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.

Signal phase · model $0$

Visibility · model $1$

01 / Each particle takes two branches and is detected once

Turn the flux to swap the exits

Choose “Swap the exits”, then send 200 electrons. Neither arm enters the magnetic field.

Ideal guides · blue / amber = coherent branches · flux tube extends beyond both ends

Enclosed flux

\Phi/(h/e)=0

No counts yet. A scan records 100 trials at each of 21 settings.

Try a comparison

Path phase offset

\phi_0/\pi

0

Coherence, noise & detection

Coherence before recombination

C_0

1

Independent phase noise

\sigma_\phi/\mathrm{rad}

0.15

Detection probability

\eta

0.95

Sampling seed

s

2026

Turning the scan knob keeps the data. Changing the preparation starts a new sample, so save first if you want a comparison. Presets do not take data.

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

Exit B

Not detected

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

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.

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 misconceptionThe colored lines show which way each detected electron went.

Better mental modelThey 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 misconceptionA small fit error proves the real instrument is accurate.

Better mental modelThe 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

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.
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.
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.