Contents
Introduction The Setup The 1959 Prediction The Tonomura Experiment Physical Meaning Berry Phase Connection Misconceptions FAQ Sources Introduction
The Aharonov-Bohm effect, predicted in 1959 by Yakir Aharonov and David Bohm, showed that charged quantum particles can be affected by electromagnetic potentials in regions where the electric and magnetic fields are zero. This was a fundamental shift in our understanding: in classical electromagnetism, the potentials A and φ are mathematical conveniences, while the fields E and B are physical. In quantum mechanics, the potentials are themselves physical — though only their gauge-invariant integrals affect observables.
The effect was experimentally confirmed in the 1960s and definitively in Tonomura's 1986 experiment. It reveals topological phases in quantum mechanics and has become a foundational tool in mesoscopic physics, geometric phases, and topological matter.
The Setup
Imagine an electron interferometer (like a double-slit experiment) with a long, thin solenoid placed between the two paths. The solenoid contains a magnetic flux Φ. Outside the solenoid, the magnetic field B is essentially zero. The electrons travel through field-free regions.
Classically: no force acts on the electrons. The interference pattern should be unchanged whether or not the solenoid carries current.
Quantum mechanically: the wave function on each path picks up an additional phase from the vector potential A surrounding the solenoid. The two paths have different line integrals of A, so the interference pattern shifts by Δφ = eΦ/ℏ.
Key Feature
The phase shift depends only on the enclosed magnetic flux Φ, not on the details of the paths. The electrons never enter a region with nonzero field — yet they "know" about the field through the vector potential they encounter [1 ].
The 1959 Prediction
Aharonov and Bohm published "Significance of Electromagnetic Potentials in the Quantum Theory" in Physical Review [1 ]. They considered both the magnetic and electric versions of the effect, demonstrating that classically-irrelevant potentials produce real quantum-mechanical effects.
The Argument
For a charged particle in a vector potential A, the canonical momentum is p̂ + eA. The wave function acquires a phase factor exp(ie/ℏ ∫A·dl) along its trajectory. For two paths enclosing a region with flux Φ, the relative phase is exp(ieΦ/ℏ). The interference pattern shifts accordingly.
Initial Reception
The 1959 paper was controversial. Some physicists argued that classical electromagnetism predicts no effect, so neither should quantum mechanics. The debate was settled by experiments.
Earlier Suggestions
Werner Ehrenberg and Raymond Siday had derived a similar result in 1949 [2 ], though their paper went largely unnoticed. The effect is sometimes called the "Ehrenberg-Siday-Aharonov-Bohm" effect.
The Tonomura Experiment
Akira Tonomura's group at Hitachi confirmed the effect in 1986 with an elegant design [3 ]. They used a superconducting toroid as the magnetic flux source, with electrons passing through holes in the toroid's center.
Why Superconductor
The Meissner effect in superconductors expels magnetic fields, ensuring no field leakage to the electron paths. A normal solenoid would have small fringing fields that critics argued could cause the observed effect through ordinary forces.
The Result
Tonomura observed phase shifts of exactly eΦ/ℏ where Φ is the enclosed flux, varying continuously with the current in the toroid. The result matched Aharonov-Bohm predictions to high precision, definitively establishing the effect.
Implications
The vector potential is physically real in quantum mechanics. The effect could not be explained by any local force model — the electrons never enter the magnetic field region. Topology matters: paths enclosing flux are physically distinct from paths that don't, even if they look geometrically similar.
Physical Meaning
Potentials vs Fields
Classically, only E and B are observable; potentials A and φ are mathematical conveniences with gauge freedom. Quantum mechanically, the gauge-invariant integrals of potentials produce observable effects. The fields don't fully capture the quantum-mechanical physics; you need the potentials [4 ].
Gauge Invariance Preserved
The Aharonov-Bohm phase eΦ/ℏ is gauge-invariant — Φ is the integral of B over the enclosed area, equivalent to ∮A·dl. Different gauges give different A values along paths, but the loop integral is the same. The observable effect respects gauge invariance.
