  Nanoscale photonic crystal cavity optomechanical devices enable detection of
nanomechanical phenomena with a sensitivity sufficient to observe quantum
effects. Here we present the design of a one-dimensional air-mode photonic
crystal cavity patterned in a silicon nitride nanobeam, and show that it forms
the basis for cavity optomechanical split-beam and paddle nanocavity devices
useful for force detection and nonlinear quantum sensing. The air-mode of this
device is advantageous for optomechanical coupling, while also having ultrahigh
optical quality factor $Q_o\sim 10^6$ despite its proximity to the light-line
and the relatively low refractive index of silicon nitride. Paddle nanocavities
realized from this device have a quadratic coupling coefficient
$g^{(2)}/2\pi$~=~10~MHz/nm$^{2}$, and their performance within the context of
quantum optomechanics experiments is analyzed.
