---
title: Laser Cooling and Optical Molasses
module: Modern Atomic Physics
moduleNumber: 9
lessonNumber: 1
order: 901
summary: >
  A near-resonant laser beam pushes an atom because every absorbed photon delivers
  one unit of momentum and the subsequent spontaneous emission averages to zero. Two
  counter-propagating red-detuned beams turn that push into friction: the Doppler
  shift brings a moving atom closer to resonance with the beam it moves against, so
  the net force opposes the velocity. Six beams give optical molasses in three
  dimensions. The random recoil of spontaneous emission heats against the friction,
  and the balance sets the Doppler cooling limit. Adding a magnetic-field gradient
  makes the force position-dependent as well, giving the magneto-optical trap.
topics: [Modern Atomic Physics]
sources:
  - book: Foot
    ref: "Ch. 9 — Laser Cooling and Trapping; §9.1 The Scattering Force, §9.2 Slowing an Atomic Beam, §9.3 The Optical Molasses Technique"
  - book: Metcalf & van der Straten
    ref: "Laser Cooling and Trapping (Springer, 1999), Ch. 3 — The Force on Two-Level Atoms; Ch. 7 — Cooling and Trapping"
  - book: Demtröder
    ref: "Ch. 11 — Laser Spectroscopy of Cold Atoms (concepts)"
  - book: Foot
    ref: "Ch. 9 §9.4 — The Magneto-Optical Trap"
draft: false
---

An atom in free space feels no force from a distant laser unless it absorbs light,
and each absorption is also a momentum transfer. A photon of wavevector $\vec k$
carries momentum $\hbar\vec k$; absorbing it recoils the atom by $\hbar\vec k$. The
atom then returns to the ground state, and if it does so by spontaneous emission the
recoil kicks point in random directions and average to zero over many cycles. The
directed absorption recoils accumulate; the undirected emission recoils cancel. A
resonant beam therefore exerts a steady force in its own direction, and the rate of
that force is set by how fast the atom can cycle absorption and emission.

This is the mechanism behind every technique in modern cold-atom physics. This
lesson derives the scattering force from the two-level saturation of the excited
state, shows how the Doppler shift converts a pair of counter-propagating beams into
a velocity-dependent friction, computes the equilibrium temperature at which
spontaneous-emission heating balances that friction, and adds a magnetic gradient to
turn the friction into a trap.

## The scattering force

Model the atom as a two-level system: ground state $\ket g$, excited state $\ket e$
separated by $\hbar\omega_0$, with the excited state decaying at rate $\Gamma$
(inverse of the natural lifetime $\tau = 1/\Gamma$). A laser of frequency $\omega$
drives the transition. The steady-state excited-state population, from the optical
Bloch equations for a two-level atom in a classical field,[^obe] is

$$
\rho_{ee} = \frac{1}{2}\,\frac{s_0}{1 + s_0 + \left(2\delta/\Gamma\right)^2},
\qquad
\delta \equiv \omega - \omega_0,
$$

where $\delta$ is the **detuning** of the laser from resonance and

$$
s_0 \equiv \frac{I}{I_{\mathrm{sat}}},
\qquad
I_{\mathrm{sat}} = \frac{\pi h c\,\Gamma}{3\lambda^3},
$$

is the on-resonance **saturation parameter**, the ratio of the intensity $I$ to the
saturation intensity $I_{\mathrm{sat}}$. For the rubidium-87 $D_2$ line
($\lambda = 780\ \mathrm{nm}$, $\Gamma/2\pi = 6.07\ \mathrm{MHz}$),
$I_{\mathrm{sat}} = 1.67\ \mathrm{mW\,cm^{-2}}$. As $s_0 \to \infty$ the population
saturates at $\rho_{ee} \to \tfrac12$: the atom spends at most half its time in the
excited state, because stimulated emission returns population to the ground state
just as fast as absorption removes it.

