paddle length of $\sim$538 nm and without any supports, as shown in Fig. \[figureThree\](d), a $Q_o$ = $1.1~\times~10^5$ is found. In a realistic device, the paddle is suspended from the unpatterned device layer by thin supports as shown in Fig. \[figureThree\](c). Adding 100 nm wide supports connected to the center of the paddle reduces $Q_o$ to $1.7~\times~10^4$.

The support and paddle dimensions determine the properties of the paddle’s mechanical resonances, which can interact optomechanically with the optical modes of the device. Here we are interested in the axial sliding mechanical resonance shown in Fig. \[figureThree\](c) that mimics membrane-in-the-middle devices [@Thompson2008; @sankey2010strong; @mason2018continuous; @rossi2018observing; @rossi2018measurement]. For the sliding resonance, the paddle nanocavity has a vanishing linear optomechanical coupling owing to the odd symmetry of the mechanical mode and the even symmetry of the optical mode intensity [@Kaviani:15]. However, it has a nonlinear quadratic optomechanical coupling that is of particular interest for quantum optomechanics experiments [@Thompson2008; @sankey2010strong; @paraiso2015position]. This coupling strength is determined by the change in local dielectric constant $\Delta\epsilon(\mathbf{r};x)$ as a function of position $\mathbf{r}$ of the paddle’s normal mechanical mode displacement amplitude $x$, overlapped and cross-coupling the electric fields $E_{\omega}$ of the nanocavity optical mode spectrum at frequencies $\omega$ 