Sub-Doppler Cooling and Atom Traps

The first optical-molasses experiments measured temperatures a factor of ten below the Doppler limit $T_D=\hslash \Gamma /2k_{\mathrm{B}}$.¹ A two-level atom cannot reach them, so the explanation had to use the real multilevel structure of the alkalis: a ground state with several magnetic sublevels, a spatially varying light polarization, and the finite time optical pumping takes to redistribute population among the sublevels. The combination extracts energy far more efficiently than the Doppler mechanism, down to a floor set by the recoil of a single photon.

This lesson derives the Sisyphus mechanism from the light shift and optical pumping, locates the recoil limit, and then turns to the conservative traps that hold atoms without scattering light — magnetic traps and optical-dipole traps — and to the evaporative cooling that carries a trapped gas the last three orders of magnitude in temperature to quantum degeneracy.

Light shifts and the polarization gradient

A ground-state sublevel immersed in non-resonant light is shifted in energy by the AC Stark shift (light shift). For a two-level coupling of Rabi frequency $\Omega$ and detuning $\delta$, second-order perturbation theory gives

proportional to the local intensity through ${\Omega }^2\propto I$ and, for red detuning $\delta <0$, negative: an atom is pulled toward high intensity. In a multilevel atom the shift depends on the sublevel and on the light polarization, because different sublevels couple to ${\sigma }^{+}$, ${\sigma }^{-}$, and $\pi$ light with different strengths.

Two counter-propagating beams with orthogonal linear polarizations (the lin$\perp$lin configuration) produce a standing wave whose polarization cycles through linear, ${\sigma }^{+}$, linear, ${\sigma }^{-}$ over half a wavelength. Where the light is ${\sigma }^{+}$, the sublevels that couple to ${\sigma }^{+}$ are shifted deepest; a quarter wavelength on, where the light is ${\sigma }^{-}$, the other sublevels are deepest. Each ground sublevel therefore sees a periodic potential — a corrugation of hills and valleys with period $\lambda /2$ — and the two sublevels' corrugations are out of phase.

Sisyphus cooling

Optical pumping ties the two corrugations together. An atom in sublevel $A$ climbing toward a hilltop of its potential arrives where the light is polarized to pump it into sublevel $B$ — and it arrives in a valley of the $B$ potential, because the two are out of phase. The photon scattered in that optical-pumping step carries away slightly more energy than was absorbed (the emitted photon is blue-shifted relative to the absorbed one by the light-shift difference), so the atom loses the potential energy it just gained climbing. It then climbs again in $B$, is pumped back to a valley of $A$, and repeats. The atom is condemned to climb hills and never descend them, like Sisyphus.²

Each cycle removes an energy of order the modulation depth $U_0=|\Delta E_{\mathrm{LS}}|$, which can be tuned small by working at large detuning and modest intensity — much smaller than the $\hslash \Gamma$ scale that governs Doppler cooling. The equilibrium temperature scales as

so cooling proceeds until the light shift is lowered to the point where an atom no longer has enough kinetic energy to climb even one hill.

The recoil limit

Sisyphus cooling cannot continue indefinitely. Every optical-pumping event ends in a spontaneous emission that recoils the atom by $\hslash k$, so the residual momentum cannot fall below one photon recoil. The associated energy is the recoil energy

and the recoil limit temperature is of the same order, conventionally $k_{\mathrm{B}}{T}_r={\hslash }^2{k}^2/M=2E_r$. This is the kinetic energy an atom initially at rest acquires by emitting a single photon.

The three temperature scales are separated by the small parameter $\hslash k/Mv$, equivalently by ratios of $\Gamma$, the light shift, and the recoil frequency:

The recoil velocity for rubidium is $v_r=\hslash k/M\approx 5.9\mathrm{mm}{\mathrm{s}}^{-1}$, and the recoil temperature $T_r\approx 0.36\mu \mathrm{K}$. Sisyphus cooling routinely reaches a few times $T_r$, roughly $1$–$10\mu \mathrm{K}$. Bose-Einstein condensation, treated in the next lesson, requires temperatures another two to three orders of magnitude lower — below the recoil limit — which forces a switch to trapping and evaporation.

Conservative traps

Below the recoil limit the atoms must be held by a potential that does not scatter photons, so no recoil heating occurs. Two conservative traps dominate.

Magnetic traps. An atom with magnetic moment $\vec{\mu }$ has energy $U=-\vec{\mu }\cdot \vec{B}=m_F{g}_F{\mu }_{B}B$ in a field. A low-field-seeking state ($m_F{g}_F>0$) has minimum energy where $|\vec{B}|$ is smallest, so a field configuration with a local minimum of $|\vec{B}|$ traps it. The simplest is the quadrupole field $\vec{B}=B^{\prime }(x\hat{ı}+y\hat{ȷ}-2z\hat{k})$, which vanishes at the centre; but a field zero is fatal, because an atom passing through it can flip its spin (a Majorana spin flip) into an untrapped state and be lost. The Ioffe-Pritchard trap and the time-averaged orbiting-potential (TOP) trap remove the zero by adding a bias or rotating field, giving a harmonic minimum at nonzero $|\vec{B}|$.

Optical-dipole traps. A far-detuned laser focus traps atoms directly through the light shift. The dipole potential is

while the residual scattering rate falls off faster with detuning, $R_{\mathrm{sc}}\propto I/{\delta }^2$. Working far to the red ($\delta <0$, so $U_{\mathrm{dip}}<0$) and at large $|\delta |$ gives a deep, nearly conservative well at the intensity maximum with negligible heating. A single focused beam is a cigar-shaped trap; two crossed beams give a tight three-dimensional well; a retro-reflected beam makes an optical lattice, a periodic array of microtraps spaced by $\lambda /2$ in which atoms are pinned one or a few per site.

Evaporative cooling

No laser method reaches degeneracy. The final stage is evaporative cooling: the trap depth is lowered so the most energetic atoms escape, and the remaining atoms rethermalise by elastic collisions to a lower temperature. Because the escaping atoms carry away more than the average energy per atom, each atom lost cools the rest.

Quantitatively, if a trap of depth $\eta k_{\mathrm{B}}T$ (with $\eta \sim 5$–$10$) is progressively lowered so $\eta$ stays roughly constant, the temperature and atom number fall together while the phase-space density rises. The figure of merit is

the number of atoms within a thermal de Broglie wavelength. Laser cooling and a MOT reach $\varrho \sim 10^{-6}$; evaporation raises it by six orders of magnitude to $\varrho \sim 1$, the threshold of quantum degeneracy. Runaway evaporation occurs when the elastic collision rate grows faster than atoms are lost, so rethermalisation accelerates as the gas shrinks — the condition that made the 1995 condensates possible.

The cost is atom number: reaching degeneracy typically discards more than 99% of the atoms loaded into the trap. A MOT of $10^9$ atoms yields a condensate of $10^4$–$10^6$. The trade of number for phase-space density is favourable because the collision rate that drives rethermalisation depends on density, which the tightening trap increases even as the number falls.

The sequence is now standard: a MOT collects and pre-cools, molasses reaches a few microkelvin, the atoms are transferred to a conservative magnetic or optical trap, and forced evaporation carries the gas across the degeneracy threshold. The physics of what happens at that threshold — the macroscopic occupation of a single quantum state — is the subject of the next lesson.