= Three-sublevel optical pumping model
{title2=$F=1\to F'=0$}
Consider an original hypothetical atom with a lower <angular momentum> multiplet $F=1$, $m=-1,0,+1$, and one opposite-<parity> excited level $F'=0,m'=0$. Stipulate that all <spontaneous emission> returns to these three lower <quantum states>, with no other decay channel. This is an isolated four-state model, not the actual <sodium> or <rubidium> D-line structure. In particular, the physical <rubidium> cell above contains multiple <hyperfine structure> levels.
The <electric-dipole selection rule> permits $\Delta F=-1$ here; $\sigma^+$ absorption with $\Delta m=+1$ excites only $m=-1$. Decay from $F'=0$ has three equal integrated branching probabilities $b_-=b_0=b_+=1/3$: rotational symmetry gives equal squared <Clebsch-Gordan coefficients>. The emitted <photons> carry the three permitted <angular momentum> projections; equal integrated branching does not imply equal intensity in every observation direction. Thus $m=0$ and $m=+1$ are both dark. The chosen beam cannot prepare a unique pure $m=+1$ state.
Let $R\ge0$ be the absorption rate of an atom in $m=-1$, and let $\Gamma$ be the excited-state <spontaneous emission> rate, both in inverse seconds. Assume $R,\gamma\ll\Gamma$, negligible <stimulated emission>, and no maintained <quantum coherence> between sublevels. Eliminating the short-lived excited population gives normalized lower populations $p_-+p_0+p_+=1$ to this approximation. At each scattering event the atom leaves $m=-1$ with probability $2/3$; the remaining $1/3$ simply returns it there.
Model depolarizing collisions and other population relaxation by resets to the unpolarized distribution at rate $\gamma\ge0$. The <rate equations> are
$$
\dot p_-=-\frac{2R}{3}p_-+\gamma\left(\frac13-p_-\right),\qquad
\dot p_0=\frac R3p_-+\gamma\left(\frac13-p_0\right),\qquad
\dot p_+=\frac R3p_-+\gamma\left(\frac13-p_+\right).
$$
This relaxation prescription assumes that <Zeeman splitting> is small compared with thermal <energy>, so the unpolarized distribution approximates <thermal equilibrium>. It is not an energy-independent relaxation law at arbitrarily low <temperature>.
Every transfer has a nonnegative rate: optical transfers $-\to0$ and $-\to+$ each have rate $R/3$, and relaxation transfers between each distinct pair have rate $\gamma/3$. These define a <continuous-time Markov chain>. The three derivatives sum to zero, and at $p_i=0$ the corresponding derivative cannot be negative. Hence the model conserves population and preserves nonnegative populations. Optical cycles that return to $m=-1$ contribute to scattering even though they cause no net population transfer.
\Image[/optical-pumping-three-sublevel-rate-dynamics.png]
{title=Three-sublevel pumping, spontaneous branching and dark-state dynamics}
{description=Original hypothetical $F=1\to F'=0$ model. Solid blue shows selective absorption; dashed purple arrows show equal spontaneous branching. Two lower states are dark to the selected beam. Curves start unpolarized and compare pure pumping with chosen $R/\gamma=6$. Rates and curves are stipulated examples, not historical data or a model of the full rubidium spectrum.}
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