  The homogeneous precession domain (HPD) of superfluid $^{3}$He has recently
been identified as a detection medium which might provide sensitivity to the
axion-nucleon coupling $g_{aNN}$ competitive with, or surpassing, existing
experimental proposals. In this work, we make a detailed study of the
statistical and dynamical properties of the HPD system in order to make
realistic projections for a full-fledged experimental program. We include the
effects of clock error and measurement error in a concrete readout scheme using
superconducting qubits and quantum metrology. This work also provides a more
general framework to describe the statistics associated with the axion gradient
coupling through the treatment of a transient resonance with a non-stationary
background in a time-series analysis. Incorporating an optimal data-taking and
analysis strategy, we project a sensitivity approaching $g_{aNN} \sim 10^{-12}$
GeV$^{-1}$ across a decade in axion mass.
