--- title: "Identical Atom Exchange Interactive Lab | Sandbox Physics" description: "When two atoms in neighboring traps tunnel at the same time, do they end up together or apart?" canonical_url: "https://sandboxphysics.com/labs/identical-atom-exchange/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/identical-atom-exchange/" last_modified: "2026-10-27" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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 $\text{—}$ 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. $\text{—}$ $\text{—}$ 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](https://arxiv.org/abs/1312.7182) · [Atomic HOM · Sections II.3, III.4, III.6](https://arxiv.org/html/1801.04670v2) Physics tutorial ## Identical Atom Exchange **Background** 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. **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 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 misconception** Fermions repel one another in this simulation. **Better mental model** There are no interactions. Antisymmetry changes the two-atom amplitudes without any repulsive force. #### 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 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. 2. 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. 3. 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. Physical conclusion 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. ## 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 **Quantum Gas Occupation** See how particles fill the energy levels of a quantum gas](https://sandboxphysics.com/labs/quantum-gas-occupation/) 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)