= Solution
A good atomic <qubit> needs two reproducible, <orthogonal> internal states whose populations and relative <quantum phase> survive for many gate times. Both should be long lived, with weak <spontaneous emission> and little <decoherence> from fluctuating fields or atomic motion. Their transition must nevertheless permit fast, selective, coherent control with little leakage to other states. Reliable initialization and state-selective <quantum measurement> are also required; <optical pumping> and fluorescence readout can supply these functions.
For <rubidium-87>, use the two <ground state> <hyperfine levels> in the $5\,{}^2S_{1/2}$ electronic configuration. The electronic <angular momentum> is $J=1/2$, and <addition of angular momentum> to the given <nuclear spin> $I=3/2$ gives $F=|I-J|,\ldots,I+J=1,2$. A suitable <Rubidium-87 clock qubit> is
$$
\boxed{|0\rangle=|5\,{}^2S_{1/2};F=1,m_F=0\rangle,\qquad
|1\rangle=|5\,{}^2S_{1/2};F=2,m_F=0\rangle.}
$$
These are coupled electron-nucleus <angular momentum eigenstates>, rather than two bare nuclear-spin projections. Their <hyperfine structure> supplies a resolvable splitting $\nu_{01}\simeq6.835\,\mathrm{GHz}$. Both belong to the electronic <ground state>, avoiding a short optical excited-state lifetime. Since the weak-field <Zeeman shift> is proportional to $g_Fm_FB$, both have zero linear shift at $B=0$: their <rubidium-87 clock transition> suppresses first-order magnetic-field <dephasing> near zero field. A <quadratic Zeeman shift of a clock transition> and differential optical shifts remain. A small stable bias field can define the <quantization axis> and help resolve unwanted transitions, at the cost of some residual magnetic sensitivity.<Rubidium 87 D Line Data by Daniel A. Steck>[Section 3, ground-state hyperfine structure and clock-transition Zeeman shift]{ref}
Drive the <magnetic dipole transition> with a resonant <microwave> field of suitable polarization, here $\Delta m_F=0$, or use two phase-coherent optical fields for a far-detuned <stimulated Raman transition> with their frequency difference resonant with $\nu_{01}$. The optical excited state is then only virtually populated, reducing <spontaneous emission>. In the rotating frame and <rotating-wave approximation>, define the drive phase $\phi$ by the effective <Hamiltonian>
$$
H_{\mathrm{rot}}=\frac{\hbar\Omega}{2}(\cos\phi\,X+\sin\phi\,Y),\qquad
R_\phi(\theta)=e^{-i\theta(\cos\phi\,X+\sin\phi\,Y)/2},\quad\theta=\Omega t.
$$
Here $X,Y,Z$ are the <Pauli matrices> on the chosen <qubit>, and $\Omega$ is the <Rabi angular frequency>. Pulse duration controls the <rotation gate> angle, while phase selects the axis in the equatorial plane of the <Bloch sphere>. A $\pi$ pulse exchanges the populations; a $\pi/2$ pulse makes an equal <quantum superposition> from either basis state. Controlled <Rabi detuning>, a differential energy shift, or compositions of these pulses give $z$ rotations. For example, $R_z(\alpha)R_y(\beta)R_z(\gamma)$ realizes any one-<qubit> <unitary operator> up to a <global phase>. Thus the atomic encoding admits arbitrary single-<qubit> <quantum gates>.
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