= Solution
{c}
An ideal instantaneous NOT about $x$ reverses each atom's accumulated <phase> relative to its subsequent free precession. Algebraically, $X\sigma_zX=-\sigma_z$, so for a static <Rabi detuning> and $F=e^{i\delta T\sigma_z/4}$,
$$
FXF=X.
$$
The entire <Ramsey interferometry> sequence is consequently $PXP=-iI$, up to the additional common <global phase> if NOT is implemented as $-iX$. Thus
$$
\boxed{\rho_{11}(T)=1\quad\hbox{for every }T.}
$$
This <spin echo> restores the initial population even for the distribution of static <Rabi detunings> in part (d). It removes reversible inhomogeneous <dephasing>; fluctuating <Rabi detuning>, finite pulse errors and irreversible loss of <quantum coherence> need not refocus. With an opposite final pulse, the return population would instead be zero; this distinguishes that alternative from the sequence actually specified.
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