Equatorial reachability of a detuned spin pulse ($|\Delta |\le \Omega$)

For an initial pole of the Bloch sphere, one fixed-amplitude harmonic pulse detuned by $\Delta$ transfers at most ${\Omega }^2/({\Omega }^2+{\Delta }^2)$ of the population. A pure equatorial state requires half transfer, so some pulse duration reaches the equator exactly when $|\Delta |\le \Omega$. Adjusting drive phase then selects its transverse azimuth. Exact inversion to the opposite pole is unavailable for nonzero detuning with that single fixed pulse. This is a restriction on the specified controls, rather than on composite-pulse or frequency-swept state preparation.