= Delayed-pumping model of Sisyphus friction
{c}
{title2=$\overline F=-U_0k^2v\gamma_p/[2(\gamma_p^2+k^2v^2)]$}
Consider an original reduced <rate equation> model using the same potentials. Write $s(z)=\cos(2kz)$ and stipulate positive <optical pumping> rates $R_{+\to-}=\gamma_p(1-s)$, $R_{-\to+}=\gamma_p(1+s)$, with $\gamma_p>0$. They pump towards $+$ at $z=0$ and towards $-$ at $z=\lambda/4$, consistently with the diagram. These rates retain the relevant symmetry and delay, while omitting excited-state dynamics, <quantum coherence> and recoil diffusion.
For populations $p_++p_-=1$, put $q=p_+-p_-$. The <rate equations> conserve population and give $\dot q=2\gamma_p(s-q)$. The stationary atom has $q=s$; a moving atom cannot follow it instantaneously. On a prescribed trajectory $z=vt$, the periodic solution is
$$
q(z)=\frac{\gamma_p^2\cos(2kz)+\gamma_pkv\sin(2kz)}{\gamma_p^2+k^2v^2}.
$$
Substitution into $vq'=2\gamma_p(s-q)$ verifies both coefficients. The mean mechanical <force> is $F=-p_+U_+'-p_-U_-'=-U_0kq\sin(2kz)$. Averaging over a spatial period, with mean squared sine $1/2$ and mean sine times cosine zero, gives
$$
\overline F=-\frac{U_0k^2v\gamma_p}{2(\gamma_p^2+k^2v^2)},\qquad
\overline F\simeq-\frac{U_0k^2}{2\gamma_p}v\quad (|kv|\ll\gamma_p).
$$
Thus the <force> opposes either sign of <velocity> and vanishes at rest. A small pumping rate makes the small-<velocity> friction slope large while narrowing its useful <velocity> range. Holding $v$ fixed while sending $\gamma_p$ to zero makes the <force> vanish; the apparently divergent zero-<velocity> slope is not unlimited cooling. Because this model has no <momentum> diffusion, it predicts no equilibrium <temperature>. It demonstrates the lag mechanism rather than replacing a multilevel quantum calculation.
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