 Solution

An ideal instantaneous NOT about $x$ reverses each atom's accumulated phase relative to its subsequent free precession. Algebraically, $X{\sigma }_{z}X=-{\sigma }_z$, so for a static Rabi detuning and $F=e^{i\delta T{\sigma }_z/4}$,The entire Ramsey interferometry sequence is consequently $PXP=-iI$, up to the additional common global phase if NOT is implemented as $-iX$. ThusThis 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.