Past exam of the atomic and Optical Physics course of the University of Cambridge / 2014 / Paper 1 / 3

An electric dipole transition has amplitude proportional to $⟨f|\mathbf{d}\cdot \mathbf{ϵ}|i⟩$. The electric dipole operator is odd under parity, so the initial and final states must have opposite parity. As a rank-one spherical tensor it also requires $\Delta J=0,\pm 1$, excluding $J=0\leftrightarrow 0$, and $\Delta m_J=0,\pm 1$. These follow from angular-momentum addition and the Wigner-Eckart theorem; vanishing Clebsch-Gordan coefficients can impose further restrictions. Relative to the quantization axis, $\pi$ polarization drives $\Delta m=0$, while ${\sigma }^{\pm }$ drive $\Delta m=\pm 1$.

For a single active electron, parity and the orbital triangle rule combine to give $\Delta l=\pm 1$. In LS coupling, the spin-independent electric dipole operator gives $\Delta S=0$ and $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$; total parity must still change. Spin-orbit mixing can weakly relax the approximate spin rule. For resolved hyperfine states the corresponding triangle rule is $\Delta F=0,\pm 1$, excluding $F=0\leftrightarrow 0$, with $\Delta m_F=0,\pm 1$. These labels require the appropriate coupling regime. There is no universal E1 rule demanding a particular change in principal quantum number. An E1-forbidden line may occur through another multipole or a multiphoton process.

The two-level Doppler limit, $k_B{T}_D=\hslash \Gamma /2$, balances velocity-dependent scattering friction with momentum diffusion. Multilevel atoms have additional, slower internal dynamics. Counterpropagating beams with orthogonal linear polarizations create a spatially varying polarization, giving different position-dependent light shifts for ground-state magnetic sublevels. A moving atom climbs a light-shift hill, loses kinetic energy and is optically pumped into a lower-energy sublevel: this repeated Sisyphus cooling cycle dissipates motion through emitted light.

Optical pumping responds with a finite delay. Its rate, rather than only the excited-state linewidth, sets the low-velocity response scale. In the opposite-circular-polarization arrangement, motion instead produces unequal sublevel populations and unbalanced radiation pressure. Both mechanisms can cool below the two-level Doppler temperature; diffusion heating still balances the cooling, and recoil matters at the lowest temperatures.^[ref]

Ground-state substructure, optical pumping and polarization gradients enable the extra cooling. Magnetic fields that disturb the required sublevel dynamics can suppress it. Optical molasses damps velocity but does not by itself provide position-restoring confinement.

Sound propagation. In the 1997 experiment, a focused laser beam abruptly modified the trapping potential of a dilute condensate, producing a local excitation. A rapid sequence of nondestructive phase-contrast imaging measurements followed the resulting density motion. Measuring propagation speed at different densities tested the sound-like collective excitation spectrum, rather than individual free-particle flight.^[ref] For a weakly interacting uniform condensate, the Bogoliubov spectrum isDensity dependence and trapped-geometry averaging matter when comparing a measured pulse speed with this local uniform formula. A strictly ideal gas would have no nonzero interaction-supported sound speed.

Moving-obstacle dissipation. In the 1999 experiment, a blue-detuned focused beam repelled atoms and acted as an obstacle swept through a sodium condensate. Heating was inferred from the condensate fraction after stirring and re-equilibration. The heating rose markedly near $1.6\mathrm{mm}{\mathrm{s}}^{-1}$, below the quoted peak sound speed $6.2\mathrm{mm}{\mathrm{s}}^{-1}$.^[ref] Landau's criterion $v_L={\inf }_pϵ(p)/p$ motivates a dissipation threshold, but density variation, flow acceleration around the obstacle and vortex nucleation make the observed threshold different from simply substituting the peak sound speed. Together these experiments probe a collective excitation spectrum and the onset of dissipative flow; condensation alone is not a complete demonstration of superfluidity.