Each atom in the excited state decays at rate $\Gamma$, so the **scattering rate**
(photons scattered per second) is

$$
R_{\mathrm{sc}} = \Gamma\,\rho_{ee}
= \frac{\Gamma}{2}\,\frac{s_0}{1 + s_0 + \left(2\delta/\Gamma\right)^2}.
$$

Every scattered photon is absorbed from the beam (momentum $+\hbar\vec k$) and
re-emitted in a random direction (momentum averaging to zero), so the mean force is
the momentum per photon times the scattering rate:

$$
\vec F_{\mathrm{sc}} = \hbar\vec k\, R_{\mathrm{sc}}
= \hbar\vec k\,\frac{\Gamma}{2}\,\frac{s_0}{1 + s_0 + \left(2\delta/\Gamma\right)^2}.
$$

> **Definition (Scattering force).** The mean radiation-pressure force on a
> two-level atom in a travelling-wave field, equal to the photon momentum $\hbar k$
> times the scattering rate $R_{\mathrm{sc}} = \Gamma\rho_{ee}$. It saturates at
> $F_{\max} = \hbar k\Gamma/2$, one photon momentum per two lifetimes, as
> $s_0 \to \infty$ and $\delta \to 0$.

The saturated force is enormous on an atomic scale. For rubidium,
$F_{\max} = \hbar k\Gamma/2$ produces an acceleration
$a_{\max} = \hbar k\Gamma/2m \approx 1.1\times10^5\ \mathrm{m\,s^{-2}}$, about
$10^4$ times gravity. A thermal rubidium atom leaving an oven at
$300\ \mathrm{m\,s^{-1}}$ can be brought to rest in under a millisecond over a
distance of tens of centimetres. This is the basis of the Zeeman slower that loads a
trap from an atomic beam.[^slower]

$$
% caption: Scattering rate versus detuning is a Lorentzian of half-width
% (Gamma/2) sqrt(1+s0); increasing intensity raises the peak toward Gamma/2 and
% power-broadens the line.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (-3.4,0) -- (3.6,0) node[right] {detuning};
  \draw[->, black] (0,0) -- (0,3.4) node[above] {scattering rate};
  \node[black, anchor=north] at (0,-0.08) {0};
  % low intensity: narrow, low
  \draw[black, thick, domain=-3.2:3.2, samples=120, smooth]
    plot (\x, {2.1/(1 + 4*\x*\x)});
  % high intensity: broad, high (power-broadened, approaches saturation)
  \draw[black, very thick, dashed, domain=-3.2:3.2, samples=120, smooth]
    plot (\x, {3.0/(1 + 0.55*\x*\x)});
  \node[black, anchor=west] at (0.9,1.9) {low $s_0$};
  \node[black, anchor=west] at (1.7,2.75) {high $s_0$};
  \draw[dashed, black] (-3.3,3.0) -- (3.5,3.0);
  \node[black, anchor=east] at (-2.6,3.15) {saturated maximum};
\end{tikzpicture}
$$

## The Doppler cooling mechanism

A single beam only pushes; it cannot slow a moving atom below zero velocity. The
trick is to use two counter-propagating beams and to detune them **below** resonance
($\delta < 0$, "red detuning"). An atom moving with velocity $v$ toward one beam sees
that beam Doppler-shifted **up** in frequency by $kv$, and the opposing beam
shifted **down**. The beam the atom moves against is therefore shifted closer to
resonance and scatters more strongly; the beam the atom moves with is shifted farther
away and scatters less. The imbalance produces a net force opposing the motion.

