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Q053 · Cool / count / compare

Quantum Gas Occupation

Prepare a small ideal gas with a fixed particle count. Record complete energy-level occupations, cool it, and compare bosons, fermions and independent distinguishable particles.

Interactive modelQuantum Gas Occupation
Records at this setting—\text{—}
Mean ground-level fraction—\text{—}

02 / EACH ROW IS A FRESH PREPARATION

One gas. A whole row of occupations.

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

—\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.

kBT/(hν),n‾0/Nk_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 · SEPARATE FROM RECORDS

Filling levels changes the cloud size.

x/μm,ρ(x)/μm−1x/\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 / KEEP A COMPARISON

Save the preparation with its evidence.

Eight frozen samples. Restore a sample to inspect its settings. New acquisition after saving starts a new sample. CSV retains 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 ground-level distribution supplies the finite-number fluctuations. A 64-versus-96 mode comparison reports cutoff sensitivity separately from shot uncertainty. Interactions, collisions, preparation time, evaporation and real mode-resolving detection errors are omitted.

Tong · Quantum Gases, Sections 3.5–3.6 · Xiong et al. · Canonical statistics

Physics tutorial

Quantum Gas Occupation

BackgroundPrepare a small ideal gas with a fixed particle count. Record complete energy-level occupations, cool it, and compare bosons, fermions and independent distinguishable particles.

Why it mattersKeep the preparation fixed while collecting independent copies. Change one cause and compare saved records.

Start with the essentials

Focus question
Cool the same trap. Who gets the lowest level?
One-sentence intuition
Fixed particle number couples the level occupations. Sample the whole configuration rather than independent Bose or Fermi mean values.

Core mathematical model

Fixed-number equilibrium

P(n)=ZN−1exp⁡[−∑jjnj/τ],∑jnj=N,τ=kBThνP(\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 allow any nonnegative occupation; single-component fermions allow at most one. This is a finite canonical distribution, not independent grand-canonical sampling.

Distinguishable control

PMB(n)=N!∏jnj!∏jpjnj,pj=e−j/τ∑ke−k/τP_{\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 obey exact total-number conservation. Their multiplicity differs from bosonic counting.

A model density, not a photograph

ρ(x)=∑j⟨nj⟩∣ϕj(x)∣2,a=ℏ2πmν\rho(x)=\sum_j\langle n_j\rangle|\phi_j(x)|^2,\quad a=\sqrt{\frac{\hbar}{2\pi m\nu}}

The cloud uses harmonic oscillator modes and mass forty atomic mass units for every statistical comparison. Recorded data are ideal mode occupations.

Common difficulties

Interpret the limit

Typical misconceptionA small cold cloud proves a bulk condensation transition.

Better mental modelThis finite one-dimensional system shows a smooth occupation crossover. No thermodynamic critical temperature is fitted.

Model and records

Typical misconceptionThe glowing cloud is measured data.

Better mental modelThe apparatus is schematic. Blue recorded charts and raw CSV come from sampled outcomes; model references are explicitly labelled.

Run the experiment

  1. 01

    Predict

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

    What to observe: A prediction is a choice, not a fitted result.
  2. 02

    Record a contrast

    Record cold bosons, save, and repeat with cold fermions. Inspect complete configurations and mean occupations.

    What to observe: Low temperature enhances ground occupation for bosons; Pauli filling preserves a broad fermionic cloud.
  3. 03

    Test the boundary

    Increase temperature and change the trap frequency at fixed reduced temperature. Separate occupation changes from physical length and temperature units.

    What to observe: Save and export zero results as well as successful detections. The source documentation separates sampling uncertainty and approximation error.