--- title: "Aharonov–Bohm Quantum Cage Interactive Lab | Sandbox Physics" description: "A cage with no walls. Release a localized particle into a diamond chain. Half a flux quantum closes every escape route by interference. Unbalance the two arms and watch the cage leak." canonical_url: "https://sandboxphysics.com/labs/aharonov-bohm-quantum-cage/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/aharonov-bohm-quantum-cage/" last_modified: "2026-10-18" --- QUANTUM CONFINEMENT / INTERFERENCE # Aharonov–Bohm Quantum Cage Release a localized particle into a diamond chain. Half a flux quantum closes every escape route by interference. Unbalance the two arms and watch the cage leak. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Finite, coherent single-particle tight-binding model with prescribed Peierls phases. The scene is a lattice probability diagram, not a waveguide apparatus. No drive micromotion, disorder, loss or measurement during evolution. LATTICE · PROBABILITY INSIDE THE ORIGINAL CAGE $0$ 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. $0$ 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 distance from launch $\text{—}$ Model boundary probability $\text{—}$ 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, ONE OUTCOME ### Where the particle was found. Gold: recorded frequency. Cyan: model probability. Linear vertical scale; all shots included. Cell coordinate 03 / READ THE STRUCTURE ### A breathing cage leaves a narrow trace. 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. 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. $$ \phi=2\pi\Phi/\Phi_0,\qquad J_{\mathrm{upper,lower}}=J(1\pm\delta) $$ 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](https://arxiv.org/html/1805.03564) 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 ## Aharonov–Bohm Quantum Cage **Background** Release a localized particle into a diamond chain. Half a flux quantum closes every escape route by interference. Unbalance the two arms and watch the cage leak. **Why it matters** A localized picture can have very different causes. Change the cause, then measure the consequence. Start with the essentials - **Focus question**: A cage with no walls. - **One-sentence intuition**: Two paths can cancel escape while motion continues inside a compact cage. ### 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 $$ P_{\mathrm{cage}}=\sum_{j\in C}|\psi_j|^2 $$ 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** Finite, coherent single-particle tight-binding model with prescribed Peierls phases. The scene is a lattice probability diagram, not a waveguide apparatus. No drive micromotion, disorder, loss or measurement during evolution. ### Run the experiment 1. 01 #### Predict Compare the first two presets at the same time. **What to observe:** Half flux holds all probability in five sites. 2. 02 #### Break the condition Choose the third preset. **What to observe:** Unbalanced paths no longer cancel escape. 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 Finite, coherent single-particle tight-binding model with prescribed Peierls phases. The scene is a lattice probability diagram, not a waveguide apparatus. No drive micromotion, disorder, loss or measurement during evolution. 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)