  Critical dressing, the simultaneous dressing of two spin species to the same
effective Larmor frequency, is a technique that can, in principle, improve the
sensitivity to small frequency shifts. The benefits of spin dressing and thus
critical dressing are achieved at the expense of generating a large (relative
to the holding field $B_{0}$,) homogeneous oscillating field. Due to inevitable
imperfections of the fields generated, the benefits of spin dressing may be
lost from the additional relaxation and noise generated by the dressing field
imperfections. In this analysis the subject of relaxation and frequency shifts
are approached with simulations and theory. Analytical predictions are made
from a new quasi-quantum model that includes gradients in the holding field
$B_{0}=\omega _{0}/\gamma $ and dressing field $B_{1}=\omega _{1}/\gamma $
where $B_{1}$ is oscillating at frequency $\omega $. The results are compared
with a Monte Carlo simulation coupled with a 5$^{\text{th}}$ order Runge-Kutta
integrator. Comparisons of the two methods are presented as well as a set of
optimized parameters that produce stable critical dressing at a range for
oscillating frequencies $\omega ,$ as well as pulsed frequency modulation
parameters for maximum sensitivity.
