---
title: Sub-Doppler Cooling and Atom Traps
module: Modern Atomic Physics
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  Optical molasses cools multilevel atoms below the Doppler limit. A polarization
  gradient plus optical pumping makes an atom repeatedly climb a light-shift hill and
  be pumped to the valley, losing kinetic energy each cycle — Sisyphus cooling. The
  floor is the recoil limit, one photon momentum of residual motion. Below it, cooling
  must avoid scattering photons: conservative magnetic and optical-dipole traps hold
  the atoms while forced evaporation removes the hot tail, driving the phase-space
  density up toward quantum degeneracy.
topics: [Modern Atomic Physics]
sources:
  - book: Foot
    ref: "Ch. 9 §9.4–9.6 — Sub-Doppler cooling; Ch. 10 — Magnetic Trapping, Evaporative Cooling and Bose-Einstein Condensation"
  - book: Metcalf & van der Straten
    ref: "Laser Cooling and Trapping (Springer, 1999), Ch. 8 — Multilevel Atoms; Ch. 11 — Magnetic Traps; Ch. 14 — Optical Dipole Traps"
  - book: Demtröder
    ref: "Ch. 11 — Optical Cooling and Trapping of Atoms (concepts)"
draft: false
---

The first optical-molasses experiments measured temperatures a factor of ten below
the Doppler limit $T_D = \hbar\Gamma/2k_{\mathrm B}$.[^lett] 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

$$
\Delta E_{\mathrm{LS}} = \frac{\hbar\Omega^2}{4\delta}
= \frac{\hbar\Gamma^2}{8\delta}\,s_0
\quad (\text{for } |\delta| \gg \Gamma,\ s_0 \ll 1),
$$

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.

$$
% caption: In a lin-perp-lin standing wave the polarization cycles between
% sigma-plus and sigma-minus every quarter wavelength, giving each ground sublevel
% a light-shift potential; the two sublevel potentials are a half-period out of
% phase.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (7.2,0) node[right] {position};
  \draw[->, black] (0,-1.7) -- (0,1.9) node[above] {light shift};
  % sublevel A potential (cosine)
  \draw[black, very thick, domain=0:6.8, samples=140, smooth]
    plot (\x, {-1.2*cos(deg(\x))});
  % sublevel B potential (out of phase)
  \draw[black, very thick, dashed, domain=0:6.8, samples=140, smooth]
    plot (\x, {1.2*cos(deg(\x))});
  \node[black, anchor=south] at (3.14,1.25) {sublevel A};
  \node[black, anchor=north] at (3.14,-1.25) {sublevel B};
  % mark the polarization character
  \node[black, anchor=north] at (0,-0.05) {$\sigma^+$};
  \node[black, anchor=north] at (3.14,-0.05) {$\sigma^-$};
  \node[black, anchor=north] at (6.28,-0.05) {$\sigma^+$};
\end{tikzpicture}
$$

## 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.[^sisyphus]

> **Definition (Sisyphus cooling).** Sub-Doppler cooling in a polarization-gradient
> field in which optical pumping preferentially transfers an atom from a potential
> hilltop of one ground sublevel to a valley of another, so it repeatedly converts
> kinetic energy into potential energy and radiates the difference. The energy lost
> per cycle is of order the light-shift depth $U_0$, not the natural linewidth
> $\hbar\Gamma$.

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 $\hbar\Gamma$ scale that governs Doppler cooling. The equilibrium
temperature scales as

$$
k_{\mathrm B}T \sim U_0 \propto \frac{\hbar\Omega^2}{|\delta|}
\propto \frac{I}{|\delta|},
$$

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.

