5.pdf){width="11.5cm"}

Although $g^{(2)}$ is reduced in Si$_3$N$_4$ paddle nanocavities in comparison to their Si implementation, due in part to the lower refractive index of Si$_3$N$_4$, applications in quantum optomechanics can benefit from the exceptionally high aspect-ratio mechanical structures realizable in Si$_3$N$_4$. To illustrate this potential, we now investigate the effect of increasing the paddle’s sliding mode per-phonon displacement amplitude by increasing the support length $L$ to reduce the mode’s spring constant. Figure \[figureFour\](a) shows a schematic of the device with a 50 $\mu$m $\times$ 100 nm support cross section. The mechanical frequency $\omega_m$ and effective mass $m_{eff}$ of the sliding mode (computed with COMSOL) as a function of support length are shown in Fig. \[figureFour\](b). For $L$ ranging 5-50 $\mu$m, $\omega_m/2\pi$ varies from $\sim10^7-10^5$ Hz, and has a $L^{-1.9}$ dependence slightly shifted from $L^{-2}$ as expected from Euler-Bernoulli nanobeam theory [@Cleland:02] due to the mass distribution of the paddle. 