= Doppler cooling force
{c}
{title2=$f(v)\simeq-\alpha v$}
Use a closed <two-level system> with <spontaneous emission> population decay $P_e(t)=P_e(0)e^{-\Gamma t}$. Here $\Gamma$ is a decay rate in inverse seconds and also the full width at half maximum of the <natural linewidth> expressed in <angular frequency>. The corresponding full width in <hertz> is $\Gamma/(2\pi)$; the angular half width is $\Gamma/2$. These conventions determine every factor of two and $2\pi$ below.
Let $s$ be the on-resonance <saturation parameter> of each beam, with total excitation weak enough that saturation and coherent coupling between beams can be neglected. The independent-beam scattering rates and net <force> are
$$
R_\pm(v)=\frac{\Gamma s}{2}\frac{1}{1+4(\delta\mp kv)^2/\Gamma^2},\qquad
f(v)=\hbar k\bigl(R_+(v)-R_-(v)\bigr).
$$
The <force> is the <photon> <momentum> per absorption multiplied by the difference in absorption rates. With dimensionless <laser detuning> $d=\delta/\Gamma$ and dimensionless <velocity> $u=kv/\Gamma$, write
$$
f(v)=\frac{\hbar k\Gamma s}{2}F(u),\qquad
F(u)=\frac{1}{1+4(d-u)^2}-\frac{1}{1+4(d+u)^2}.
$$
For the example $d=-0.5$ and $u=0.25$, the positive and negative beam factors are respectively $4/13$ and $4/5$. Thus $F(0.25)=-32/65\simeq-0.4923$: the atom is moving right and the net <force> points left. All values in this example are dimensionless.
Subtracting the two fractions gives an especially useful sign check:
$$
F(u)=\frac{16du}{\bigl(1+4(d-u)^2\bigr)\bigl(1+4(d+u)^2\bigr)},\qquad
F'(0)=\frac{16d}{(1+4d^2)^2}.
$$
For red detuning, the <force> opposes nonzero $v$ throughout this ideal model; its useful magnitude eventually falls at large $|u|$. Near zero, a <Taylor series> gives <damping>
$$
f(v)=-\alpha v+O(v^3),\qquad
\alpha=-\frac{8\hbar k^2 s d}{(1+4d^2)^2}>0.
$$
At $d=-0.5$, $F'(0)=-2$ and $\alpha=\hbar k^2s$. For blue detuning the sign reverses and the small-<velocity> motion is amplified; at exact resonance the opposing mean <forces> cancel in this symmetric model. Its derivation assumes weak excitation, a closed <two-level system>, equal beam intensities and nonrelativistic <velocities>; real multilevel atoms need additional physics.
\Image[/laser-cooling-dimensionless-force-velocity.png]
{title=Calculated Doppler cooling force versus velocity}
{description=The equal-beam weak-field model at $d=-0.5$. The right panel compares the exact calculated force with its tangent $F=-2u$ near zero velocity. Neither panel depicts measured data, and the horizontal axis is velocity rather than displacement.}
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