[@Kaviani:15]. The quadratic optomechanical coupling coefficient: $$g^{(2)}=\frac{\omega}{2} \frac{|\langle E_{\omega}|\frac{\delta \epsilon}{\delta x}|E_{\omega}\rangle|^2}{|\langle E_{\omega}| \epsilon |E_{\omega}\rangle|^2} -\sum\limits_{\omega^{\prime}\neq\omega}\left(\frac{\omega^{3}}{\omega^{\prime 2}-\omega^{2}}\right) \frac{|\langle E_{\omega^{\prime}}|\frac{\delta \epsilon}{\delta x} |E_{\omega}\rangle|^2}{\langle E_{\omega^{\prime}}| \epsilon |E_{\omega^{\prime}}\rangle \langle E_{\omega}| \epsilon |E_{\omega}\rangle}
\label{eq:second_order}$$ describes the strength of the nonlinear photon-phonon interactions in the system [@Johnson02; @Rodriguez:11; @Eichenfield:09OE]. As discussed above, the first self-term describing the nonlinear contribution from the linear optomechanical coupling vanishes [@Kaviani:15]. However, in practice there is some amount of linear coupling [@Healey:15] generated by imperfections in the fabricated device. Linear coupling can also be introduced with an external waveguide used to couple light into and out of the cavity [@Hryciw:15]. Following Ref. [@Kaviani:15], a $g^{(2)}/2\pi\approx$ 10 MHz/nm$^{2}$ was calculated by inputting the mechanical sliding resonance profile and the optical field profiles of the fundamental ($E_\omega$) and higher order ($E_{\omega'}$) optical modes of the paddle nanocavity into Eq. \[eq:second\_order\]. These calculations where performed using finite element method simulations (COMSOL), and contributions from higher order modes over a spectral range of tens of terahertz were considered. Including this extended higher order mode spectrum was aided by automating the calculations that were performed manually in Ref. [@Kaviani:15].

![PSD of the mechanical resonance for (a) low $Q_o$ = $4.0~\times~10^3$, $Q_m~=~10^2$ and (b) high $Q_o$ = 