Q052 · Two atoms / two traps

Identical Atom Exchange

Load one atom into each trap. Apply the same tunneling pulse, count both outputs, then mark one internal state. Separate exchange interference from ordinary single-particle motion.

Records at this setting $\text{—}$

Detected split fraction $\text{—}$

01 / TWO WAYS TO COUNT ATOMS

Same pulse. Different company.

Record the default bosons and save. Switch to fermions: does the pair stay together? Mark one atom to erase the exchange contrast.

Schematic optical tweezers · latest detected atoms shown after readout · no particle paths · number-resolving readout is idealized

Prepare fresh atoms
Apply tunnel pulse
Record occupations

Tunnel pulse duration

t/\mathrm{ms}

1

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

TRY A CONTRAST

Statistical preparation

Tunnel coupling

J/h\,/\,\mathrm{kHz}

0.125

Trap energy imbalance

\Delta/h\,/\,\mathrm{kHz}

0

Internal-state marking angle

\alpha/\mathrm{deg}

0

Sampling & detection

Single-atom detection efficiency

\eta

1

Sampling seed

s_0

13721

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

Predict the more likely result

Save before changing preparation. Moving the scan setting preserves records.

Noninteracting two-mode model, one atom per input. Ideal number-resolving detection with equal independent losses. Bosonic and fermionic statistics are alternative model preparations, not a switch of atomic species. No collisions, heating or parity-only readout.

02 / EACH ROW IS A FRESH PREPARATION

Count both outputs. Keep every outcome.

$(N_L,N_R),\quad \widehat P$ · At the selected setting. Blue: current sample; amber: the most recent other saved sample.

Show separate model predictions

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Recent raw detections: two output counts per row. Empty and missed detections stay in the denominator.

03 / CHANGE ONE SETTING

Find where the pair changes its mind.

$t/\mathrm{ms},\quad \widehat P_{\rm det}(1,1)$ · Drag to inspect a setting; use the slider for keyboard input.

Whiskers: pointwise 95% Wilson intervals across all preparations. Each scan setting uses 200 fresh pairs.

MODEL INSPECTION · SEPARATE FROM RECORDS

Same one-atom motion. Different joint statistics.

$t/\mathrm{ms},\quad R,\;P_{11}$

Gray: single-atom transfer. Blue: selected two-atom split probability before loss. Amber dashed: distinguishable pair.

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There is no force between the atoms in this model. A spin-unresolved detector cannot recover interference erased by orthogonal internal labels.

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

Two noninteracting spatial modes begin with one atom each. A constant tunnel coupling and trap bias act for the selected time. The marking angle sets the squared internal-state overlap. Fermionic double occupation at partial overlap uses distinct internal components, not two identical fermions in one complete quantum state. Both detection arms have equal, independent single-atom efficiency. Ideal number resolution omits light-assisted collisions and parity projection. The tweezer separation and atom sizes are schematic; coupling is controlled directly, not inferred from the drawing.

Kaufman et al. · Science 345, 306–309 · Atomic HOM · Sections II.3, III.4, III.6

Start with the essentials

Focus question: Will two atoms leave together—or apart?
One-sentence intuition: The single-particle tunnel probability can be identical while the two-particle outcomes are opposite. An internal label removes the exchange cross term.

Core mathematical model

A coherent tunnel pulse

H=\begin{pmatrix}\Delta/2&-J\\-J&-\Delta/2\end{pmatrix},\quad R=\frac{J^2}{J^2+\Delta^2/4}\sin^2\!\left(\frac{t\sqrt{J^2+\Delta^2/4}}{\hbar}\right)

One atom starts in each mode. Bias changes the maximum transfer; pulse duration changes the rotation.

Exchange changes joint outcomes

P_{11}=1-2R(1-R)(1+s\mathcal I),\quad \mathcal I=\cos^2\alpha

The sign is positive for bosons, negative for fermions and zero for the distinguishable control. Internal overlap multiplies only the exchange term.

Keep missed detections

P_{\rm det}(1,1)=\eta^2P_{11},\quad P_{\rm det}(0,0)=(1-\eta)^2

Both outputs use the same independent efficiency. The primary plot counts all shots; the conditional two-detected fraction is reported separately.

Common difficulties

Interpret the limit

Typical misconceptionFermions repel one another in this simulation.

Better mental modelInteractions are zero. Antisymmetry changes the joint amplitudes without a repulsive force.

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

01
Predict
At the default pulse, predict whether both atoms leave the same trap.
What to observe: A prediction is a choice, not a fitted result.
02
Record a contrast
Scan bosons and save. Repeat for fermions, then mark one internal state.
What to observe: The loss of exchange interference brings both statistics toward the distinguishable control.
03
Test the boundary
Reduce efficiency. Compare the all-shot split fraction and the conditional fraction among two detected atoms.
What to observe: Save and export zero results as well as successful detections. The source documentation separates sampling uncertainty and approximation error.