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Technical Physics Letters

Influence of phase and frequency modulation of a light beam on electron diffraction as a result of the Aharonov-Bohm effect

Sergei Yu. Davydov, A. N. Ageev

Published Mar 1, 1997• 1 citation
Publisher

Abstract

It is shown that the phase and frequency modulation of the vector potential responsible for the Aharonov-Bohm effect causes changes in the interference pattern that may be used to study this effect under conditions of an ac magnetic flux.
DOI
10.1134/1.1261596

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The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
In the well-known Aharonov-Bohm and Ehrenberg-Siday effects, the diffraction envelope of an electron is not affected by Lorentz forces in a region of zero magnetic field; instead, a phase difference—created by the magnetic vector potential—shifts the constructive- and destructive-interference fringe positions when electron wavefunctions are overlapped. This shift in fringe positions has been previously derived as resulting from potential-induced phase differences, without being clear on the physical mechanism behind it. In this paper, we show that the de Broglie wavelength of the electron is changed locally by its interaction with the vector potential. The vector potential thus acts as a quantum “phase plate,” changing the phase difference between interfering electron wavefunctions in a non-dispersive, gauge-invariant manner.
When an electron beam passes outside a confined magnetic flux, the motional magnetic field of the electrons overlaps the confined flux. The potential energy of the magnetic dipolar interaction of the motional magnetic field with the confined magnetic flux causes the velocity of the electrons to change slightly, which shifts the phase of the electron de Broglie wave. When portions of an electron beam pass on different sides of a confined flux and are then mixed, they produce an interference pattern. There is a similar magnetic dipolar interaction, with resulting velocity and phase changes, when a beam of magnetic particles interacts with the motional magnetic flux produced by their crossing a strong electric field. These are classical electromagnetic dipolar (not Lorentz) interactions that causes changes in Berry's geometrical phase, but are not ESAB or AC effects. Data that were suggested to demonstrate the ESAB and AC effects have been quantitatively interpreted in terms of motional magnetic fields and velocity changes.
When the electromagnetic potentials are expressed in the Coulomb gauge in terms of the electric and magnetic fields rather than the sources responsible for these fields they have a simple form that is nonlocal (i.e., the potentials depend on the fields at every point in space). It is this nonlocality of classical electrodynamics that is primarily responsible for the puzzle associated with the static Aharonov–Bohm effect: that its interference pattern is affected by fields in a region of space that the electron beam never enters.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics. While extensively studied in the static case, the impact of a time-varying magnetic flux on the electron's phase shift remains an open and debated question. In this paper, we derive the AB phase shift for a time-dependent magnetic vector potential and show that it is proportional to the time average of enclosed magnetic flux. Our analysis reveals that the AB phase is continuously accumulated as the electron traverses its path, challenging the conventional view that it emerges instantaneously at the point of interference. This generalized AB effect may provide deeper insight into the role of gauge-dependent potentials in quantum mechanics and also suggest novel experimental tests using alternating or pulsed magnetic flux.