= Solution
Fix the <qubit polarization> convention by $\rho=(I+\vec P\cdot\vec\sigma)/2$, where the <Pauli matrices> are ordered in the basis $(|1\rangle,|2\rangle)$ and $\sigma_z=|1\rangle\langle1|-|2\rangle\langle2|$. Thus the <ground state> is the north pole of the <Bloch sphere>. In a <rotating frame> and within the <rotating-wave approximation>, choose the drive phase and detuning sign so that
$$
H_R=\frac\hbar2(\Omega\sigma_x+\Delta\sigma_z),\qquad
\dot{\vec P}=(\Omega,0,\Delta)\times\vec P.
$$
For example, with this ground-state convention one may take $\Delta=\omega_L-\omega_0$ and choose the transverse coupling positive. The <Bloch vector> precesses around the effective field $(\Omega,0,\Delta)$ at <angular frequency> $\Omega_R=\sqrt{\Omega^2+\Delta^2}$, tracing a circle at fixed projection on that axis. Its length is conserved under coherent <unitary time evolution>. On resonance the axis is transverse and the <Rabi oscillation> can fully invert an initially ground-state atom. Away from resonance the axis tilts, and
$$
P_{1\to2}(t)=\frac{\Omega^2}{\Omega^2+\Delta^2}\sin^2\!\left(\frac{\Omega_Rt}{2}\right).
$$
There is no relaxation towards an equilibrium <Bloch vector> in this ideal model. An inhomogeneous ensemble can nevertheless lose its \i[averaged] <qubit polarization> through <dephasing> even without <spontaneous emission>; the circle description assumes a common drive and detuning.
For $|a|^2+|b|^2=1$, the stated <quantum state> gives
$$
\rho=|\Psi\rangle\langle\Psi|=
\begin{pmatrix}|a|^2&ab^*\\a^*b&|b|^2\end{pmatrix},\qquad
\vec P=(2\operatorname{Re}(a^*b),\,2\operatorname{Im}(a^*b),\,|a|^2-|b|^2).
$$
The diagonal elements are the two state populations. The off-diagonal elements are <quantum coherences>: they retain the <relative quantum phase> and determine <interference> under a subsequent pulse. \b[This is a <pure state>], even if both populations are nonzero: $\rho^2=\rho$, $\operatorname{Tr}\rho^2=1$, and $|\vec P|=1$. A <mixed state> averages projectors over preparation <probabilities> and generally has $\operatorname{Tr}\rho^2<1$. Equal populations alone do not distinguish an equal coherent <superposition> from an unpolarized mixture.
For use in all three cases, multiplication of the gates gives the <Hadamard-phase-Hadamard qubit rotation>
$$
U=HP_\alpha H=\frac12\begin{pmatrix}1+e^{i\alpha}&1-e^{i\alpha}\\1-e^{i\alpha}&1+e^{i\alpha}\end{pmatrix}
=e^{i\alpha/2}\exp(-i\alpha\sigma_x/2).
$$
The <global phase> cancels in $\rho'=U\rho U^\dagger$. Thus the <Bloch vector> map is
$$
\boxed{(P_x,P_y,P_z)\longmapsto(P_x,\,P_y\cos\alpha-P_z\sin\alpha,\,P_y\sin\alpha+P_z\cos\alpha).}
$$
The final unlabelled <Ramsey spectroscopy> comparison applies to the three cases as follows. In an <atomic clock>, an initial coherent pulse prepares a <superposition>, a free interval $T$ accumulates a <relative quantum phase> proportional to detuning times $T$, and a final coherent pulse converts that phase to a population. Here $\alpha$ plays the role of that accumulated phase, with sign and fringe offset set by pulse conventions. In (a), the first <Hadamard gate> creates the needed <superposition> and the final gate reveals full-contrast <Ramsey interferometry>. In (b), the first <Hadamard gate> instead puts the state in $|1\rangle$, so the intervening <phase gate> has no <relative quantum phase> to imprint: the output is independent of $\alpha$. In (c), the unpolarized <density matrix> remains unchanged under every gate, so the ensemble has no <Ramsey interferometry> contrast. A coherent <unitary gate> alone cannot initialize a maximally mixed ensemble into a pure clock state.
The <Ramsey interferometry> analogy concerns preparation, phase accumulation, and readout; \b[a <Hadamard gate> is not literally a transverse $\pi/2$ <rotation> on every input]. In this basis $H=R_y(\pi/2)\sigma_z$, and its <Bloch sphere> action is a $\pi$ <rotation> about $(\hat x+\hat z)/\sqrt2$, up to a <global phase>. The phase conventions of physical <Ramsey spectroscopy> pulses must be specified to compare their outputs for arbitrary initial states.
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