Q006 · Split / recombine / count

Aharonov–Bohm Flux Interferometer

Turn the enclosed flux. Send electrons one at a time and watch the output counts exchange, although both paths stay outside the magnetic field.

Signal phase · model $0$

Visibility · model $1$

01 / TWO BRANCHES. ONE DETECTION.

Turn the flux. 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 CONTRAST

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

The scan knob preserves records. Changing preparation starts a new sample: save first to keep a comparison. Presets do not acquire data.

Ideal balanced two-path electron interferometer around an inaccessible, infinitely extended flux tube. Two-port simulated counting; no spatial hologram, stray fields or real superconducting flux control.

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. Errors use a conservative variance bound and a local approximation, not an instrument accuracy claim.

04 / CONNECT THE PATHS TO THE PHASE

A phase around an inaccessible region.

\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 / KEEP THE EVIDENCE

Save, change one thing, 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)

Ideal balanced two-port electron interferometer around an inaccessible infinitely extended flux tube. Its finite drawing is schematic. The continuously variable flux is not a superconducting flux-quantization simulation. 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 hidden magnetic flux change the exit?
One-sentence intuition: Compare settings, retain raw records, and fit only the recorded counts.

Core mathematical model

Closed-loop phase

\Delta\phi_{\rm AB}=\frac{q\Phi}{\hbar}=-2\pi\frac{\Phi}{h/e}

Electron charge is negative. The loop goes along the blue arm and back along the amber arm; positive flux follows its right-hand normal.

Gauge independence

\oint(\mathbf A+\nabla\chi)\cdot d\mathbf l=\oint\mathbf A\cdot d\mathbf l

A single-valued gauge change adds opposite endpoint terms to the two arms. It cannot alter the closed-loop phase.

Output counts

P_A=\tfrac12[1+C\cos(\phi_0+\Delta\phi_{\rm AB})],\quad P_B=1-P_A

Balanced coherent recombination is a two-port idealization. There is no local magnetic bending on either path.

One electron period

\Phi_e=h/e

This is the interference period. A real superconducting loop has a different flux-quantization constraint; the continuous knob is a teaching control.

Loss and phase stability

C=C_0e^{-\sigma_\phi^2/2},\quad P_{\varnothing}=1-\eta

Independent Gaussian phase noise is averaged analytically. Equal loss reduces accepted counts, not conditional visibility.

Common difficulties

A path is not a trajectory

Typical misconceptionThe colored lines reveal the route taken by each detected particle.

Better mental modelThey show coherent branches or wavepacket centers; no which-path record is generated.

An ideal instrument

Typical misconceptionA narrow fit error establishes real instrument accuracy.

Better mental modelThe stated uncertainty is statistical within this model. Omitted apparatus effects and ambiguity remain.

Run the experiment

01
Predict
Try the first two presets and predict which exit gains counts.
What to observe: The presets set conditions but do not secretly acquire data.
02
Measure
Detect once, then collect 200 trials. Scan all 21 positions.
What to observe: Misses are retained; the frequency and its pointwise Wilson interval use accepted detections.
03
Compare
Save the scan, change coherence or path phase, and scan again. Export both runs.
What to observe: Low contrast suppresses the phase estimate; the fixed-period fit estimates an offset, not the flux period.