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Conference on Precision Electromagnetic Measurements

Experimental confirmation of the aharonov-bohm effect

Tonomura

Published Jan 1, 1988• 1 citation
Publisher

Abstract

Experimental evidence was obtained for the existence of the AB effect. This effect has been a subject of long controversy regarding the physical significance of gauge fields, -- in the present case, the vector potentials.
DOI
10.1109/cpem.1988.671180

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Whether the time-dependent Aharonov-Bohm (AB) effect even exists or not has been the subject of long-standing debate. There are two factors complicating the problem. First, in the closed spacetime line integral of the vector potential that is thought to give the AB-phase shift, how to treat the time-varying vector potential is highly nontrivial. Second, the time-varying magnetic flux generates induced electric field even outside the solenoid. In the present paper, motivated by a recent work by Gao, we re-investigate the role of the induced electric field with the utmost care. This analysis reveals a highly nontrivial effect of the induced electric field, which turns out to be useful for verifying the very existence of the time-dependent AB-effect.

quant-pharXiv:2511.07775•Nov 12, 2025•0 citations
It is often said that the Aharonov-Bohm (AB) effect (1959) is an effect that is due to the vector potential. This is considered remarkable since a quantity that is not uniquely determined has a measurable effect. The authors show that, in the symmetric Coulomb gauge, (e/c)A(r) is simply the momentum of the electromagnetic field. The effect may equally well be considered an effect of the electromagnetic momentum. Thus, it is possible to discuss the AB effect strictly in terms of observables. The idea that the effect is due to the vector potential is correct, but sheds little light on the physics of the effect.
The seminal paper of Aharonov and Bohm [Phys. Rev. 115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Phys. Rev. Lett. 48, 1443 (1982); Phys. Rev. Lett. 56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Phys. Rev. Lett. 99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura et al. experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys. 285, 345 (2009)10.1007/s00220-008-0579-1; J. Math. Phys. 50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys. 303, 175 (2011)]10.1007/s00220-010-1166-9 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura et al. and of Caprez et al. The electric Aharonov-Bohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric Aharonov-Bohm effect.
Whether the time-dependent Aharonov-Bohm (AB) effect even exists or not has been the subject of long-standing debate. There are two factors complicating the problem. First, in the closed spacetime line integral of the vector potential that is thought to give the AB-phase shift, how to treat the time-varying vector potential is highly nontrivial. Second, the time-varying magnetic flux generates induced electric field even outside the solenoid. In the present paper, motivated by a recent work by Gao, we re-investigate the role of the induced electric field with the utmost care. This analysis reveals a highly nontrivial effect of the induced electric field, which turns out to be useful for verifying the very existence of the time-dependent AB-effect.