---
title: "Quantum Gas Occupation Interactive Lab | Sandbox Physics"
description: "When the same trap is cooled, which particles crowd into the lowest energy level? Bosons. As the gas cools they crowd into the lowest level."
canonical_url: "https://sandboxphysics.com/labs/quantum-gas-occupation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/quantum-gas-occupation/"
last_modified: "2026-10-27"
---

Q053 · Cool, count and compare

# Quantum Gas Occupation

**When the same trap is cooled, which particles crowd into the lowest energy level?** Bosons. As the gas cools they crowd into the lowest level. Fermions can take only one per state, so they fill a ladder of levels from the bottom up, and distinguishable particles land in between. With the total number fixed, filling one level leaves fewer particles for the others, so whole arrangements must be sampled, not one level at a time.

Prepare a small ideal gas with a fixed number of particles and record how many sit in every energy level. Cool it down and compare bosons, fermions and independent distinguishable particles.

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

The gas is a finite canonical ensemble in a one-dimensional harmonic trap with 64 nondegenerate modes. Occupations are read out by ideal projective measurement, and the particle mass is a fixed hypothetical value. The model has no interactions and no thermalization dynamics, and it does not claim a bulk condensation transition at finite temperature.

02 / Each row is a fresh preparation

## Each gas gives a whole row of occupations.

$j,\quad \overline n_j$ · At the selected setting. Blue: current sample; amber: the most recent other saved sample.

Show separate model predictions

$\text{—}$

Recent raw occupations: first sixteen levels, then a summed tail. Every complete record has the chosen total particle number.

03 / Change one setting

## Cooling changes the distribution, shot by shot.

$k_BT/(h\nu),\quad \overline n_0/N$ · Drag to inspect a setting; use the slider for keyboard input.

Whiskers: approximate pointwise 95% intervals from the variance between preparations. Each temperature uses 100 independent equilibrium preparations; this is not a cooling trajectory.

Model inspection · kept separate from the records

## Filling levels changes the cloud size.

$x/\mathrm{\mu m},\quad \rho(x)/\mathrm{\mu m}^{-1}$

Blue: selected statistics. Gray dashed: distinguishable control with the same particle count and temperature. Density is a model expectation, not a reconstruction of recorded occupations.

$\text{—}$ $\text{—}$

A finite one-dimensional ideal gas has a smooth ground-occupation crossover here. Do not read a bulk critical temperature from this plot.

04 / Save a sample for comparison

## Save the preparation with its evidence.

You can save up to eight samples. Restore one to see its settings. Acquiring again after saving starts a new sample. The CSV keeps every shot and every zero channel.

Units, approximations & primary sources

Exactly fixed particle number in a 64-mode, one-dimensional harmonic spectrum. Transverse motion is frozen. The canonical dynamic program samples complete Fock occupations; the distinguishable control includes multinomial multiplicity. The readout module denotes an ideal energy-basis projective measurement, not a simulated absorption camera or a calibrated laboratory sorter. The displayed cloud is the model one-body density. All statistics use a hypothetical mass of forty atomic mass units. At fixed reduced temperature a tighter trap changes physical temperature and length, while level occupations stay the same.

The finite-number fluctuations come from the ground-level distribution. Comparing 64 and 96 modes shows how sensitive the result is to the cutoff, reported separately from shot uncertainty. Interactions, collisions, preparation time, evaporation and the errors of a real mode-resolving detector are omitted.

[Tong · Quantum Gases, Sections 3.5–3.6](https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html) · [Xiong et al. · Canonical statistics](https://arxiv.org/abs/cond-mat/0111125)

Physics tutorial

## Quantum Gas Occupation

**Background** Prepare a small ideal gas with a fixed number of particles and record how many sit in every energy level. Cool it down and compare bosons, fermions and independent distinguishable particles.

**Why it matters** Collect many independent runs with the same preparation. Then change one thing at a time and compare the saved results.

Start with the essentials

- **Focus question**: When the same trap is cooled, which particles crowd into the lowest energy level?
- **One-sentence intuition**: With the particle number fixed, the occupations of different levels depend on each other, so the whole configuration has to be sampled at once. Drawing each level from its own Bose or Fermi average would be wrong.

### Core mathematical model

#### Equilibrium at fixed particle number

$$
P(\mathbf n)=Z_N^{-1}\exp[-\sum_j j n_j/\tau],\quad \sum_j n_j=N,\quad \tau=\frac{k_BT}{h\nu}
$$

Bosons can put any number of particles in a mode; single-component fermions allow at most one. The distribution is finite and canonical, so the levels are not sampled independently as in the grand-canonical ensemble.

