---
title: "Absorption, Spontaneous & Stimulated Emission Interactive Lab | Sandbox Physics"
description: "What decides whether atoms absorb resonant light or amplify it? It depends on which level holds more atoms."
canonical_url: "https://sandboxphysics.com/labs/absorption-spontaneous-stimulated-emission/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/absorption-spontaneous-stimulated-emission/"
last_modified: "2026-10-28"
---

Optics 101 · Gain or absorption

# Absorption, Spontaneous & Stimulated Emission

**What decides whether atoms absorb resonant light or amplify it?** It depends on which level holds more atoms. Atoms in the lower level absorb resonant light; atoms in the upper level are stimulated by it to emit coherent copies of the photons. If the lower level is more crowded, absorption wins; if the upper level is, the light is amplified. Spontaneous emission happens at random either way, with or without the light.

A probe beam crosses a gas cell just after a pulse has excited most of its atoms. Watch the beam grow, then fade as the atoms decay, and count each absorbed, stimulated and spontaneous photon.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Two-level atoms with equal degeneracy and one homogeneously broadened Lorentzian line, in the rate-equation limit: dephasing is much faster than spontaneous decay, so the atomic polarization follows the field and coherent Rabi oscillation is left out. A short preparation pulse sets the upper-level fraction at time zero. How that pulse prepares the atoms is outside the model; a resonant coherent pulse can do it, while continuous incoherent light on two levels cannot. Time is in spontaneous lifetimes. The probe enters with a fixed intensity and is followed through 64 slices of the cell. In each slice it grows or decays exponentially with the local population difference, and the atoms there see the slice-mean intensity, so the photons added to the beam equal stimulated minus absorbed events. Absorption and stimulated emission share one rate per atom, set by the local intensity and the Lorentzian line factor. Spontaneous emission goes into every direction and mode; its share in the probe mode is neglected. The phase delay comes from the same Lorentzian susceptibility and equals minus half the gap offset times the natural log of the transmission. The 120 tagged atoms are a random sample drawn with the selected seed. Each follows the ensemble rates of its slice and its events are counted one by one; the ensemble record does not depend on the seed. Doppler broadening, hyperfine structure, radiation trapping, optical pumping among sublevels, beam diffraction and the probe's travel time are not included. The wavelength, the drawn atoms and the photon markers are schematic. The drawn wave amplitude follows the square root of the transmission and is clipped above e times the input and below a tenth.

Physics tutorial

## Watch a gas switch from amplifying light to absorbing it

**Background** An atom with two levels can do three things with light at its transition frequency: absorb a photon, be stimulated by one to emit a copy, or emit a photon on its own. Einstein showed in 1916 that all three are needed for matter and thermal radiation to reach equilibrium.

**Why it matters** Whether a gas brightens a beam or dims it is a race between the first two processes. The record follows a probe beam through a cell whose atoms start mostly excited and then decay.

Start with the essentials

- **Focus question**: When does a beam gain more photons from stimulated emission than it loses to absorption?
- **One-sentence intuition**: Absorption and stimulated emission have the same rate per atom, so the beam grows exactly when more atoms are in the upper level than in the lower one. The probe itself drives the atomic dipole that radiates a stimulated photon, so that photon joins the probe in the same mode, with the same frequency, phase and polarization.

### Core mathematical model

#### Three processes, one shared rate

$$
\frac{dn}{dt}=W(1-n)-W\,n-\frac{n}{\tau}
$$

Here n is the upper-level fraction, W the stimulated rate per atom set by the local probe intensity and the line shape, and tau the spontaneous lifetime. The two levels have equal degeneracy, so absorption and stimulated emission share W.

#### The stimulated rate

$$
W\tau=\frac{\ell(\delta)}{2}\,\frac{I}{I_{\rm sat}},\qquad \ell(\delta)=\frac{1}{1+\delta^{2}}
$$

Delta is the level gap minus the photon energy, in half linewidths. Moving the gap off the photon energy lowers the rate along the Lorentzian line.

#### Gain or absorption along the cell

$$
\frac{dI}{dz}=\alpha_0\,\ell(\delta)\,(2n-1)\,I
$$

The population difference 2n minus 1 decides the sign. The beam grows wherever more than half the atoms are in the upper level.

#### The phase follows the gain

$$
\Delta\phi=-\frac{\delta}{2}\,\ln T
$$

For a Lorentzian line the same population difference sets the refractive index. Below resonance an absorbing gas delays the wave and an excited gas advances it; on resonance the added light is exactly in phase or out of phase.

