Short introduction to Doppler cooling

Need three ingredients to understand Doppler cooling:
1. Three elementary processes of atom-light interaction:

2. Momentum conservation

v typically a few cm/s 3. Doppler shift

$\omega_{A}=\omega\pm \vec{k}\cdot\vec{v}$

Construct velocity-dependent dissipative force on atom with two red detuned laser beams:

Red detuned
$\Rightarrow$ absorption preferentially from laser beam opposing velocity (Doppler shift towards resonance)
$\Rightarrow$ slows atom down towards v=0 if de-excitation mainly due to isotropic spontaneous emission (low intensity limit)

($\Gamma$=natural line width)
Force $\propto (-v) \Rightarrow$ energy dissipation


Ultimate limit for Doppler cooling
Atom cannot be cooled to v=0 because of random recoil from spontaneous emission (random walk in momentum space with increments $\hbar k$)
$\Rightarrow$ heating rate associated with recoil

Landmark energy: Recoil energy $E_{R}=\frac{\hbar^{2}k^{2}}{2m}$

E_R corresponds to temperature $\frac{1}{2}kT=\frac{\hbar^{2}k^{2}}{2m}$ ( $\approx 20\mu K$ for Na)

Note: Omitted discussion of so-called Sub-doppler cooling