= Solution
Use the ordered basis $(|1\rangle,|2\rangle)$ and $\sigma_z=\operatorname{diag}(1,-1)$, with
$$
\rho=\tfrac12(I+\mathbf R\cdot\boldsymbol\sigma),\qquad
R_z=p_1-p_2,\quad p_2=\frac{1-R_z}{2}.
$$
A pure <quantum superposition> $|+x\rangle=(|1\rangle+|2\rangle)/\sqrt2$ has $\mathbf R=(1,0,0)$ on the <Bloch sphere>. In contrast, a <mixed state> $\rho=p|1\rangle\langle1|+(1-p)|2\rangle\langle2|$ has $\mathbf R=(0,0,2p-1)$, inside the sphere for $0<p<1$. At equal populations its vector is zero. Equal populations alone therefore do not distinguish a coherent superposition from a mixture; their off-diagonal <density matrix> elements differ.
Choose the drive <phase> so the rotating-frame <Hamiltonian> is $H_{\rm rot}=\hbar(\Omega\sigma_x+\delta\sigma_z)/2$, where $\delta=\omega-\omega_0$ as specified. The <Rabi vector> is the angular-velocity vector
$$
\boxed{\mathbf W=(\Omega,0,\delta),\qquad \dot{\mathbf R}=\mathbf W\times\mathbf R.}
$$
Here $\Omega$ is the on-resonance population oscillation <angular frequency>, not the unhalved coefficient of a cosine coupling. From $\mathbf R(0)=\hat z$, rotation about $\mathbf W$ at $W=\sqrt{\Omega^2+\delta^2}$ gives
$$
\mathbf R(t)=\left(\frac{\Omega\delta}{W^2}[1-\cos Wt],\,-\frac{\Omega}{W}\sin Wt,\,
\frac{\delta^2+\Omega^2\cos Wt}{W^2}\right),
$$
$$
\boxed{p_2(t)=\frac{\Omega^2}{\Omega^2+\delta^2}\sin^2\left(\frac{\sqrt{\Omega^2+\delta^2}\,t}{2}\right).}
$$
For $\Omega\gg|\delta|$, the rotation axis is nearly transverse, the trajectory nearly a great circle, and transfer is almost complete. For $\Omega=\delta>0$, the axis bisects $+x$ and $+z$, the orbit lies in $R_x+R_z=1$, and $p_2=\tfrac12\sin^2(\sqrt2\Omega t/2)$ never exceeds $1/2$. Each member remains a <pure state> in this coherent evolution.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2015/paper-1-rabi-bloch.png]
{title=Pure and mixed Bloch vectors, resonant and detuned trajectories, and their population oscillations}
For the Ramsey free interval, let atom $j$ have detuning $\delta_j=\delta+\eta_j$. Instantaneous ideal pulses give the <Ramsey ensemble coherence> factor $\chi(T)=\langle e^{i\eta_jT}\rangle$. Averaging the free-evolution vector obtained in step (ii) gives
$$
\overline{\mathbf R}(T)=\left(\operatorname{Im}[e^{i\delta T}\chi(T)],\,-\operatorname{Re}[e^{i\delta T}\chi(T)],\,0\right),
\qquad
\boxed{\overline p_2(T)=\tfrac12\{1+\operatorname{Re}[e^{i\delta T}\chi(T)]\}.}
$$
Different atoms fan out around the equator, shortening the ensemble <Bloch vector> even though every individual's vector remains unit length. A broad smooth detuning distribution usually makes the average spiral towards the centre and damps the Ramsey fringes towards $1/2$. The <dephasing> time is of order $1/\Delta\omega_0\sim5/\delta$. <inhomogeneous ensemble dephasing> is not necessarily irreversible decoherence of individual atoms; an appropriate <spin echo> can rephase static <frequency> offsets.
