= Solution
Treat $|a\rangle=|0\rangle$ and $|b\rangle=|1\rangle$ as an isolated <ground-state hyperfine qubit>. Their transition angular frequency is $\omega_0=2\pi\times320.7\,\mathrm{MHz}$. The radiofrequency <magnetic field> couples them through an allowed <magnetic dipole transition>. After choosing its phase reference, take the two-level laboratory <Hamiltonian operator> to be
$$
H(t)=-\frac{\hbar\omega_0}{2}Z+\hbar\Omega\cos(\omega t)X,
$$
where $X,Z$ are <Pauli operators>. Here $\Omega$ is defined by the on-resonance rotating-frame off-diagonal matrix element $\hbar\Omega/2$.
Use a <quantum rotating frame> and the <rotating-wave approximation>, valid for $\Omega,|\delta|\ll\omega_0$, with <Rabi detuning> $\delta=\omega_0-\omega$. Ignoring off-resonant coupling to other levels gives
$$
H_R=\frac\hbar2(\Omega X-\delta Z).
$$
At resonance $\delta=0$, the <unitary time evolution> is
$$
U_R(t)=\cos\frac{\Omega t}{2}I-i\sin\frac{\Omega t}{2}X,
\qquad
|\psi_R(t)\rangle=\cos\frac{\Omega t}{2}|a\rangle-i\sin\frac{\Omega t}{2}|b\rangle.
$$
Thus the <Born rule> gives coherent <Rabi oscillations>,
$$
\boxed{P_a(t)=\cos^2\frac{\Omega t}{2},\qquad P_b(t)=\sin^2\frac{\Omega t}{2}.}
$$
The stated amplitudes are in the rotating frame; the laboratory carrier phases are tracked by the drive reference.
These logical states are orthogonal, individually resolvable <hyperfine levels> in the same electronic <ground state>. They avoid fast optical decay, can be coherently coupled by radiofrequency control, and have a state-selective optical readout through $|c\rangle$. The excited level $|c\rangle$ is useful for <fluorescence> detection rather than for the storage superposition in this model. Long lifetimes alone do not guarantee a useful qubit: control isolation and adequate <quantum coherence> are also required.
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