= Two-level saturation population model
{title2=$p_e=\frac{s}{2(1+s+4\Delta^2/\Gamma^2)}$}
Choose a closed <two-level system>, equal stimulated absorption and emission strengths, a monochromatic pump, and no extra <dephasing>. Let $p_g+p_e=1$ be the lower and upper populations, and let <spontaneous emission> remove upper population at rate $\Gamma=1/\tau_e$. In this convention the <natural linewidth> is <full width at half maximum> $\Gamma$ in <angular frequency>, its half width at half maximum is $\Gamma/2$, and its <full width at half maximum> in <hertz> is $\Gamma/(2\pi)$. All $\Delta$ below are angular detunings.
Let $s=I/I_{\rm sat}$ be the on-resonance <saturation parameter>. An effective stimulated transition rate that reproduces the monochromatic steady response is
$$
R(\Delta)=\frac{\Gamma s}{2(1+4\Delta^2/\Gamma^2)},\qquad
\dot p_e=Rp_g-(R+\Gamma)p_e=R(1-2p_e)-\Gamma p_e.
$$
Here $R$ describes each stimulated direction, while $\Gamma$ describes irreversible <spontaneous emission>. The rate equation gives the steady populations; it is not a description of transient <Rabi oscillations>. Setting $\dot p_e=0$ gives
$$
p_e=\frac{R}{\Gamma+2R}
=\frac{s}{2(1+s+4\Delta^2/\Gamma^2)},\qquad p_g=1-p_e,
$$
and the population difference relevant to net weak-probe <photon absorption> is
$$
d=p_g-p_e=\frac{1+4\Delta^2/\Gamma^2}{1+s+4\Delta^2/\Gamma^2},\qquad
q=1-d=\frac{s}{1+s+4\Delta^2/\Gamma^2}.
$$
The pump depletes the lower population by $p_e$, but reduces the population difference by $q=2p_e$: <stimulated emission> contributes as well as missing ground-state absorbers. On resonance, $s=1$ gives $p_e=1/4$, $p_g=3/4$, $d=1/2$; $s=9$ gives $p_e=0.45$, $p_g=0.55$, $d=0.10$. As $s\to\infty$, the two populations approach $1/2$, rather than producing a <population inversion>. Far off resonance, $p_e\to0$ and $d\to1$.
For moving atoms use $\Delta=\delta-kv$ in these pump-prepared populations. A weak counterpropagating probe interrogates them near $\delta+kv=0$. A simple population-only approximation to its <absorption coefficient> averages the weak-probe <Lorentzian line shape> times $d(\delta-kv)$ over the <Maxwell-Boltzmann velocity distribution>. It neglects interference between the beams, changes of <velocity> during illumination, and multilevel <optical pumping>. Those restrictions matter for real alkali <chemical atoms>.
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