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.

TWO POSITIONS · ONE STATE

ON THE SAME SITE

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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.

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Drag the gold launch marker, or use the arrow keys.

REPEAT. FREEZE. READ THE POSITIONS.

Does the picture survive a measurement?

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

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95% Wilson interval

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Model center-of-mass distance

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Model boundary probability

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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
Valiente & Petrosyan — two-particle states, Eqs. 6–8 & 18

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.

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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 misconceptionRows show the path of one particle.

Better mental modelEach shot starts again. The simulation is unmeasured until the selected final time.

Model boundary

Typical misconceptionA localized picture proves a pure bound state.

Better mental modelTwo 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

01
Predict
Compare the first two presets at the same time.
What to observe: Strong repulsion keeps much more probability on the diagonal.
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Break the condition
Choose the third preset.
What to observe: Initially separated particles are not the same as a prepared pair.
03
Measure
Collect positions, pin the result, then compare another preparation.
What to observe: The estimate fluctuates, and its interval narrows with more shots.