$$
% caption: Sisyphus cycle: an atom climbs a light-shift hill in sublevel A, is
% optically pumped near the top into a valley of sublevel B losing energy U-nought,
% climbs again, and is pumped back, radiating the difference each time.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % potential A
  \draw[black, very thick, domain=0:6.4, samples=140, smooth]
    plot (\x, {-0.9*cos(deg(\x))});
  % potential B
  \draw[black, very thick, dashed, domain=0:6.4, samples=140, smooth]
    plot (\x, {0.9*cos(deg(\x))});
  % atom climbing A from valley to hilltop
  \fill[acc] (0,-0.9) circle (2.4pt);
  \draw[->, acc, thick] (0,-0.9) .. controls (1.2,-0.6) and (2.2,0.4) .. (3.0,0.85);
  % pump down (dashed) to B valley
  \draw[->, black, thick, dashed] (3.14,0.85) -- (3.14,-0.85);
  \node[black, anchor=west] at (3.25,0) {pump, radiate};
  \fill[acc] (3.14,-0.9) circle (2.4pt);
  % climb again in B
  \draw[->, acc, thick] (3.14,-0.9) .. controls (4.3,-0.6) and (5.3,0.4) .. (6.1,0.82);
  \node[acc, anchor=south] at (1.4,0.75) {climb};
\end{tikzpicture}
$$

## The recoil limit

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

$$
E_r = \frac{\hbar^2 k^2}{2M},
$$

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

> **Definition (Recoil limit).** The temperature scale $k_{\mathrm B}T_r \sim
> \hbar^2 k^2/M$ corresponding to one photon recoil, the floor of any cooling method
> whose last step is a spontaneous emission. Reaching lower temperatures requires
> sub-recoil methods (velocity-selective coherent population trapping, Raman cooling)
> or evaporative cooling, none of which end each atom's history with a scattered
> photon.

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

| Scale | Formula | Rubidium-87 (780 nm) | Set by |
| --- | --- | --- | --- |
| Doppler limit | $\hbar\Gamma/2k_{\mathrm B}$ | $146\ \mathrm{\mu K}$ | natural linewidth $\Gamma$ |
| Sub-Doppler (Sisyphus) | $\sim U_0/k_{\mathrm B}$ | few $\mathrm{\mu K}$ | light-shift depth, tunable |
| Recoil limit | $\hbar^2 k^2/M k_{\mathrm B}$ | $0.36\ \mathrm{\mu K}$ | photon momentum $\hbar k$ |

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

$$
% caption: The three cooling floors on a logarithmic temperature scale, spanning
% from the Doppler limit through the Sisyphus regime to the recoil limit; quantum
% degeneracy lies below all three and needs evaporative cooling.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % horizontal log axis, colder to the right
  \draw[->, black] (0,0) -- (9.2,0) node[right] {colder};
  % ticks
  \foreach \x/\lab in {0.8/{$100\ \mathrm{\mu K}$}, 3.0/{$10\ \mathrm{\mu K}$},
      5.2/{$1\ \mathrm{\mu K}$}, 7.4/{$100\ \mathrm{nK}$}} {
    \draw[black] (\x,-0.1) -- (\x,0.1);
    \node[black, anchor=north] at (\x,-0.15) {\lab};
  }
  % Doppler
  \fill[black] (1.0,0) circle (2.6pt);
  \node[black, anchor=south] at (1.0,0.15) {Doppler};
  % Sisyphus band
  \draw[black, very thick] (3.0,0.55) -- (5.4,0.55);
  \node[black, anchor=south] at (4.2,0.6) {Sisyphus};
  % recoil
  \fill[acc] (6.3,0) circle (2.6pt);
  \node[acc, anchor=south] at (6.3,0.15) {recoil};
  % degeneracy region
  \draw[black, thick, dashed] (7.4,-0.55) -- (9.0,-0.55);
  \node[black, anchor=north] at (8.2,-0.6) {degeneracy};
\end{tikzpicture}
$$

## 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 $\vec B = B'(x\,\hat\imath + y\,\hat\jmath - 2z\,\hat k)$, 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
$|\vec B|$.

