--- title: "Repulsively Bound Atom Pair Interactive Lab | Sandbox Physics" description: "Repulsion can keep a pair together. Prepare two bosons on one lattice site. Increase their repulsion and a traveling diagonal ridge survives in the joint position map. Start the atoms apart to expose the role of preparation." canonical_url: "https://sandboxphysics.com/labs/repulsively-bound-atom-pair/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/repulsively-bound-atom-pair/" last_modified: "2026-10-18" --- QUANTUM CONFINEMENT / INTERACTION # Repulsively Bound Atom Pair Prepare two bosons on one lattice site. Increase their repulsion and a traveling diagonal ridge survives in the joint position map. Start the atoms apart to expose the role of preparation. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Two identical bosons on 31 open sites in the single-band Bose–Hubbard model. A local double occupation is a superposition of bound and scattering states, not a pure bound eigenstate. Ideal number-resolved snapshots; no atom loss, higher bands, thermal bath or hardware calibration. TWO POSITIONS · ONE STATE ON THE SAME SITE $0$ Model probability Same site does not mean same place forever. Each pillar is a pair of possible readout positions. Height: square root of joint probability; gold diagonal: same site. This is configuration space, not a physical surface. $0$ Drag the gold launch marker, or use the arrow keys. REPEAT. FREEZE. READ THE POSITIONS. ## Does the picture survive a measurement? 0 Every shot evolves a fresh preparation to the selected time, then measures position once. Ideal unit efficiency and number resolution. Records do not track a trajectory or reconstruct phase. Recorded fraction $\text{—}$ 95% Wilson interval $\text{—}$ Model center-of-mass distance $\text{—}$ Model boundary probability $\text{—}$ 01 / THE WHOLE EVOLUTION ### Together, not motionless. Cyan: model retained fraction. Coral: matched reference with flux or interaction removed. Gold marker: current time. Drag to inspect. $Jt/\hbar$ 02 / ONE SHOT, ONE OUTCOME ### How far apart were they? Gold: recorded frequency. Cyan: model probability. Linear vertical scale; all shots included. Site separation 03 / READ THE STRUCTURE ### A diagonal is a pair moving together. Model joint probability. Coordinates are symmetrized, not labels for distinguishable atoms. Opposite cells share one unordered outcome equally. Color uses a fixed square-root gain. 04 / RAW SNAPSHOTS ### Every row starts fresh. Latest 20 independent shots. A double mark is two atoms on one site. Full records are retained in the CSV. Read the raw ledger | Trial | Position | Retained | | --- | --- | --- | Model, units & sources $$ \tau=Jt/\hbar,\qquad x_j=ja $$ Static, closed, finite tight-binding systems with open ends. Time is in inverse hopping units; distance is in cell or site spacings. Both the chosen state and matched reference are evolved numerically. Edge probability warns when finite ends matter; it is not an error estimate. $$ \frac{H}{J}=-\sum_j(b_j^\dagger b_{j+1}+\mathrm{h.c.})+\frac{U}{2J}\sum_j n_j(n_j-1)+\frac{Fa}{J}\sum_j jn_j $$ Exactly two identical bosons on 31 sites: 496 number states, with bosonic hopping factors. Strong repulsion suppresses separation for a same-site preparation; it does not bind arbitrary initially separated atoms. The same-site fraction is not the spectral bound-state weight. Higher bands, loss, thermal noise and experimental readout errors are omitted. $$ E_{\mathrm{bound}}(K)=\sqrt{U^2+16J^2\cos^2(Ka/2)}\quad(F=0,\ U>0) $$ This dispersion is an infinite-chain reference, not the finite-chain solver. A nonzero slope can also limit motion; compare against the matched zero-interaction curve. [Winkler et al. — repulsively bound pairs, Eq. 1 & Fig. 1](https://arxiv.org/html/cond-mat/0605196) [Valiente & Petrosyan — two-particle states, Eqs. 6–8 & 18](https://arxiv.org/html/0805.1812) Numerical evolution uses norm-limited exponential-action steps without renormalization. The diagnostic reports norm drift and energy drift. Statistical intervals describe finite ideal samples only. The 3D view and smooth maps show the model; records are independent Born-rule draws. $0$ $0$ Physics tutorial ## Repulsively Bound Atom Pair **Background** Prepare two bosons on one lattice site. Increase their repulsion and a traveling diagonal ridge survives in the joint position map. Start the atoms apart to expose the role of preparation. **Why it matters** A localized picture can have very different causes. Change the cause, then measure the consequence. Start with the essentials - **Focus question**: Repulsion can keep a pair together. - **One-sentence intuition**: A bounded kinetic-energy band can protect a repulsively interacting pair from separating. ### Core mathematical model #### Unitary evolution $$ |\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle $$ The plotted probability and the recorded outcomes use the same state. #### What is counted $$ P_{\mathrm{same}}=\sum_j|\langle 2_j|\psi\rangle|^2 $$ This is a position-based event, not a measurement of phase or a bound-state projector. ### Common difficulties #### Separate preparations **Typical misconception** Rows show the path of one particle. **Better mental model** Each shot starts again. The simulation is unmeasured until the selected final time. #### Model boundary **Typical misconception** A localized picture proves a pure bound state. **Better mental model** Two identical bosons on 31 open sites in the single-band Bose–Hubbard model. A local double occupation is a superposition of bound and scattering states, not a pure bound eigenstate. Ideal number-resolved snapshots; no atom loss, higher bands, thermal bath or hardware calibration. ### Run the experiment 1. 01 #### Predict Compare the first two presets at the same time. **What to observe:** Strong repulsion keeps much more probability on the diagonal. 2. 02 #### Break the condition Choose the third preset. **What to observe:** Initially separated particles are not the same as a prepared pair. 3. 03 #### Measure Collect positions, pin the result, then compare another preparation. **What to observe:** The estimate fluctuates, and its interval narrows with more shots. Physical conclusion Two identical bosons on 31 open sites in the single-band Bose–Hubbard model. A local double occupation is a superposition of bound and scattering states, not a pure bound eigenstate. Ideal number-resolved snapshots; no atom loss, higher bands, thermal bath or hardware calibration. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)