---
title: "Population Inversion Workbench Interactive Lab | Sandbox Physics"
description: "Why can stronger pumping still fail to produce gain? Because gain needs population inversion, and pumping harder does not guarantee it."
canonical_url: "https://sandboxphysics.com/labs/population-inversion/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/population-inversion/"
last_modified: "2026-10-28"
---

L03 · Laser foundations

# Population Inversion Workbench

**Why can stronger pumping still fail to produce gain?** Because gain needs population inversion, and pumping harder does not guarantee it. With only two levels, the pump light drives atoms down as readily as up, so the upper level can never outnumber the lower one. Three- and four-level schemes get around this, but inversion still fails if the lower laser level empties too slowly or atoms leak out of the upper level without emitting light.

Build a pumping scheme with two, three or four levels. Store excitation in the upper laser level, empty the lower level, and watch a bottleneck or a nonradiative leak destroy the inversion.

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

The model is a closed rate network of levels with equal degeneracy, and the incoherent optical pump drives atoms both up and down. All atoms start in the ground state. The population fractions are deterministic; the 80 markers are a rounded display and do not follow individual atoms. The density and cross section used for the weak-probe gain are given in the lesson formulas below. Probe depletion, cavity loss, thermal effects and the lasing threshold are not included.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Engineer a population route

**Background** A pump pushes the atoms away from thermal equilibrium. Whether an inversion builds up depends on three steps working together: the pump transition, relaxation into the upper laser level, and removal from the lower laser level.

**Why it matters** More pump power on the wrong transition cannot make up for a poor route between levels. A four-level scheme helps only when its lower laser level really empties fast enough.

Start with the essentials

- **Focus question**: Can a four-level medium lose its inversion while the pump is still on?
- **One-sentence intuition**: A long-lived upper level can store excitation, and a lower level that empties quickly absorbs less. Both effects have to compete with leakage and with the pump driving atoms back down.

### Core mathematical model

#### Rate equations that conserve atoms

$$
\dot n_i=\sum_{j\ne i}(k_{ji}n_j-k_{ij}n_i),\qquad\sum_i n_i=1
$$

All rates are per millisecond. Optical pumping moves atoms up and down at equal rates. Relaxation and leakage send atoms to definite levels, so no atom is ever lost.

#### Continuous two-level limit

$$
n_{u,\infty}=\frac{W_p}{2W_p+\Gamma}<\frac12,\qquad\Gamma=\tau_u^{-1}+q
$$

This bound holds for continuous incoherent pumping of levels with equal degeneracy and a positive relaxation rate, starting from the ground state. It does not rule out a brief inversion made by a coherent pulse.

#### Inversion and gain with assumed parameters

$$
\Delta n=n_u-n_l,\qquad g_0=\sigma_0N\Delta n
$$

The coefficient is positive for gain and negative for absorption. Radiative decay from the upper level feeds the lower laser level, and nonradiative leakage goes to the ground state.

#### Teaching values for density and cross section

$$
N=10^{24}\,\mathrm{m}^{-3},\quad\sigma_0=2\times10^{-24}\,\mathrm{m}^{2}
$$

These values are assumed; they are not fitted to ruby or to any particular commercial gain medium. In the three-level scheme, the lower laser level is the ground state; the four-level scheme adds a separate lower level.

### Common difficulties

#### Transparency is not the lasing threshold

**Typical misconception** Any positive gain means the medium is already a laser.

**Better mental model** A laser needs a resonator that feeds light back into a mode, and the gain must beat all the losses of that mode. This experiment has no resonator, so its gain readout alone cannot show oscillation.

#### Four levels do not guarantee inversion

**Typical misconception** A four-level system inverts at any pump rate, however low.

**Better mental model** That ideal limit assumes the lower level empties fast and relaxation sends atoms where they need to go. Finite rates and leakage can change or remove the advantage.

### Run the experiment

1. 01

   #### Try to beat the two-level limit

   Choose Two-level limit and advance 10 ms. Increase lifetime or remove leakage, then compare the upper and lower fractions.

   **What to observe:** A stronger continuous pump brings the populations closer to equal, but under the model assumptions it never sustains a positive inversion.

2. 02

   #### Store excitation

   Choose Three levels and advance. Lower the pump-to-upper relaxation rate or raise the nonradiative leak; the populations carry over when a rate changes.

   **What to observe:** Storage in the upper level competes with atoms returning through the pump and with losses from the upper level. Putting atoms into a high pump level does not by itself invert the chosen laser transition.

3. 03

   #### Clear a bottleneck

   Choose Lower-state bottleneck and advance 10 ms. Change only the lower-to-ground emptying rate to 20 per millisecond, then advance again.

   **What to observe:** When atoms pile up in the lower laser level, even a four-level medium can absorb. Emptying that level restores a useful population difference without raising the pump rate.

4. 04

   #### Switch the pump off

   Build an inversion, uncheck Pump on, and step through the decay. Compare the time trace with the updated steady-state value.

   **What to observe:** The stored population relaxes once pumping stops. The dashed steady-state line belongs to the current settings, not to the earlier history. The total population stays the same.

Physical conclusion

Population inversion gives small-signal gain. A laser also needs feedback and enough gain to cover the cavity loss; this Lab measures transparency and inversion, not a lasing threshold.

## Related experiments and articles

Continue exploring

### Create an inversion

What prevents a pumped medium from maintaining a useful inversion?

1. [Lab **Pumping & Population Inversion** Pump two-, three- and four-level gases with one lamp and see which one amplifies.](https://sandboxphysics.com/labs/pumping-population-inversion/)
2. [You are here **Population Inversion Workbench** Follow two-, three- and four-level transients, leakage and bottlenecks.](https://sandboxphysics.com/labs/population-inversion/)

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)