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Proposal for a Time‑Resolved Superconducting‑Toroid Aharonov–Bohm Experiment: Incremental‑Cost Discriminator Measurement of Phase Propagation and Relaxation

James E Beecham

Published Feb 14, 2026• 0 citations

Abstract

The Aharonov–Bohm effect has been conclusively established through electron interferenceexperiments using toroidal magnetic geometries fully shielded by superconductors,demonstrating measurable phase shifts in electrons traveling exclusively through regionswhere magnetic field B = 0. However, these experiments have primarily investigated staticor quasi‑static flux conditions. This paper proposes a time‑resolved extension of thesuperconducting‑toroid Aharonov–Bohm experiment designed to measure the temporalresponse of electron interference phase following controlled flux modulation within a fullyshielded superconducting toroid. The proposed experiment exploits moderninterferometric instrumentation and superconducting flux control to test whether the phaseresponse exhibits purely circuit‑limited behavior or additional propagation or relaxationdynamics. Because existing interferometry and superconductivity laboratories alreadypossess the required infrastructure, the incremental cost of implementing the experiment isestimated at $10,000–$50,000 USD. The proposed experiment therefore represents auniquely accessible and decisive extension of a foundational quantum interferencemeasurement.

DOI: 10.5281/zenodo.18637847

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The Aharonov–Bohm effect has been conclusively established through electron interferenceexperiments using toroidal magnetic geometries fully shielded by superconductors,demonstrating measurable phase shifts in electrons traveling exclusively through regionswhere magnetic field B = 0. However, these experiments have primarily investigated staticor quasi‑static flux conditions. This paper proposes a time‑resolved extension of thesuperconducting‑toroid Aharonov–Bohm experiment designed to measure the temporalresponse of electron interference phase following controlled flux modulation within a fullyshielded superconducting toroid. The proposed experiment exploits moderninterferometric instrumentation and superconducting flux control to test whether the phaseresponse exhibits purely circuit‑limited behavior or additional propagation or relaxationdynamics. Because existing interferometry and superconductivity laboratories alreadypossess the required infrastructure, the incremental cost of implementing the experiment isestimated at

10,000–

50,000 USD. The proposed experiment therefore represents auniquely accessible and decisive extension of a foundational quantum interferencemeasurement.

The Aharonov-Bohm (AB) effect was tested under conditions where an electron wave and a magnetic field did not overlap: the Meissner effect of the superconductor shielding a toroidal ferromagnet eliminated the leakage field. Using the newly-developed technique of electron holography, conclusive evidence for the AB effect was obtained by detecting a relative phase shift of π between the two electron waves passing through the hole and outside the toroid. The detected phase shift value of exactly π comes from the quantization of the magnetic flux within the superconductor. This quantization assures the complete shielding of the magnetic field.

This technical note gives a self-contained exposition of the Aharonov–Bohm effect, aimed at advanced undergraduate students. Starting from the role of electromagnetic potentials in classical electrodynamics, it introduces gauge freedom, the quantum Hamiltonian with minimal coupling, and the gauge covariance of the Schrödinger equation. It then derives the Aharonov–Bohm phase for both the time-dependent scalar configuration and the magnetic configuration, showing how a charged particle acquires an observable phase shift while travelling only through regions where the electromagnetic field vanishes. The final part reviews the experimental evidence, from the early electron-interference experiments of Chambers and of Möllenstedt and Bayh to the 1986 toroidal-magnet experiment of Tonomura and collaborators. In that experiment, superconducting shielding confined the magnetic flux completely and closed the leakage-field loophole, while flux quantization produced the characteristic half-fringe shift. The note is expository and pedagogical: it does not present original research results and has not been peer reviewed. Originally developed from study notes prepared for the PHYS02 scientific talk at Lady Margaret Hall, Oxford, in July 2026; an earlier version of the notes has been publicly available on GitHub since August 2026. Corrections and comments are more than welcome at giancarminesparso@gmail.com.

We propose a nonlocal flux-transfer experiment in which a quantized magnetic-field excitation confined within one toroidal superconducting structure is coherently transferred to a spatially separated toroid without magnetic-field occupation of the intervening region. The transfer arises from quantized Aharonov-Bohm-type vector potential coupling mediated by a superconducting loop threading the toroids, which, however, remains in the ground state, acting only through a global fluxoid constraint. A direct experimental signature would be the observation of correlated, time-resolved flux exchange between remote toroids in a SQUID readout. We analyze an apparent signaling paradox related to this interaction as a probe of the broader question of whether spatiotemporal quantum coherence is fundamentally bounded. The proposed setup can provide an experimental testbed for addressing foundational questions such as the existence of an objective collapse of a wavefunction or the fundamental limits of macroscopic quantum coherence which is relevant to large scale quantum computers.

Evidence for the Aharonov-Bohm effect was obtained with magnetic fields shielded from the electron wave. A toroidal ferromagnet was covered with a superconductor layer to confine the field, and further with a copper layer for complete shielding from the electron wave. The expected relative phase shift was detected with electron holography between two electron beams, one passing through the hole of the toroid, and the other passing outside. The experiment gave direct evidence for flux quantization also.

The electron holography technique was employed to make a crucial test of the existence of the Aharonov-Bohm (AB) effect. The relative phase shift was measured between two electron waves passing through spaces inside and outside a tiny toroidal ferromagnet, covered completely with a superconductor layer and a Cu layer. Below the transition temperature the relative phase shift was measured to be 0 or \ensuremath{\pi} due to the magnetic-flux quantization in units of h/2e. The results directly demonstrated the existence of the AB effect even when the magnetic field was confined by the surrounding superconductor due to the Meissner effect and an electron beam was prevented from penetrating the magnet.

Predictions made by Aharonov and Bohm concerning the effects of electromagnetic potentials on electrons in field-free regions of space, have been demonstrated in superconductors with the use of the flux produced by a very fine toroidal solenoid. The use of a toroidal geometry sidesteps the objections raised by Roy concerning the use of sources of flux with linear geometries. The use of superconductors excludes the possibility that stray electrons can pass through the flux-filled region of space to produce the observed phenomena.