Optimizing Spin Dressing Sensitivity for the nEDMSF Experiment

nEDMSF aims to measure the neutron electric dipole moment ($d_n$) with unprecedented precision. In this paper we explore the experiment's sensitivity when operating with an implementation of the critical dressing method in which the angle between the neutron and Helium-3 spins ($\phi_{3n}$) is subjected to a square modulation by an amount $\phi_d$ (the "dressing angle"). Several parameters can be tuned to optimize sensitivity. We find roughly 10% improvement over a previous estimate, resulting primarily from the addition of a waiting period between the $\pi/2$ pulse that initiates $d_n$-driven $\phi_{3n}$ growth and the start of $\phi_{3n}$ modulation. We find negligible further improvement by allowing $\phi_d$ to vary continuously over the course of a run, and no degradation resulting from the addition of an in situ background measurement into each $\phi_{3n}$ modulation sequence. A complete simulation confirms a 300 live-day sensitivity of $\sigma = 1.45\times10^{-28}\,e \cdot \rm{cm}$. At this level of sensitivity, $\sigma_{\phi_{3n0}} = 1\,{\rm mrad}$ precision on the initial $n/^3{\rm He}$ angle difference is not negligible.