Solution (source code)

= Solution

With the coupling off, $\mathbf W=\delta\hat z$. Precession during the dark interval gives
$$
\boxed{\mathbf R_2(T)=(\sin\delta T,-\cos\delta T,0).}
$$
The populations remain $p_1=p_2=1/2$ during the dark interval; only their relative <phase> changes. For $T=2\pi/\delta$, the vector makes one complete equatorial circuit and returns to $(0,-1,0)$.