Q052 · Two atoms in two traps

Identical Atom Exchange

When two atoms in neighboring traps tunnel at the same time, do they end up together or apart? Together, if they are identical bosons and the tunneling pulse is balanced. The route where the atoms swap places interferes with the route where each stays put, and the outcome with one atom on each side cancels. Mark one atom's internal state and they become distinguishable: the interference vanishes, though each atom tunnels just as before.

Load one atom into each trap, apply the same tunneling pulse and count the atoms at both outputs. Then give one atom a different internal state. The comparison separates exchange interference from ordinary single-particle motion.

Records at this setting $\text{—}$

Detected split fraction $\text{—}$

01 / Two ways to count atoms

The pulse is the same; the atoms pair up differently.

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 comparison

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

The model has two modes, one atom in each input and no interactions. Detection resolves atom number perfectly, with equal, independent losses on both sides. Bosonic and fermionic statistics are two separate model preparations; choosing one does not switch the atomic species. Collisions, heating and parity-only readout are not included.

02 / Each row is a fresh preparation

Count both outputs and 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 when the pair's outcome changes.

$t/\mathrm{ms},\quad \widehat P_{\rm det}(1,1)$ · 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 · kept separate from the records

Single-atom motion is the same; the pair's joint statistics differ.

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

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 / 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

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: When two atoms in neighboring traps tunnel at the same time, do they end up together or apart?
One-sentence intuition: The single-atom tunneling probability can be exactly the same while the two-atom outcomes are opposite. Giving one atom a different internal state 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. The bias changes the largest possible transfer, and the pulse length changes how far the state rotates.

Exchange changes where the two atoms end up

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 distinguishable particles. The overlap of the internal states multiplies only the exchange term.

Count missed detections too

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

Both outputs have the same detection efficiency and miss atoms independently. The main plot counts every shot; the fraction among shots where both atoms are detected is reported separately.

Common difficulties

Reading the result correctly

Typical misconceptionFermions repel one another in this simulation.

Better mental modelThere are no interactions. Antisymmetry changes the two-atom amplitudes without any repulsive force.

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
At the default pulse, predict whether both atoms leave the same trap.
What to observe: Your prediction is a guess to be tested, not a fitted result.
02
Record a comparison
Scan bosons and save. Repeat for fermions, then mark one internal state.
What to observe: Once exchange interference is lost, bosons and fermions both move toward the distinguishable-particle result.
03
Test the limits
Lower the detection efficiency. Compare the split fraction over all shots with the fraction among shots where both atoms are detected.
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.

Identical Atom Exchange

Count both outputs and keep every outcome.

Find when the pair's outcome changes.

Single-atom motion is the same; the pair's joint statistics differ.

Save the preparation with its evidence.