= Doppler cooling
{c}
{wiki}
Consider equal opposing <laser> beams along the $x$ axis and an atom with <velocity> $v$. Let $\omega_L$ be their <angular frequency>, $\omega_0$ the atomic resonance <angular frequency>, $k=2\pi/\lambda>0$ their <wavenumber>, and $\delta=\omega_L-\omega_0$ the <laser detuning>. Red detuning means $\delta<0$. To first order in $v/c$, the atom sees
$$
\omega'_+=\omega_L-kv,\qquad \omega'_- =\omega_L+kv,\qquad
\delta_+=\delta-kv,\qquad \delta_- =\delta+kv.
$$
For small positive $v$, the beam travelling towards negative $x$ is closer to resonance and causes more <photon absorption>; its <momentum> kicks point against $v$. The beam travelling towards positive $x$ is further from resonance. Reversing $v$ exchanges the roles of the two beams. This is a <Doppler effect> on the absorption rates, rather than a need to follow each atom with a moving beam.
An absorbed <photon> gives the atom <momentum> $\pm\hbar k$, where $\hbar=h/(2\pi)$ is the reduced <Planck constant>. Subsequent <spontaneous emission> gives a recoil opposite to the emitted <photon>. For an emission pattern symmetric under reversal of direction, that recoil has zero <expected value>, but nonzero <variance>. Its mean therefore leaves the directional absorption <force> intact, while its randomness adds <recoil energy>. Random absorption times and choices of beam add further <momentum> fluctuations. Zero mean recoil does not mean zero heating.
\Image[/laser-cooling-opposed-beams-doppler-absorption.png]
{title=Opposing beams and Doppler-selective photon absorption}
{description=An original schematic for an atom moving right. Both beams have the same red detuning in the laboratory. The calculated weak-field line shape below shows their different atom-frame scattering rates for $d=-0.5$ and $u=0.25$; it is not experimental data.}
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