The corresponding $m_{eff}$ varies from $\sim1-7$ pg. Shown in Fig. \[figureFour\](c), the zero-point fluctuation amplitude $x_{zpf}~=~\sqrt{\hbar/2m_{eff}\omega_{m}}$ varies from 20-120 fm (where $\hbar$ is the reduced Planck’s constant), and the corresponding single-photon to two-phonon coupling rate $\Delta\omega_{o}\equiv|g^{(2)}x^{2}_{zpf}|$ varies from 2$\pi~\times$ (0.25-0.9) Hz. These plots illustrate that increasing $L$ results in a larger per-phonon nonlinear optomechanical coupling due to the increase in $x_{zpf}$.

Given these simulated parameters we can make predictions regarding the suitability of Si$_3$N$_4$ paddle nanocavities for nonlinear optomechanical detection of motion driven by thermal and quantum effects. If photons are coupled into and out of the cavity with an external waveguide (e.g. a fiber taper or end-fire coupled waveguide) and measured with a photodetector, the optical response of the paddle nanocavity transducing the nonlinear mechanical displacement can be predicted [@Kaviani:15]. For input power $P_i$, the photodetected optical power spectral density (PSD) $S_{P}^{(2)}(\omega) = \frac{1}{4}P_i^2 G_2^{2} S_{x^{2}}(\omega)$, where $G_2$ is the quadratic coefficient of transduction of the fluctuating waveguide output, and $S_{x^{2}}$ is the PSD of the $x^2$ mechanical motion, given in Ref. [@Kaviani:15] for the case of a thermally driven mechanical resonator. Here $G_2$ describes the cavity optical response measured by the photodetector, and includes the influence of the 