 Delayed-pumping model of Sisyphus friction

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 isSubstitution into $v{q}^{\prime }=2{\gamma }_p(s-q)$ verifies both coefficients. The mean mechanical force is $F=-p_{+}{U}_{+}^{\prime }-p_{-}{U}_{-}^{\prime }=-U_{0}kq\sin (2kz)$. Averaging over a spatial period, with mean squared sine $1/2$ and mean sine times cosine zero, givesThus 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.