Let the two beams propagate along $\pm\hat x$, each of intensity $I$ giving the same
$s_0$. In the atom's rest frame the beam it moves against has effective detuning
$\delta - kv$ (moving toward a beam raises its frequency, moving red-detuned laser
closer to resonance when $\delta<0$ requires the sign care below) and the co-moving
beam has $\delta + kv$. The net force along $\hat x$ is the difference of two
scattering forces:

$$
F(v) = \hbar k\,\frac{\Gamma}{2}
\left[
\frac{s_0}{1 + s_0 + \left(\tfrac{2(\delta - kv)}{\Gamma}\right)^2}
-
\frac{s_0}{1 + s_0 + \left(\tfrac{2(\delta + kv)}{\Gamma}\right)^2}
\right].
$$

The first term (from the $-\hat x$ beam, pushing in $+\hat x$) and the second (from
the $+\hat x$ beam, pushing in $-\hat x$) enter with opposite signs. For $\delta < 0$
and $v > 0$ the co-moving-against beam is nearer resonance, the bracket is negative,
and the force opposes $v$: the motion is damped.

$$
% caption: An atom moving right sees the left-going beam blue-shifted toward
% resonance (stronger push) and the right-going beam red-shifted away (weaker
% push); the imbalance opposes the velocity.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % atom
  \shade[ball color=acc!30] (0,0) circle (0.32);
  \node[anchor=south] at (0,0.45) {atom};
  \draw[->, black, very thick] (0,-0.05) -- (1.5,-0.05) node[right, black] {velocity $v$};
  % left beam (going right, pushes right)
  \foreach \x in {-3.3,-2.9,...,-1.1} \draw[acc, thick] (\x,0.75) -- (\x+0.25,0.75);
  \draw[->, acc, very thick] (-3.4,0.75) -- (-0.5,0.75);
  \node[acc, anchor=west, align=left] at (-3.4,1.25)
    {left beam: blue-shifted,\\nearer resonance, strong};
  % right beam (going left, pushes left)
  \foreach \x in {1.1,1.5,...,3.3} \draw[black, thick] (\x,-0.75) -- (\x+0.25,-0.75);
  \draw[->, black, thick, dashed] (3.4,-0.75) -- (0.5,-0.75);
  \node[black, anchor=east, align=right] at (3.4,-1.25)
    {right beam: red-shifted,\\farther off, weak};
\end{tikzpicture}
$$

### The friction coefficient

Near $v = 0$ the force is linear in velocity. Expand $F(v)$ to first order. With
$x_0 \equiv 2\delta/\Gamma$ and treating $2kv/\Gamma$ as small,

$$
F(v) \approx \frac{8\hbar k^2\,\delta\, s_0}{\Gamma\left(1 + s_0 + x_0^2\right)^2}\,v
\;\equiv\; -\alpha\, v,
\qquad
\alpha = -\frac{8\hbar k^2\, s_0\,(\delta/\Gamma)}{\left(1 + s_0 + (2\delta/\Gamma)^2\right)^2}.
$$

For red detuning $\delta < 0$ the coefficient $\alpha > 0$, so $F = -\alpha v$ is a
**friction force**: the medium of light acts like a viscous fluid. This configuration
is **optical molasses**. A displaced or moving atom decelerates on a timescale
$m/\alpha$, which for typical parameters is tens of microseconds.

$$
% caption: Net force in a two-beam molasses versus velocity is the difference of
% two Lorentzians; near v=0 it is linear with negative slope (friction), and it
% captures atoms within the capture range set by the detuning.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.6,0) -- (3.8,0) node[right] {velocity};
  \draw[->, black] (0,-2.2) -- (0,2.4) node[above] {force};
  % dispersive-shaped net force: F(v) = A[ 1/(1+(v-d)^2) - 1/(1+(v+d)^2) ] scaled
  \draw[acc, very thick, domain=-3.4:3.4, samples=160, smooth]
    plot (\x, {1.9*( 1/(1+ (\x-1.1)*(\x-1.1)) - 1/(1+ (\x+1.1)*(\x+1.1)) )});
  % linear region tangent
  \draw[black, dashed, thick] (-1.15,1.9) -- (1.15,-1.9);
  \node[black, anchor=west] at (0.55,1.35) {slope $-\alpha$};
  \node[black, anchor=south west] at (-3.3,-1.9) {captured range};
\end{tikzpicture}
$$

The linear range extends only to velocities where $kv \sim \Gamma$; beyond that the
Doppler shift outruns the linewidth and the force falls off. The **capture velocity**
$v_c \sim \Gamma/k$ is a few metres per second for rubidium, so molasses cools atoms
already slowed to that range but does not stop a thermal beam by itself.