#### When the gain disappears

$$
t_{\times}=\tau\ln(2p_0)
$$

With a weak probe every atom decays freely from the prepared fraction p0, and the gain vanishes when half the atoms are left up. A stronger probe drives atoms down faster, so the crossing comes earlier.

#### Continuous light never inverts two levels

$$
n\to\frac{s/2}{1+s}<\frac{1}{2},\qquad s=\ell(\delta)\,\frac{I}{I_{\rm sat}}
$$

The steady upper fraction under any constant intensity approaches one half from below.

#### How far a strong probe bleaches the cell

$$
\ln T+\ell(\delta)\,\frac{I_{\rm in}}{I_{\rm sat}}\,(T-1)=-\alpha_0 L\,\ell(\delta)
$$

Steady transmission of an absorbing cell. The record's end is compared with this curve in the collapsed diagnostic.

### Common difficulties

#### A stimulated photon does not go its own way

**Typical misconception** A stimulated photon leaves in a random direction, like spontaneous light.

**Better mental model** The probe drives the atomic dipole, so the light it radiates adds to the same beam mode with the same frequency, phase and polarization. Only spontaneous emission spreads into every direction.

#### Strong light does not pass the halfway mark

**Typical misconception** Bright enough light can push more than half the atoms up.

**Better mental model** Light at the transition drives both directions at the same rate. Under constant light the upper fraction only approaches one half; inversion needs another route through extra levels.

#### Spontaneous light does not add to the beam on average

**Typical misconception** Spontaneous photons strengthen the probe beam.

**Better mental model** They go into every direction with random phase, so they add no coherent field to the beam. The tiny share that lands in the beam mode is noise, which this model leaves out.

### Run the experiment

1. 01

   #### Watch the gas amplify, then absorb

   Keep the default preset, Excited gas amplifies, then absorbs, and watch one replay.

   **What to observe:** The output wave starts taller than the input. Purple atoms go dark as amber photons fly off, and after about half a lifetime the output is weaker than the input.

2. 02

   #### Start with every atom down

   Load Ground-state gas absorbs.

   **What to observe:** Absorption events outnumber stimulated ones and the gain stays below zero for the whole record.

3. 03

   #### Push the probe hard

   Load Strong probe: clear, never gain.

   **What to observe:** Stimulated events almost match absorptions. The upper fraction stalls just below one half and the cell becomes nearly transparent, but never amplifies.

4. 04

   #### Detune and compare the phase

   Load the two off-resonance presets one after the other and compare the output wave with the faint empty-cell wave.

   **What to observe:** The absorbing gas shifts the crests back; the excited gas shifts them forward. The phasor plot turns the opposite way.

Physical conclusion

Gain needs an inversion. Continuous light on a two-level gas cannot make one: it pushes atoms down as often as it lifts them, so the gas only bleaches toward transparency.

## Related experiments and articles

Continue exploring

### Events and their average

How do random transition counts approach a smooth population curve?

1. [You are here **Absorption, Spontaneous & Stimulated Emission** Send a probe through an excited gas and watch gain turn into absorption.](https://sandboxphysics.com/labs/absorption-spontaneous-stimulated-emission/)
2. [Lab **Three Light–Matter Transitions** Count individual stochastic events and compare with the expected population.](https://sandboxphysics.com/labs/light-matter-transitions/)

Probes and measurement · Reading path

### Controlled light and precision measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How an ammonia beam became a microwave oscillator**](https://sandboxphysics.com/stories/probe-evidence/maser-inversion-and-feedback/)
2. [Lab **Pumping & Population Inversion** Pump three gases with one lamp and see which one amplifies](https://sandboxphysics.com/labs/pumping-population-inversion/)
3. [Lab **Passive Optical Resonator** Compare stored light with self-sustained oscillation](https://sandboxphysics.com/labs/passive-optical-resonator/)
4. [Lab **Stable Resonator & Transverse Modes** Curve two mirrors and size an iris until one beam shape is left](https://sandboxphysics.com/labs/stable-resonator-transverse-modes/)
5. [Lab **Build & Debug Your First Laser** Test gain against cavity loss](https://sandboxphysics.com/labs/build-debug-laser/)
6. [Lab **Laser Noise Startup Workbench** Follow a cavity mode from its first seed](https://sandboxphysics.com/labs/laser-noise-startup/)
7. [Lab **Laser Startup from Noise** Watch many cavity frequencies compete for one gain](https://sandboxphysics.com/labs/laser-startup-from-noise/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)