Past exam of the atomic and Optical Physics course of the University of Cambridge / 2015 / Paper 1 / 3

In the ideal point-nucleus Dirac-Coulomb problem, hydrogen levels depend on $n$ and $j$, so $2S_{1/2}$ and $2P_{1/2}$ are degenerate despite their different orbital angular momenta. The Lamb shift removes this degeneracy: the $2S_{1/2}$ level lies about $h\times 1.06\mathrm{GHz}$ above $2P_{1/2}$, an energy of roughly $4.4\mu \mathrm{eV}$. This is distinct from the much larger $2P_{3/2}-2P_{1/2}$ fine-structure splitting.

quantum electrodynamics changes the electron's interaction with the Coulomb field. electron self-energy and vacuum polarization give state-dependent radiative corrections; a complete prediction also keeps specified recoil and nuclear-size contributions. An $S$ wave samples the nucleus differently from a $P$ wave, so equal $n,j$ no longer ensures equal energy. A leading-log estimate for an $nS$ radiative contribution scales as $\alpha (Z\alpha {)}^4{m}_e{c}^2\ln [(Z\alpha {)}^{-2}]/n^3$, with a numerical prefactor and further state-dependent terms. This is a scale estimate, not a replacement for renormalized bound-state QED. The measurable shift is a difference relative to a specified reference Hamiltonian, rather than an unregulated absolute vacuum energy.^[ref]

The original microwave experiment used a tungsten oven to dissociate hydrogen, an electron beam to populate metastable $2S$ atoms, and electron ejection from a metal target to measure surviving metastable atoms. Microwaves transferred population to rapidly decaying $2P$, reducing detector current. At a fixed microwave frequency, interruption of the radiation and a scan of static magnetic field revealed Zeeman-tuned resonances. Comparing resonance locations with the Dirac-level prediction showed the $S$ level shifted upward. Field calibration and homogeneity, excessive microwave intensity, unresolved hyperfine structure and stray-electric-field quenching limit the measurement; the signal is not direct absorption by a large equilibrium excited population.^[ref]

A two-photon transition is a second-order atom-field process. For two-photon absorption, energy conservation requires $E_f-E_g=\hslash ({\omega }_1+{\omega }_2)$. A virtual intermediate state in two-photon absorption need not be populated as a real stationary state. In the electric dipole approximation, the two time orderings give an amplitude proportional toThe electric-field amplitudes multiply this expression. Far from intermediate resonance, the denominators are large relative to linewidths and one-photon couplings, keeping real intermediate occupation small. Near an intermediate resonance, sequential absorption, linewidths and saturation require a fuller treatment. For weak unsaturated absorption the rate scales as $I_1{I}_2$, or $I^2$ when both photons come from one field. Absorption followed by stimulated emission instead gives a difference-frequency Raman transition; it is not two-photon absorption at the sum frequency.

Two electric dipole interactions change parity twice, so initial and final states have the same parity. For a one-electron orbital, the usual changes are $\Delta \ell =0,\pm 2$, subject to angular-momentum and polarization constraints. For identical equal-frequency photons, exchange symmetry removes the antisymmetric rank-one tensor: ranks zero and two survive. Thus $J=0\to 0$ is possible, whereas $J=0\to 1$ is forbidden in this degenerate electric-dipole approximation. It would be too strong to prohibit every $\Delta J=1$ transition between nonzero angular momenta, since a rank-two tensor can connect some such pairs. Spin remains unchanged to leading electric-dipole order.

A central example is hydrogen $1S\to 2S$, forbidden as a one-photon electric dipole transition but accessible with two photons near $243\mathrm{nm}$. Absorption of one photon from each of equal-frequency counterpropagating beams makes the first-order Doppler shift $({\mathbf{k}}_1+{\mathbf{k}}_2)\cdot \mathbf{v}$ vanish and suppresses net recoil. The two same-direction absorption paths can still give a broad Doppler background. A primary demonstration used a frequency-doubled pulsed dye laser near this wavelength and compared with a simultaneously recorded Balmer reference.^[ref]

Experimentally, scan the sum frequency and detect the excited state, its subsequent fluorescence or its ionization. Counterpropagating and single-beam spectra distinguish the narrow Doppler-free contribution from its background. Varying intensity tests weak-field scaling and extrapolates AC Stark shifts and power broadening; frequency calibration, beam overlap, residual angle and second-order Doppler shifts still matter. “Doppler free” here cancels the leading velocity term, not every systematic shift.

Two complementary signatures are suppressed low-speed dissipation and quantized circulation. Condensation alone establishes macroscopic occupation, not all of these flow properties. In a uniform weakly interacting condensate, the Landau criterion givesfor the Bogoliubov spectrum. Creating an excitation in a flow changes energy by $E(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$; a negative value allows energy transfer. Boundaries and vortex nucleation can make an actual measured threshold lower than this ideal bulk sound speed.

In the moving-obstacle test of atomic superfluidity, a blue-detuned focused laser repelled atoms in a sodium condensate and was scanned at different speeds. After a fixed exposure, time-of-flight absorption imaging measured the reduction of condensate fraction as a heating indicator. Low-speed heating was small, with a pronounced onset near $1.6mm/s$, below the peak-density sound speed near $6.2mm/s$. A stationary-beam control showed no heating; above-transition clouds constrained heating through the noncondensed fraction. Changing scan frequency and amplitude at common obstacle velocity helped separate a flow threshold from a particular trap resonance. Density inhomogeneity, obstacle geometry and vortex nucleation limit identification of the observed onset with a unique homogeneous Landau velocity.^[ref]

A second experiment stirred rubidium-87 in a magnetic trap by rotating a small transverse anisotropy generated with a far-detuned attractive laser. After stirring, the cloud expanded for $27\mathrm{ms}$ and absorption imaging along the rotation axis revealed vortex cores. Below a threshold no cores appeared; faster rotation produced one or several. Ramping the stirrer down before release reduced additional excitation, and waiting after removing it tested vortex survival. Beam-centre stability and residual trap anisotropy are important controls; an isolated density dip without the rotation and persistence evidence would be less conclusive. This stirred-condensate vortex experiment is a response to rotation, distinct from merely imaging the appearance of a condensate.^[ref]

The superfluid velocity is ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$. A closed loop avoiding a core must return a single-valued order parameter, soA singly quantized vortex therefore has $2\pi$ phase winding and circulation $h/m$, with depleted density in a core of order the healing length. This quantized rotational response differs from ordinary rigid-body flow. The experiment supports that interpretation through controlled stirring and vortex formation; imaging cores alone does not independently measure every phase winding or an absolute zero dissipative rate.