## Momentum diffusion and the Doppler limit

Friction alone would cool to absolute zero. It does not, because the same photon
scattering that provides the friction also delivers random momentum kicks. Two
sources of randomness heat the atom:

- **Absorption fluctuations.** The number of photons absorbed in a time interval
  fluctuates; each absorption is a discrete kick $\hbar k$.
- **Spontaneous-emission recoil.** Each emitted photon leaves in a random direction,
  so the emission recoils execute a random walk in momentum with step $\hbar k$.

Both contribute a **momentum diffusion**: $\langle p^2\rangle$ grows linearly in
time, $\d\langle p^2\rangle/\d t = 2D$, with diffusion constant $D$ of order
$(\hbar k)^2 R_{\mathrm{sc}}$ summed over the beams. The friction removes energy at
rate $\alpha v^2$ (averaged, $\alpha\langle v^2\rangle$), while diffusion adds it at
rate $D/m$. In steady state the two balance:

$$
\frac{\d}{\d t}\!\left\langle \tfrac12 m v^2\right\rangle
= -\frac{\alpha}{m}\langle p^2\rangle_{\text{cool}} + \frac{D}{m} = 0
\;\Longrightarrow\;
\tfrac12 k_{\mathrm B} T = \frac{D}{2\alpha}
\quad\text{per degree of freedom.}
$$

Carrying the low-intensity limit ($s_0 \ll 1$) through with the explicit $D$ and
$\alpha$ gives the equilibrium temperature[^doplimit]

$$
k_{\mathrm B} T = \frac{\hbar\Gamma}{4}\,
\frac{1 + (2\delta/\Gamma)^2}{2|\delta|/\Gamma}.
$$

Minimising over detuning, $\d T/\d\delta = 0$ gives $2|\delta|/\Gamma = 1$, i.e.
$\delta = -\Gamma/2$, and the minimum temperature is the **Doppler cooling limit**:

$$
k_{\mathrm B} T_D = \frac{\hbar\Gamma}{2},
\qquad
T_D = \frac{\hbar\Gamma}{2k_{\mathrm B}}.
$$

> **Definition (Doppler limit).** The lowest temperature achievable by
> Doppler cooling of a two-level atom, $T_D = \hbar\Gamma/2k_{\mathrm B}$, reached at
> detuning $\delta = -\Gamma/2$. It is set by the natural linewidth: a broader
> transition cools less deeply because each spontaneous photon carries a larger
> random recoil relative to the friction.

Numerically $T_D$ depends only on the linewidth. For the alkali cooling transitions:

| Atom | Transition | $\Gamma/2\pi$ | $T_D = \hbar\Gamma/2k_{\mathrm B}$ |
| --- | --- | --- | --- |
| Sodium | $D_2$, 589 nm | $9.79\ \mathrm{MHz}$ | $235\ \mathrm{\mu K}$ |
| Rubidium-87 | $D_2$, 780 nm | $6.07\ \mathrm{MHz}$ | $146\ \mathrm{\mu K}$ |
| Cesium | $D_2$, 852 nm | $5.22\ \mathrm{MHz}$ | $125\ \mathrm{\mu K}$ |

The corresponding root-mean-square speed for rubidium at $146\ \mathrm{\mu K}$ is
about $12\ \mathrm{cm\,s^{-1}}$, five orders of magnitude below the thermal speed at
room temperature. When the first molasses experiments measured temperatures
**below** $T_D$, the two-level model had to be extended; that sub-Doppler physics is
the subject of the [next lesson](/atomic-physics/modern-atomic-physics/sub-doppler-trapping).

