Quantum confinement / Interaction

Repulsively Bound Atom Pair

How can two atoms that repel each other move together as a pair? Because a lattice caps how much kinetic energy the atoms can carry. Two atoms on one site hold extra repulsion energy, and when it exceeds that cap they cannot turn it into motion by flying apart. They hop through the lattice together, usually found on the same site. Start them on neighboring sites and most of the probability spreads out; the pair forms mainly when the atoms begin together.

Put two bosons on the same lattice site and increase their repulsion. In the map of both atoms' positions, a ridge along the diagonal survives and travels outward. Start the atoms on different sites to see what the preparation contributes.

Two atoms' positions in one joint state

Both particles still 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 the run and read out positions

Do measurements show the same picture?

Each shot starts from a fresh preparation, evolves to the selected time and measures position once. Detection is ideal: every atom is seen and atom numbers are resolved exactly. The records do not follow a trajectory or recover the phase.

Recorded fraction

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95% Wilson interval

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Model RMS center-of-mass distance from launch

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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 gives one outcome

How far apart were they?

Gold: recorded frequency. Cyan: model probability. Linear vertical scale; all shots included.

Site separation

03 / See the structure

A diagonal ridge means the two atoms move as a pair.

Model joint probability. The atoms are identical, so the coordinates are symmetrized and do not label atom one and atom two. Each unordered outcome is split equally between its two mirror cells. Color uses a fixed square-root gain.

04 / Raw snapshots

Every row starts fresh.

The latest 20 independent shots. A double mark means two atoms on one site. The CSV keeps every record.

Show the raw records

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: How can two atoms that repel each other move together as a pair?
One-sentence intuition: On a lattice the kinetic-energy band has an upper limit, and that limit can keep a repelling pair from separating.

Core mathematical model

Unitary evolution

|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle

The plotted probability and the recorded outcomes both come from this same state.

What is counted

P_{\mathrm{same}}=\sum_j|\langle 2_j|\psi\rangle|^2

This event is defined by positions only. It does not measure phase, and it is not a projection onto the bound state.

Common difficulties

Each shot is a fresh start

Typical misconceptionRows show the path of one particle.

Better mental modelEach shot starts from a fresh preparation, and nothing is measured until the chosen final time.

Limits of the model

Typical misconceptionA localized picture proves a pure bound state.

Better mental modelTwo identical bosons move on an open chain of 31 sites in the single-band Bose–Hubbard model. Two atoms on one site start in a superposition of bound and scattering states, not a pure bound eigenstate. The snapshots resolve atom numbers ideally. Atom loss, higher bands, a thermal bath and hardware calibration are not included.

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: Atoms that start apart do not behave like a pair prepared on one site.
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Measure
Collect positions, pin the result, then compare another preparation.
What to observe: The estimate fluctuates, and its interval narrows with more shots.