Topological Nature
The phase depends on the topology of the path-flux configuration, not on local geometry. Paths that enclose the flux differently produce different phases; deforming a path without crossing the flux source doesn't change the phase. This is the prototype of topological phases in quantum mechanics.
Berry Phase Connection
Michael Berry (1984) generalized the Aharonov-Bohm phase to arbitrary slow (adiabatic) parameter variation [5 ]. A quantum system carried slowly around a closed loop in parameter space acquires a "Berry phase" that depends on the loop's geometry, not just the dynamics.
Geometric Phases Everywhere
Berry's phase appears in many systems:
Polarized light traveling along curved paths.
Molecular wave functions during slow internal motions.
Spin precession in magnetic fields.
Electrons in crystals (Brillouin-zone geometry).
Topological Materials
Modern topological insulators and topological superconductors exploit the Aharonov-Bohm/Berry phase structure. Their characteristic edge states are protected by topology — by the same kind of phase considerations that govern the Aharonov-Bohm effect.
Common Misconceptions
"The Aharonov-Bohm effect violates locality"
It depends on the definition. The vector potential A is local in the sense that the electron at any point feels only A at that point. The effect is non-local in the sense that the phase depends on flux enclosed globally. It's not a violation of relativity; it's a feature of how potentials work in quantum mechanics.
"The electrons feel a force from the magnetic field"
No. The electrons remain in field-free regions throughout their journey. The effect is purely from the vector potential.
"The effect is just classical induction"
No. The electron paths don't enclose a changing flux; the flux is static. Faraday induction doesn't apply.
"The Aharonov-Bohm phase depends on the path"
It depends on which topological class the path falls into (which flux it encloses). Within a class, smooth deformations don't change the phase. Cross the flux line, and the phase shifts by a full eΦ/ℏ.
"Only electrons can show this effect"
Any charged particle. Demonstrations have been done with neutrons, atoms, and even neutral systems with magnetic moments (a variant of the effect for neutral particles in electromagnetic fields).
FAQ
What's the smallest measurable phase shift?
Modern electron interferometers can detect phase shifts well below 0.01 radians. For an electron, this corresponds to flux of ~10⁻¹⁶ Wb — fewer than 10⁻⁵ flux quanta.
Are there electric and gravitational analogs?
Yes. The electric Aharonov-Bohm effect uses a time-varying scalar potential in a field-free region. Gravitational versions have been proposed but are harder to demonstrate experimentally.
Is the Aharonov-Bohm phase a Berry phase?
Closely related. The AB phase is a specific example of a geometric phase. Berry's generalization shows similar phases arise in many adiabatic processes.
What if the flux is quantized?
For half-integer or integer multiples of the flux quantum Φ₀ = h/e, the AB phase is multiples of π. Half-integer flux quanta produce destructive interference. Superconducting flux quanta are Φ₀/2 = h/(2e) because of Cooper pair charge.
Could the AB effect be used for technology?
Yes — flux qubits in superconducting quantum computers exploit AB-like phase structure. The AB effect is part of the foundation of mesoscopic quantum devices.
Sources
Aharonov, Y., Bohm, D. (1959). "Significance of electromagnetic potentials in the quantum theory." Physical Review , 115(3), 485–491.
Ehrenberg, W., Siday, R. E. (1949). "The refractive index in electron optics and the principles of dynamics." Proc. Phys. Soc. B , 62, 8–21.
Tonomura, A., et al. (1986). "Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave." Physical Review Letters , 56(8), 792–795.
Healey, R. (2007). Gauging What's Real . Oxford University Press.
Berry, M. V. (1984). "Quantal phase factors accompanying adiabatic changes." Proceedings of the Royal Society A , 392(1802), 45–57.
Peshkin, M., Tonomura, A. (1989). The Aharonov-Bohm Effect . Springer.