The Holstein equation

An excited atom in a vapour decays by emitting a photon. The photon flies in a straight line and may be reabsorbed by another atom — which then re-emits, and so on. After many such hops, the photon either escapes the cell or is destroyed by a quenching collision. The question is: how long does it take, and where does the excitation distribution settle?

The naive answer is "diffusion." Photons hop between atoms much like neutrons in a moderator. Compton (1922) and Milne (1926) wrote diffusion equations. Their equations gave excitation decaying faster than the natural lifetime — physically impossible.

The frequency-averaged mean free path of a photon in an atomic vapour is not finite. For every lineshape with extended wings — Doppler, Lorentz, Voigt — the integral $\int dx / k(x)$ diverges. There is no diffusion equation. Holstein (1947) and Biberman (1947) independently realized this and wrote an integral equation instead:

\[ \frac{\partial n(\mathbf{r}, t)}{\partial t} = -\frac{1}{\tau} n(\mathbf{r}, t) + \frac{1}{\tau} \int_V n(\mathbf{r}', t)\, G(\mathbf{r}, \mathbf{r}')\, d\mathbf{r}' \]

Here $n(\mathbf{r}, t)$ is the excited-state density, $\tau$ is the natural lifetime, and $G(\mathbf{r}, \mathbf{r}')$ is the Kernel function — the probability that a photon emitted at $\mathbf{r}'$ is reabsorbed at $\mathbf{r}$. The Kernel encodes everything: the lineshape, the geometry, the wing physics.

The solutions of this integral equation are eigenmodes:

\[ n(\mathbf{r}, t) = \sum_j \alpha_j\, \psi_j(\mathbf{r})\, \exp\!\left(-\frac{t}{g_j \tau}\right) \]

Each mode $\psi_j(\mathbf{r})$ decays with its own time constant $g_j \tau$. The trapping factor $g_j$ is the mean number of absorption-reemission processes a photon belonging to mode $j$ undergoes before escaping. At late times only the slowest-decaying mode $j=0$ survives, with $g_0$ the dominant figure of merit.

Explore

Below: change the lineshape, the geometry, the opacity. Watch $g_0$ scale linearly with opacity for Doppler (wings die fast), as $\sqrt{k_0 L}$ for Lorentz (wings die slowly). Watch the mode shape change. Watch the wing-escape region — shaded in red — shrink as opacity grows.

Lineshape

Geometry

Opacity $k_0 L$ 10

Mode index j 0 (j>0 shapes: schematic — see Layer 2)

Lineshape $k(x)$ — wings shaded where photons escape

Red shading: $|x| > x_\text{esc}$ where $k(x)\,k_0 L \le 1$. Photons here escape on first emission.

Trapping factors $g_j(k_0 L)$ — logarithmic

Solid: $g_0$. Dashed: $g_1$ … $g_3$. Dot: current opacity.

Mode shape $\psi_j$ in space

Slab: $\psi$ vs $z/L$. Cyl/sph: $\psi$ vs $r/R$. Ground mode $j{=}0$ is the correct Holstein eigenfunction. $j{\geq}1$ shapes are schematic placeholders ($z_j \approx z_0 + j\pi$, not the exact eigenvalues) — Layer 2 implements the full multi-mode solver.

FEYNMAN's worked example. Set Doppler · slab · $k_0 L = 10$. Read off $g_0 \approx 12.9$. That means a photon born at the cell centre bounces around thirteen times before escaping. At $k_0 L = 100$, $g_0 \approx 177$. At $k_0 L = 1000$, $g_0 \approx 2350$ — close to linear in $k_0 L$, modulated by $\sqrt{\ln(k_0 L)}$. Now switch to Lorentz · slab · $k_0 L = 1000$: $g_0 \approx 49$. Forty-eight times fewer reabsorptions despite identical opacity, because Lorentz wings $1/x^2$ give photons an escape route Doppler wings $e^{-x^2}$ deny. The lineshape is the load-bearing parameter — not the strength.

What the equations say

The Kernel $G(\mathbf{r}, \mathbf{r}')$ depends on the geometry. For a slab of thickness $L$:

\[ G(|z-z'|) = \frac{C_x}{2} \int_1^\infty \int_{-\infty}^\infty \frac{k^2(x)}{u}\, \exp\!\left[-k(x)|z-z'|u\right]\, dx\, du \]

where $u = 1/|\cos\vartheta|$ over the half-sphere of photon directions. Equivalent forms exist for cylinder and sphere (M&O Insert 4.1).