The given spread does not specify a <probability distribution> or whether it denotes an RMS width. An exact envelope is therefore not determined. The figure illustrates, rather than infers, a <Gaussian distribution> with RMS $\sigma_\delta=0.2\delta$, for which $\chi=e^{-\sigma_\delta^2T^2/2}$. A <uniform distribution> would instead give a sinc envelope and side lobes. Both have the same qualitative initial loss of contrast on the quoted inverse-spread scale.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2015/paper-1-ramsey.png]
{title=An ideal Ramsey path and an explicitly illustrative Gaussian ensemble-dephasing envelope}
In the <rubidium-87 ground-state hyperfine structure>, $J=1/2$ and $I=3/2$, so $F=1,2$. Taking $H_{\rm hf}=A\mathbf I\cdot\mathbf J/\hbar^2$ gives
$$
E_F=\frac A2[F(F+1)-I(I+1)-J(J+1)],\qquad
E_1=-\frac54A,\quad E_2=\frac34A.
$$
The zero-field splitting is $\Delta_{\rm hf}=2A\simeq h\times6.8\,\mathrm{GHz}$. Ignoring the small nuclear magnetic term, the weak-field <Landé g-factors> are $g_{F=1}\simeq-1/2$ and $g_{F=2}\simeq+1/2$. The diagram labels all three $F=1$ and five $F=2$ substates, with linear shifts $g_F\mu_Bm_FB$.
The pair \b[$|1\rangle=|F=1,m_F=0\rangle$ and $|2\rangle=|F=2,m_F=0\rangle$] is connected by a <magnetic dipole transition> with $\Delta F=1$, $\Delta m_F=0$, driven with its oscillating magnetic component along the quantization axis. Both first-order shifts vanish, giving the <rubidium-87 clock transition> first-order insensitivity near zero field. Its insensitivity does not imply absence of quadratic shifts.
For these two $m_F=0$ levels the <Breit-Rabi formula> gives the <quadratic Zeeman shift of a clock transition>:
$$
\Delta E(B)=\sqrt{(2A)^2+[(g_J-g_I)\mu_BB]^2}
\simeq2A+\frac{\mu_B^2B^2}{A},
$$
where the last expression neglects the nuclear term and sets $g_J\simeq2$. Hence
$$
\frac{\partial\Delta E}{\partial B}\simeq\frac{2\mu_B^2B}{A},\qquad
\boxed{\frac{|\mu_2-\mu_1|}{\mu_B}\simeq\frac{2\mu_BB}{A}\simeq8.2\times10^{-3}\quad(B=10\,\mathrm G).}
$$
This uses $\mu_B/h\simeq1.40\,\mathrm{MHz/G}$ and the supplied $A/h\simeq3.4\,\mathrm{GHz}$. With physical moments defined by $\mu_i=-\partial E_i/\partial B$, the <differential magnetic moment> is \b[$\mu_2-\mu_1\simeq-8.2\times10^{-3}\mu_B$]. If effective magnetic moment denotes the positive <energy> slope instead, that slope difference is positive. The magnitude and physical <frequency> sensitivity are the same. Retaining the full atomic factors gives the familiar quadratic clock-frequency coefficient about $575\,\mathrm{Hz/G^2}$; the simplified supplied parameters give about $576\,\mathrm{Hz/G^2}$.<Rubidium 87 D Line Data by Daniel A. Steck>[§3, equations (26)–(29); table of magnetic factors and clock-transition quadratic shift]{ref}
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2015/paper-1-rubidium-levels.png]
{title=Weak-field rubidium-87 hyperfine manifolds and the quadratic clock-transition shift}
\b[Use a far-detuned <optical dipole trap>], such as a red-detuned focused-beam trap, to confine both clock states. Their first-order magnetic moments vanish; their residual quadratic magnetic slopes are opposite, so a simple static magnetic-field minimum does not confine both in the same way. An optical potential acts through the electronic <polarizability> and can trap both ground hyperfine states. It is not perfectly state independent: differential <AC Stark shifts>, intensity noise and residual magnetic gradients still need control during Ramsey evolution.
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