$$
% caption: Cooling by friction (removing energy as alpha times v-squared) balances
% heating by photon-recoil diffusion (adding energy at rate D over m); their
% steady state fixes the temperature, minimized at detuning minus half-Gamma.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % energy axis
  \draw[->, black] (0,0) -- (0,3.6) node[above] {kinetic energy};
  % cooling arrow down
  \draw[->, acc, very thick] (1.2,3.2) -- (1.2,1.4);
  \node[acc, anchor=west, align=left] at (1.4,2.4) {friction removes\\energy};
  % heating arrow up
  \draw[->, black, very thick, dashed] (3.6,0.4) -- (3.6,1.4);
  \node[black, anchor=west, align=left] at (3.8,0.9) {recoil adds\\energy};
  % equilibrium level
  \draw[black, thick, dashed] (0.4,1.4) -- (4.9,1.4);
  \node[black, anchor=south east] at (4.9,1.45) {equilibrium: Doppler limit};
\end{tikzpicture}
$$

## The magneto-optical trap

Molasses damps velocity but exerts no restoring force on position: an atom diffuses
out of the beam overlap region over time. A spatial confinement is added by making
the detuning depend on position through the Zeeman effect. Superpose a
**quadrupole magnetic field** from a pair of coils carrying opposite currents
(anti-Helmholtz), which vanishes at the centre and grows linearly,
$B(x) = B'x$. The Zeeman shift of the excited-state magnetic sublevels then grows
linearly with displacement, and by choosing circular polarizations for the two
beams the atom is always pushed back toward the centre.

Consider one dimension with a $J=0 \to J=1$ transition. The excited state splits into
$m_J = -1, 0, +1$ sublevels with energies shifting as $\pm\mu_B g B'x$. The beam from
$+x$ carries $\sigma^-$ polarization (driving $\Delta m = -1$) and the beam from $-x$
carries $\sigma^+$ ($\Delta m = +1$). At a displacement $x > 0$ the $m_J = -1$
sublevel is Zeeman-shifted **toward** resonance with the red-detuned $\sigma^-$ beam,
which pushes the atom back toward the centre; at $x < 0$ the mirror-image geometry
pushes the other way. The force acquires a position-dependent term:

$$
F = -\alpha v - \kappa x,
\qquad
\kappa = \frac{\mu_B g B'}{\hbar k}\,\alpha,
$$

the equation of a **damped harmonic oscillator**. The friction $\alpha$ cools; the
spring constant $\kappa$ confines. This is the **magneto-optical trap** (MOT), the
standard first stage of nearly every cold-atom experiment.

$$
% caption: A magneto-optical trap: three orthogonal pairs of counter-propagating
% circularly polarized beams overlap a quadrupole field from anti-Helmholtz coils,
% giving both velocity damping and a position-dependent restoring force.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % coils (anti-Helmholtz), drawn as ellipses above and below
  \draw[black, very thick] (-2.4,1.7) ellipse (0.9 and 0.28);
  \draw[black, very thick] (-2.4,-1.7) ellipse (0.9 and 0.28);
  \node[black, anchor=east] at (-3.4,1.7) {coil};
  \node[black, anchor=east] at (-3.4,-1.7) {coil};
  % trapped cloud
  \shade[ball color=acc!35] (0,0) circle (0.32);
  \node[anchor=north] at (0,-0.45) {cold cloud};
  % six beams: horizontal pair
  \draw[->, black, thick] (-3.0,0) -- (-0.5,0);
  \draw[->, black, thick] (3.0,0) -- (0.5,0);
  % vertical pair
  \draw[->, black, thick] (0,2.6) -- (0,0.5);
  \draw[->, black, thick] (0,-2.6) -- (0,-0.5);
  % diagonal pair (depth)
  \draw[->, black, thick] (2.2,1.9) -- (0.45,0.4);
  \draw[->, black, thick] (-2.2,-1.9) -- (-0.45,-0.4);
  \node[black, anchor=west] at (2.4,0.25) {six beams};
\end{tikzpicture}
$$

