linear optomechanical coupling between the optical cavity modes that is discussed in detail in Refs. [@miao2009standard; @ludwig2012enhanced; @paraiso2015position]. This coupling, which originates from the non-zero cross-coupling terms in Eq. , is a form of decoherence that imposes a strong coupling requirement for QND measurements [@miao2009standard]. Applying the theory from Ref. [@miao2009standard] to our system and assuming that the nonlinear coupling is dominated by a single cross-coupling term, this condition roughly requires $\kappa_1\kappa_2/|\Delta\omega_o (\omega'~-~\omega)|~<~1$, where $(\omega'~-~\omega)$ is the detuning between the cavity modes whose cross-coupling is the dominant contributor to $g^{(2)}$ in Eq. , and $\kappa_i = \omega_i/Q_i$ are the optical loss rates of the modes. For our system, $|\omega' - \omega|/2\pi \sim 10$ THz, $\kappa_{1,2}/2\pi \sim 1$ GHz ($Q_o~\sim~10^5)$, and $g^{(2)}/2\pi \sim 1$ Hz, indicating that the device is four to five orders of magnitude away from satisfying this condition, and that increasing $Q_o$ as well as $g^{(2)}$ and/or $|\omega - \omega'|$ is required before QND measurements will be possible. Increasing $Q_o$ to $10^7$, as achieved in silicon photonic crystals [@sekoguchi2014photonic], is one such path towards this goal. In parallel, further increasing $g^{(2)}$ while maintaining a large $|\omega~-~\omega'|$ would contribute to reaching this regime.

![(a) Predicted phononic Fock state measurement 