The trapping factor for the ground mode follows closed-form fitting equations developed by Molisch & Oehry (Ch 7). For Doppler in any geometry:

\[ g_0^D = 1 + \frac{k_0 L}{m_0^D}\sqrt{\ln\!\left(\frac{k_0 L}{2} + e\right)} - \frac{c_{00}\, k_0 L \ln(k_0 L) + c_{10}\, k_0 L + c_{20}\,(k_0 L)^2}{1 + c_{30}\, k_0 L + c_{40}\,(k_0 L)^2} \]

For Lorentz:

\[ g_0^L = \frac{1}{m_0^L}\sqrt{\pi\, k_0 L + (m_0^L)^2} - \text{(rational correction)} \]

The constants $m_j$, $c_{ij}$ are tabulated in M&O Appendix D, Tables D.1–D.9 (slab/cylinder/sphere × Doppler/Lorentz, first 10–20 modes). They are baked into the explorable above.

The wing-escape picture (Zanstra, via Irons 1979): at high opacity, only photons at frequencies $x$ where the local opacity $k(x) k_0 L \le 1$ can escape on first emission. Solving $k(x_\text{esc}) k_0 L = 1$ gives

Doppler: $x_\text{esc} = \sqrt{\ln(k_0 L)}$ — logarithmic growth of escape window
Lorentz: $x_\text{esc} \approx \sqrt{k_0 L}$ — power-law growth, much faster
Voigt: numerical root-find on $H(a, x)/H(a, 0) = 1/k_0 L$

The escape-frequency window controls the trapping factor scaling. Wings dominate.

What this layer does NOT do

Multi-mode initial conditions — Layer 2.
Higher-mode eigenfunctions $\psi_{j\geq 1}$ with the exact transcendental eigenvalues (M&O Eq 7.7–7.9) — Layer 2. (Higher-mode trapping factors $g_{j\geq 1}$ are correct here; only the shape functions are placeholders.)
Partial frequency redistribution (PFR) — Layer 3. The Holstein equation as written above assumes complete frequency redistribution (CFR), valid only when collisions or Doppler reshuffling dominate natural broadening.
Multi-level atoms, hyperfine splitting, isotope mixing — Layer 4.
Saturation, ground-state depletion — Layer 5.
Self-consistent NLTE radiation transport (CRETIN's regime) — Layer 6.
2D / 3D geometry — Layer 7.
The LightCell 819 nm extraction problem — Layer 8.

Validation

Numerical spot-checks against M&O Appendix D fitting equations, computed live by this page. (If any of these is off by >3 % the layer is broken.)

Setup	$g_0$ (fitting.js)	$g_0$ (M&O Eq D.1, hand-traced)	$\Delta$

Read further

M&O Ch 4 (derivation of the Holstein equation; the nine assumptions)
M&O Ch 5 (PCA, variational, propagator-function solution methods)
M&O Ch 7 (fitting equations, physical interpretation, escape via wings)
M&O Appendix D (constants $m_j$, $c_{ij}$, $d_i$)
Bezuglov, Klyucharev, Kazansky, Molisch, Fuso, Allegrini, PRA 57, 2612 (1998) — geometric-quantization analytical solution of Holstein. Molisch's own post-book extension.

FEYNMAN ✓ worked example present (lines 110) · Victor ✓ reactive sliders · Kay ⏳ Layer 2 needs to compose from these objects without rewriting · Scott ⏳ Doppler/Lorentz × g₀ verified against M&O Eq D.1 to <0.01% (9 rows); g_{j>0} monotonicity verified j=0..19; Voigt path (Eq 7.16) cross-check + higher-mode shape correctness deferred to Layer 2/3

Adversarial review: AUDIT-LAYER1-v2.md — PASS-WITH-CAVEATS (2026-05-27). Three v1 defects fixed; five new findings (mode-shape oversell scoped to Layer 2; voigtH0 latent at a≥5 patched; signoff de-puffed; Voigt validation row added; dead `asymptoticG0` accepted).