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Physical Review Letters

Observation of Aharonov-Bohm Effect by Electron Holography

Akira Tonomura, Tsuyoshi Matsuda, Ryō Suzuki, Akira Fukuhara, Nobuyuki Osakabe, Hiroshi Umezaki, Junji Endo, Kohsei Shinagawa, Yutaka Sugita, Hideo Fujiwara

Published May 24, 1982• 266 citations
Publisher

Abstract

In this experiment, an electron- and optical-holographic technique is employed with small toroidal ferromagnets each forming a magnetic-flux closure. The holographic interferometry proves that a phase difference between two electron beams having passed through the field-free regions agrees well with the fundamental relation known as the Aharonov-Bohm effect. It is also confirmed from the same hologram that flux leakage from the toroids does not affect the conclusion.

DOI
10.1103/physrevlett.48.1443

Related papers

ELECTRON HOLOGRAPHY AND AHARONOV-BOHM EFFECT

Akira Tonomura

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.

International Journal of Modern Physics B•Apr 1, 1989•3 citations

Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor

Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki +4 more

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.

Physical Review A•Aug 1, 1986•205 citations

Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave

Akira Tonomura, Nobuyuki Osakabe, Tsuyoshi Matsuda +4 more

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.

Physical Review Letters•Feb 24, 1986•725 citations

Verification of the Aharonov-Bohm effect in superconductors by use of a toroidal flux geometry

I. R. Walker

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.

Physical review. B, Condensed matter•Apr 1, 1986•2 citations

The Aharonov–Bohm Effect: Gauge Potentials, Quantum Phase, and Experimental Evidence

Giancarmine Sparso

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.

•Jul 1, 2026•0 citations

The electric Aharonov-Bohm effect

Ricardo Weder

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.

arXiv:1006.1385•Journal of Mathematical Physics•May 1, 2011•16 citations
In this experiment, an electron- and optical-holographic technique is employed with small toroidal ferromagnets each forming a magnetic-flux closure. The holographic interferometry proves that a phase difference between two electron beams having passed through the field-free regions agrees well with the fundamental relation known as the Aharonov-Bohm effect. It is also confirmed from the same hologram that flux leakage from the toroids does not affect the conclusion.
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.
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.
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.
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.
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.
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.