= Solution
For an unpolarized ensemble $\rho_i=I/2$. Every <unitary gate> satisfies $U(I/2)U^\dagger=I/2$, so
$$
\boxed{\rho_f=\tfrac12I,\qquad \vec P_f=(0,0,0),\qquad p_1=p_2=\tfrac12.}
$$
The <purity> remains $1/2$. Equal populations here come with zero <quantum coherence>, unlike the state in (b). There is \b[no <Ramsey interferometry> fringe]: the two orthogonal components of any equal-weight ensemble decomposition give cancelling population responses. <Quantum state preparation> that reduces the ensemble <entropy> is needed for the usual polarized clock interrogation.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2012/paper-1-gate-fringes.png]
{title=Gate sequence populations}
The original plot compares the three excited-state populations. The constant $1/2$ lines coincide despite the very different <purities> in (b) and (c).
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