$$
% caption: A magnetic trap holds a low-field-seeking atom near a nonzero minimum of
% the field magnitude; a bare quadrupole (dashed) has a zero at the center where
% spin flips eject atoms, so a bias field lifts the bottom to a harmonic well.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.4,0) -- (3.6,0) node[right] {position};
  \draw[->, black] (0,0) -- (0,3.2) node[above] {potential};
  % quadrupole: V-shaped, zero at center (dashed)
  \draw[black, thick, dashed] (-3.0,2.7) -- (0,0) -- (3.0,2.7);
  \node[black, anchor=south] at (-2.3,2.0) {quadrupole (zero)};
  % harmonic well with lifted bottom (solid)
  \draw[acc, very thick, domain=-2.7:2.7, samples=120, smooth]
    plot (\x, {0.35*\x*\x + 0.35});
  \node[acc, anchor=west] at (1.5,2.3) {bias trap};
  \fill[acc] (0,0.35) circle (2.4pt);
  \node[acc, anchor=west] at (0.15,0.45) {lifted bottom};
\end{tikzpicture}
$$

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

$$
U_{\mathrm{dip}}(\vec r) = \frac{\hbar\Omega^2(\vec r)}{4\delta}
= \frac{3\pi c^2}{2\omega_0^3}\,\frac{\Gamma}{\delta}\,I(\vec r),
$$

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.

$$
% caption: A red-detuned focused beam makes an attractive dipole potential well at
% its intensity maximum; a retro-reflected standing wave makes an optical lattice, a
% periodic array of such wells spaced by half a wavelength.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % single well (left)
  \draw[->, black] (-3.6,0) -- (-0.4,0) node[right] {$x$};
  \draw[acc, very thick, domain=-3.3:-0.7, samples=120, smooth]
    plot (\x, {1.3*exp(-3*(\x+2)*(\x+2)) - 1.7});
  \node[acc, anchor=south] at (-2.0,0.45) {single focus};
  % lattice (right)
  \draw[->, black] (0.6,0) -- (3.8,0) node[right] {$x$};
  \draw[black, very thick, domain=0.7:3.6, samples=180, smooth]
    plot (\x, {-0.9 - 0.7*cos(deg((\x-0.7)*6.5))});
  \node[black, anchor=south] at (2.1,0.45) {optical lattice};
\end{tikzpicture}
$$

## 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

$$
\varrho = n\,\lambda_{\mathrm{dB}}^3,
\qquad
\lambda_{\mathrm{dB}} = \sqrt{\frac{2\pi\hbar^2}{Mk_{\mathrm B}T}},
$$

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.

$$
% caption: Evaporative cooling truncates the high-energy tail of the trapped
% distribution by lowering the trap depth; the remaining atoms rethermalize to a
% narrower, colder distribution, raising the phase-space density.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.6,0) node[right] {energy};
  \draw[->, black] (0,0) -- (0,3.0) node[above] {number};
  % broad hot distribution
  \draw[black, thick, domain=0:6.2, samples=140, smooth]
    plot (\x, {2.4*exp(-0.28*\x)});
  \node[black, anchor=west] at (3.0,1.2) {before};
  % trap depth cut
  \draw[black, thick, dashed] (3.4,0) -- (3.4,2.6);
  \node[black, anchor=south] at (3.4,2.6) {trap depth};
  % narrower colder distribution after rethermalization
  \draw[acc, very thick, domain=0:3.4, samples=120, smooth]
    plot (\x, {2.7*exp(-0.75*\x)});
  \node[acc, anchor=west] at (1.1,1.9) {after};
\end{tikzpicture}
$$

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.

[^lett]: **Foot**, _Atomic Physics_, §9.4, recounts the 1988 sodium-molasses measurement of temperatures below $T_D$ (P. D. Lett et al., _Phys. Rev. Lett._ **61**, 169 (1988)) and the polarization-gradient explanation. See also **Metcalf & van der Straten**, Ch. 8.
[^sisyphus]: **Foot**, _Atomic Physics_, §9.5, gives the lin$\perp$lin light-shift potentials and the Sisyphus energy balance; the mechanism is due to J. Dalibard and C. Cohen-Tannoudji, _J. Opt. Soc. Am. B_ **6**, 2023 (1989). Recoil data for rubidium-87: $v_r = \hbar k/M = 5.88\ \mathrm{mm\,s^{-1}}$, $T_r = \hbar^2 k^2/Mk_{\mathrm B} = 0.362\ \mathrm{\mu K}$ (D. A. Steck, _Rubidium 87 D Line Data_, https://steck.us/alkalidata/).