Laser cooling is one of the clearest demonstrations of how light can control atomic motion. In the simplest picture,
a two-level atom interacts with two counter-propagating laser beams tuned slightly below resonance. This is called
red detuning. Because of the Doppler effect, an atom moving toward one beam sees that beam shifted
closer to resonance, so it scatters more photons from that direction. Each absorbed photon transfers momentum
\(\hbar k\) to the atom, and on average this produces a force opposite to the atom’s motion. That is the essence of
Doppler cooling.
Wave number and linewidth
For a laser of wavelength \(\lambda\), the wave number is
Wave number.
\[
\begin{aligned}
k &= \frac{2\pi}{\lambda}.
\end{aligned}
\]
The atomic transition is characterized by a natural linewidth \(\Gamma\), which sets the spontaneous-decay and
scattering scale of the excited state. In experiments and data tables, one often quotes \(\Gamma/2\pi\) in MHz.
The linewidth is crucial because it determines both the force scale and the Doppler temperature limit.
Single-beam scattering rate and force
For a two-level atom driven by one laser beam, the scattering rate is
Scattering rate.
\[
\begin{aligned}
R &= \frac{\Gamma}{2}\,
\frac{s}{1+s+\left(2\delta/\Gamma\right)^2},
\end{aligned}
\]
where \(s\) is the saturation parameter and \(\delta\) is the detuning from resonance. The corresponding single-beam
radiation-pressure force magnitude is
\[
\begin{aligned}
F_{\mathrm{single}} &= \hbar k R
= \frac{\hbar k \Gamma}{2}\,
\frac{s}{1+s+\left(2\delta/\Gamma\right)^2}.
\end{aligned}
\]
This force points along the direction of the beam. On its own, one beam mainly pushes the atom. Cooling appears when
two opposite beams are used together.
Optical molasses force
In one-dimensional optical molasses, one beam propagates along \(+x\) and the other along \(-x\). If the atom moves
with velocity \(v\), the Doppler shift changes the effective detuning seen by each beam. A standard two-level model
gives
Net Doppler cooling force.
\[
\begin{aligned}
F(v)
&= \frac{\hbar k \Gamma}{2}
\left[
\frac{s}{1+s+\left(2(\delta-kv)/\Gamma\right)^2}
-
\frac{s}{1+s+\left(2(\delta+kv)/\Gamma\right)^2}
\right].
\end{aligned}
\]
Near zero velocity, the force is approximately linear:
\[
\begin{aligned}
F(v) &\approx -\alpha v.
\end{aligned}
\]
If \(\alpha > 0\), the force opposes motion and the atom is cooled. This is why red detuning is essential: with
negative detuning, the beam opposing the motion is closer to resonance and scatters more strongly.
Doppler temperature limit
Cooling cannot continue indefinitely because photon scattering also introduces momentum diffusion. In the simple
Doppler-cooling model, the balance between damping and diffusion gives the minimum achievable temperature
Doppler limit.
\[
\begin{aligned}
T_D &= \frac{\hbar \Gamma}{2k_B}.
\end{aligned}
\]
This temperature depends only on the linewidth. Broader transitions give stronger scattering forces, but also a higher
Doppler limit. Narrower transitions can cool to lower temperatures, although they usually produce weaker forces.
Sample estimate for rubidium
For the common rubidium-87 D2 transition, \(\Gamma/2\pi \approx 6.065\ \mathrm{MHz}\). Converting to \(\Gamma\) and
applying the Doppler-limit formula gives
\[
\begin{aligned}
T_D &\approx 145\ \mathrm{\mu K}.
\end{aligned}
\]
This is one of the classic benchmark numbers in laser cooling. If the laser is red detuned by several MHz and the
saturation parameter is moderate, the force-versus-velocity curve has a negative slope near \(v=0\), which means the
atom experiences viscous damping.
Physical interpretation
The force curve is most important near small velocities, because that is where the atom spends more time as it cools.
Far from resonance, the scattering rate drops because the detuning term dominates the denominator, so the force becomes
weaker. That is why the optical-molasses force is strongest over a limited range of velocities rather than over all
speeds.
This preview is intentionally restricted to a simple two-level picture. At university level, one goes beyond Doppler
cooling by considering polarization gradients, multilevel hyperfine structure, magneto-optical trapping, and
sub-Doppler mechanisms such as Sisyphus cooling. Those effects are essential in real experiments, but
the two-level Doppler model remains the fundamental starting point for understanding why red-detuned light can cool
atoms.
| Concept |
Main relation |
Meaning |
| Wave number |
\(k = 2\pi/\lambda\) |
Momentum per photon is \(\hbar k\) |
| Scattering rate |
\(R = (\Gamma/2)\, s/[1+s+(2\delta/\Gamma)^2]\) |
Photon scattering from one beam |
| Single-beam force |
\(F_{\mathrm{single}} = \hbar k R\) |
Radiation-pressure force magnitude |
| Molasses force |
\(F(v)\) from the two-beam Doppler-shifted rates |
Net cooling or heating force |
| Low-velocity form |
\(F \approx -\alpha v\) |
Viscous damping when \(\alpha > 0\) |
| Doppler limit |
\(T_D = \hbar\Gamma/(2k_B)\) |
Minimum temperature in the basic Doppler model |