#### Distinguishable particles for comparison

$$
P_{\rm MB}(\mathbf n)=\frac{N!}{\prod_j n_j!}\prod_j p_j^{n_j},\quad p_j=\frac{e^{-j/\tau}}{\sum_k e^{-k/\tau}}
$$

Independent particles still conserve the total number exactly, but they count arrangements differently from bosons.

#### Cloud density built from mode occupations

$$
\rho(x)=\sum_j\langle n_j\rangle|\phi_j(x)|^2,\quad a=\sqrt{\frac{\hbar}{2\pi m\nu}}
$$

The cloud is built from harmonic-oscillator modes, with a mass of forty atomic mass units in every comparison. The recorded data are ideal mode occupations, not a photograph.

### Common difficulties

#### Reading the result correctly

**Typical misconception** A small cold cloud proves a bulk condensation transition.

**Better mental model** This finite one-dimensional system shows a smooth crossover in occupation. No thermodynamic critical temperature is fitted.

#### What is data and what is drawn

**Typical misconception** The glowing cloud is measured data.

**Better mental model** The apparatus is a schematic. The blue charts and the raw CSV come from sampled outcomes, and model references are labeled as such.

### Run the experiment

1. 01

   #### Make a prediction

   Predict whether cooling eight fermions can put them all in the lowest level.

   **What to observe:** Your prediction is a guess to be tested, not a fitted result.

2. 02

   #### Record a comparison

   Record cold bosons and save, then repeat with cold fermions. Look at both the complete configurations and the mean occupations.

   **What to observe:** At low temperature, more bosons pile into the ground level. Fermions must fill the levels one at a time under the Pauli principle, so their cloud stays broad.

3. 03

   #### Test the limits

   Raise the temperature, then change the trap frequency while holding the reduced temperature fixed. Tell real changes in occupation apart from changes in the physical length and temperature units.

   **What to observe:** Save and export the shots with zero detections along with the successful ones. The source notes treat sampling uncertainty and approximation error separately.

Physical conclusion

The gas is a finite canonical ensemble in a one-dimensional harmonic trap with 64 nondegenerate modes. Occupations are read out by ideal projective measurement, and the particle mass is a fixed hypothetical value. The model has no interactions and no thermalization dynamics, and it does not claim a bulk condensation transition at finite temperature.

## Related experiments and articles

Quantum evidence · Reading path

### Probability and exclusion

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

1. [Article **Pauli: count the states before filling the shells**](https://sandboxphysics.com/stories/quantum-evidence/pauli-spectra-exclusion-principle/)
2. [Lab **Identical Atom Exchange** Compare identical atoms with distinguishable ones](https://sandboxphysics.com/labs/identical-atom-exchange/)

Condensed matter · Reading path

### Cold liquids and superconducting order

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

1. [Article **How Landau explained superfluid helium through its excitations**](https://sandboxphysics.com/stories/condensed-matter/landau-superfluid-helium/)
2. [Lab **BEC–BCS Crossover** Pairing across the crossover](https://sandboxphysics.com/labs/bec-bcs-crossover/)
3. [Lab **Quantum Vortex Studio** Quantum vortex dynamics](https://sandboxphysics.com/labs/quantized-vortex-dynamics/)

Condensed matter · Reading path

### The evidence for dilute-gas condensation

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

1. [Article **How expanding atom clouds revealed Bose–Einstein condensation**](https://sandboxphysics.com/stories/condensed-matter/bec-time-of-flight/)
2. [Lab **BEC–BCS Crossover** Pairing across the crossover](https://sandboxphysics.com/labs/bec-bcs-crossover/)
3. [Lab **Matter-Wave Collision** Interference between condensate clouds](https://sandboxphysics.com/labs/condensate-matter-wave-collision/)

1 more group

Condensed matter · Reading path

### Magnetism, disorder and topological response

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

1. [Article **How vortex pairs unbind and destroy two-dimensional superfluidity**](https://sandboxphysics.com/stories/condensed-matter/kt-vortex-unbinding/)
2. [Lab **Superconductors in a Magnetic Field** Superconducting critical fields](https://sandboxphysics.com/labs/superconductor-critical-fields/)
3. [Lab **Quantum Vortex Studio** Quantum vortex dynamics](https://sandboxphysics.com/labs/quantized-vortex-dynamics/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

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