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.

Records at this setting $\text{—}$

Mean ground-level fraction $\text{—}$

01 / Two ways to count atoms

Lower the temperature and watch the lowest level.

Record a cold bosonic gas and save. Keep eight particles but choose fermions. Compare the occupied levels before opening the model reference.

Schematic crossed dipole trap · glow shows model mean density, not an image · ideal mode readout is a conceptual measurement module

Prepare fresh atoms
Prepare equilibrium
Record occupations

Temperature relative to level spacing

k_BT/(h\nu)

0.8

No records yet. Every readout starts with a fresh preparation.

Try a comparison

Statistical preparation

Particles per preparation

N

8

Axial trap frequency

\nu/\mathrm{Hz}

500

Sampling & detection

Sampling seed

s_0

13721

The same seed repeats a sample; repeated seeds are not independent evidence.

Predict the mean ground fraction

Save before you change the preparation. Moving the scan setting keeps the records.

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

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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.

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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 · Xiong et al. · Canonical statistics

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 misconceptionA small cold cloud proves a bulk condensation transition.

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

What is data and what is drawn

Typical misconceptionThe glowing cloud is measured data.

Better mental modelThe 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

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.
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.
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.

Quantum Gas Occupation

Each gas gives a whole row of occupations.

Cooling changes the distribution, shot by shot.

Filling levels changes the cloud size.

Save the preparation with its evidence.