--- title: "Aharonov–Bohm Flux Interferometer Interactive Lab | Sandbox Physics" description: "Can a magnetic field the electrons never pass through change which exit they take? It can." canonical_url: "https://sandboxphysics.com/labs/aharonov-bohm-flux/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/aharonov-bohm-flux/" last_modified: "2026-10-27" --- Q006 · Split / recombine / count # Aharonov–Bohm Flux Interferometer **Can a magnetic field the electrons never pass through change which exit they take?** It can. Both electron paths stay outside the magnetic field, yet the flux they enclose shifts their relative phase. As the flux changes, the counts swap back and forth between the two exits. Change the magnetic flux enclosed between two electron paths and send electrons through one at a time. The counts swap between the two exits, even though neither path enters the magnetic field. > Interactive simulation: open this page in a web browser to change the parameters and run the model. An ideal, balanced two-path electron interferometer surrounds an infinitely long flux tube that the electrons cannot enter. The simulation counts electrons at two exits only. It does not compute a spatial hologram, stray fields or the flux control of a real superconductor. 02 / Count at this setting ## Which exit lights up? Model reference Exit A 0 Exit B 0 Not detected 0 Latest 60 trials at this setting. Blue: A · amber: B · grey: missed. Each trial starts with a fresh particle. Fraction in A · detected trials $\text{—}$ Pointwise 95% Wilson interval. Misses are retained in the raw records; the fraction conditions on detection. $\text{—}$ 03 / Read the fringe ## One flux period brings the counts back. Fraction in A versus scan setting $\Phi/(h/e)$ Blue points: recorded counts. Thin lines: Wilson intervals. Amber: model. Dashed blue: a fixed-period harmonic fit to the records. Click the chart to choose a setting. Estimated from the records $\text{—}$ At least eight settings with ten detections each are needed. An unresolved fringe gives no phase estimate. The errors come from a conservative variance bound and a local approximation; they do not describe instrument accuracy. 04 / Connect the paths to the phase ## The paths pick up a phase around a region they never enter. $\text{—}$ The closed loop follows the blue arm and returns along amber. Positive flux points along its right-hand normal. Neither arm crosses the field-filled tube. The two lines represent coherent amplitudes, not measured electron tracks. Moving the flux shifts the relative phase without a magnetic force on the arms. Guides and beam splitters are idealized; leakage fields and spatial diffraction are omitted. 05 / Save the evidence ## Save a sample, change one setting, then compare. Up to eight frozen samples. Saving freezes the run; the next acquisition starts a new run. Export includes active and saved trials with all preparation settings. Model, conventions & sources $$ P_A=\tfrac12[1+C_0e^{-\sigma_\phi^2/2}\cos\Delta\phi],\quad P_B=1-P_A $$ $$ \Delta\phi=\phi_0-2\pi\Phi/(h/e) $$ An ideal, balanced electron interferometer with two exits surrounds a flux tube that the electrons cannot enter and that extends infinitely. The finite tube in the drawing is schematic. The flux varies continuously; superconducting flux quantization is not simulated. No spatial electron hologram is reconstructed. [Tonomura et al. · PRL 1986](https://stuff.mit.edu/afs/athena/course/8/8.06/spring06/handouts/Tonomura_1986.pdf) Noise is averaged analytically before independent binary sampling. Detection loss is equal for both states and is recorded without a hidden outcome. The harmonic regression fixes the known period; it does not measure that period. It is unconstrained and descriptive, so finite samples can give fitted contrast above one. Physics tutorial ## Aharonov–Bohm Flux Interferometer **Background** A phase you can control shows up as a difference in counts between the two outputs. **Why it matters** Where a single electron exits is random. A scan over many electrons traces out a fringe that you can reproduce. Start with the essentials - **Focus question**: Can a magnetic field the electrons never pass through change which exit they take? - **One-sentence intuition**: Compare settings, keep the raw counts and fit only the counts you recorded. ### Core mathematical model #### Phase around the closed loop $$ \Delta\phi_{\rm AB}=\frac{q\Phi}{\hbar}=-2\pi\frac{\Phi}{h/e} $$ The electron charge is negative. The loop runs out along the blue arm and back along the amber arm, and positive flux points along the loop's right-hand normal. #### The phase does not depend on the gauge $$ \oint(\mathbf A+\nabla\chi)\cdot d\mathbf l=\oint\mathbf A\cdot d\mathbf l $$ A single-valued gauge change adds endpoint terms to each arm, and around the loop they cancel. The closed-loop phase cannot change. #### Probabilities at the two exits $$ P_A=\tfrac12[1+C\cos(\phi_0+\Delta\phi_{\rm AB})],\quad P_B=1-P_A $$ The two paths recombine coherently and in balance, an idealization with exactly two exits. No local magnetic field bends either path. #### Flux period for an electron $$ \Phi_e=h/e $$ This is the period of the interference. A real superconducting loop quantizes flux under a different rule; the flux knob turns continuously here for teaching. #### Detector loss and phase noise $$ C=C_0e^{-\sigma_\phi^2/2},\quad P_{\varnothing}=1-\eta $$ Independent Gaussian phase noise is averaged analytically. Loss that is equal for both exits reduces the number of detected electrons, not the visibility among those detected. ### Common difficulties #### A path is not a trajectory **Typical misconception** The colored lines show which way each detected electron went. **Better mental model** They show the coherent branches of the wave or the centers of the wave packets. Nothing records which path an electron took. #### A small fit error is not real accuracy **Typical misconception** A small fit error proves the real instrument is accurate. **Better mental model** The quoted uncertainty is only the statistical error within this model. Apparatus effects the model leaves out, and the ambiguity, are still there. ### Run the experiment 1. 01 #### Predict Try the first two presets and predict which exit gains counts. **What to observe:** A preset only sets the conditions; it does not record any data. 2. 02 #### Measure Detect once, then collect 200 trials. Scan all 21 positions. **What to observe:** Missed electrons stay in the record. The exit frequency and its pointwise Wilson interval use only the detected electrons. 3. 03 #### Compare Save the scan, change coherence or path phase, and scan again. Export both runs. **What to observe:** When the contrast is low, no phase estimate is given. The fit holds the period fixed, so it estimates a phase offset, not the flux period. Physical conclusion Keep three things apart: the model's reference curve, what you measured and the phase ambiguity that remains. Quantum & fields · Starter path ## Start from a single measurement Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many 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)