A MOT loaded from a slowed beam or from background vapour collects $10^7$ to $10^{10}$
atoms in a fraction of a second, at temperatures near the Doppler limit and densities
of $10^{10}$–$10^{11}\ \mathrm{cm^{-3}}$. Its density is capped by two effects:
re-absorption of scattered photons produces an outward radiation-pressure repulsion
between atoms, and light-assisted collisions eject atom pairs. Reaching quantum
degeneracy therefore requires the trap to be turned off and the cooling continued by
mechanisms that do not scatter photons, developed in the following lessons.

$$
% caption: One-dimensional MOT level scheme: the excited sublevels split linearly
% with position, so at positive x the m=minus-one sublevel Zeeman-tunes toward the
% red-detuned sigma-minus beam and the resulting scattering pushes the atom back.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
  \definecolor{acc}{HTML}{4A6FA5}
  % position axis
  \draw[->, black] (-3.4,0) -- (3.6,0) node[right] {position};
  \node[black, anchor=north] at (0,-0.05) {center};
  % ground level (flat)
  \draw[black, thick] (-3.2,-0.9) -- (3.2,-0.9);
  \node[black, anchor=east] at (-3.2,-0.9) {ground};
  % three excited sublevels: m=+1 rising, m=0 flat, m=-1 falling
  % (signs kept in text mode; math-mode minus garbles in tikz nodes)
  \draw[black, very thick] (-3.2,1.4) -- (3.2,3.0);
  \node[black, anchor=west] at (3.2,3.0) {$m$ = +1};
  \draw[black, thick] (-3.2,2.2) -- (3.2,2.2);
  \node[black, anchor=west] at (3.2,2.2) {$m$ = 0};
  \draw[acc, thick, dashed] (-3.2,3.0) -- (3.2,1.4);
  \node[acc, anchor=west] at (3.2,1.4) {$m$ = -1};
  % laser detuning line (below all sublevels)
  \draw[dashed, black] (-3.2,1.05) -- (3.2,1.05);
  \node[black, anchor=east] at (-3.2,1.05) {laser (red detuned)};
  % at x>0 the m=-1 sublevel meets the laser -> resonance dot
  \fill[acc] (1.85,1.74) circle (2.2pt);
  \node[acc, anchor=south west] at (0.5,2.35) {resonance};
  \draw[black, thin] (0.95,2.35) -- (1.75,1.82);
\end{tikzpicture}
$$

The MOT combines the two ingredients built in this lesson: the velocity dependence of
the scattering force (friction, from the Doppler shift) and a position dependence
(confinement, from the Zeeman shift). The Doppler limit sets the temperature of the
loaded cloud, and the recoil kicks that impose that limit are the same kicks that,
uncontrolled, prevent Doppler cooling from reaching the ground state of motion.

[^obe]: **Foot**, _Atomic Physics_, §9.1 and §7.3–7.6, derives $\rho_{ee}$ and the scattering rate from the optical Bloch equations; the saturation intensity $I_{\mathrm{sat}} = \pi hc\Gamma/3\lambda^3$ is the standard two-level result. See also **Metcalf & van der Straten**, _Laser Cooling and Trapping_, Ch. 2–3.
[^slower]: **Metcalf & van der Straten**, _Laser Cooling and Trapping_, Ch. 6 — Deceleration of an Atomic Beam; the Zeeman slower uses a spatially varying magnetic field to hold the Doppler-shifted atom on resonance as it decelerates. Rubidium-87 $D_2$ data: $\Gamma/2\pi = 6.065\ \mathrm{MHz}$, $\lambda = 780.24\ \mathrm{nm}$ (D. A. Steck, _Rubidium 87 D Line Data_, https://steck.us/alkalidata/).
[^doplimit]: **Foot**, _Atomic Physics_, §9.3, gives the diffusion–friction balance $k_{\mathrm B}T = D/\alpha$ and the minimum $k_{\mathrm B}T_D = \hbar\Gamma/2$ at $\delta = -\Gamma/2$; the same result is in **Metcalf & van der Straten**, Ch. 7 §7.1. Linewidths: NIST Atomic Spectra Database, https://www.nist.gov/pml/atomic-spectra-database.