Quantum confinement / Interference

Aharonov–Bohm Quantum Cage

How can a magnetic flux trap a particle in a diamond chain without walls? Through interference. With half a flux quantum threading each diamond, the routes over the upper and lower arms reach the next hub with opposite phases and cancel. The particle keeps sloshing between its starting hub and the four neighboring arm sites but never gets farther. Make the two arms unequal and the cancellation is incomplete, so probability leaks out along the chain.

Release a particle from one site of a diamond-shaped chain. At half a flux quantum, interference closes every escape route. Make the two arms unequal and watch the cage start to leak.

LATTICE · PROBABILITY

Still inside the original five-site cage

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Model probability
Motion can continue inside.

Node height: square root of site probability. Gold outline: the original five-site cage. Links show coupling geometry, not particle tracks.

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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 distance from launch

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Model boundary probability

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01 / The whole evolution

Confined, or merely slow?

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

Where the particle was found.

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

Cell coordinate

03 / See the structure

The probability breathes inside the cage and stays in a narrow strip.

Model site probabilities over time, grouped by cell. Horizontal: position; vertical: time. Color uses a fixed square-root probability 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.

\phi=2\pi\Phi/\Phi_0,\qquad J_{\mathrm{upper,lower}}=J(1\pm\delta)

73 sites: 25 hubs and 24 pairs of arms. Each rightward upper or lower path gains opposite half-flux phases. At half flux and balanced arms, escape amplitudes cancel exactly from an interior hub preparation; the original hub and four neighbors form the measured cage. No disorder, loss or finite-frequency synthetic-field drive is simulated.

P_{A_0}(\tau)=\cos^2(2\tau)\quad(\Phi/\Phi_0=1/2,\ \delta=0)

Mukherjee et al. — rhombic lattice and effective flux, Eqs. 1 & 4

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 a magnetic flux trap a particle in a diamond chain without walls?
One-sentence intuition: Two paths can cancel so the particle cannot escape, yet it keeps moving inside a small cage.

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{cage}}=\sum_{j\in C}|\psi_j|^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 modelThe model is a finite tight-binding chain for one coherent particle, with fixed Peierls phases on the bonds. The scene is a diagram of probability on the lattice, not a waveguide apparatus. Drive micromotion, disorder, loss and measurement during the evolution are not included.

Run the experiment

01
Predict
Compare the first two presets at the same time.
What to observe: At half flux, all the probability stays on five sites.
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Break the condition
Choose the third preset.
What to observe: When the two paths are unequal, they no longer cancel, and the particle escapes.
03
Measure
Collect positions, pin the result, then compare another preparation.
What to observe: The estimate fluctuates, and its interval narrows with more shots.