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On the role of induced electric field in the time-dependent Aharonov-Bohm effect

Masashi Wakamatsu

TL;DR

The paper reexamines the time-dependent Aharonov-Bohm effect and the role of the induced electric field produced by a time-varying magnetic flux inside a solenoid. Using Cao's elementary approach with careful 4D Stokes accounting, it derives the time-dependent AB-phase as a closed-space-time line integral of the four-vector potential and shows the induced field's nontrivial contributions can be handled consistently. Its main results include that the AB-phase equals $\phi_{AB} = e \frac{1}{t_f} \int_0^{t_f} \Phi(t) dt$, matching prior findings even when the induced field is retained, and that the re-encounter azimuth $\phi_f$ is not topologically fixed but depends on system parameters. The work suggests an experimentally accessible route—monitoring the interference peak position as a function of arrival time—to verify the time-dependent AB effect and clarifies the theoretical conditions under which the effect should exist.

Abstract

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.

On the role of induced electric field in the time-dependent Aharonov-Bohm effect

TL;DR

The paper reexamines the time-dependent Aharonov-Bohm effect and the role of the induced electric field produced by a time-varying magnetic flux inside a solenoid. Using Cao's elementary approach with careful 4D Stokes accounting, it derives the time-dependent AB-phase as a closed-space-time line integral of the four-vector potential and shows the induced field's nontrivial contributions can be handled consistently. Its main results include that the AB-phase equals , matching prior findings even when the induced field is retained, and that the re-encounter azimuth is not topologically fixed but depends on system parameters. The work suggests an experimentally accessible route—monitoring the interference peak position as a function of arrival time—to verify the time-dependent AB effect and clarifies the theoretical conditions under which the effect should exist.

Abstract

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.

Paper Structure

This paper contains 3 sections, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The role of induced electric field in the time-dependent AB-effect
  3. Questions still remain and outlook

Figures (3)

  • Figure 1: Schematic picture to consider the Aharonov-Bohm effect. The left panel corresponds to the familiar case of time-independent magnetic flux, whereas the right panel corresponds to the case of time-dependent magnetic flux.
  • Figure 2: The auxiliary function $f (\Omega \,T)$ defined in Eq.(\ref{['Eq:f']}) corresponding to the time-dependent magnetic flux $\Phi (t) = \Phi_0 + \Phi_1 \,\cos \Omega t$. The figure corresponds to the choice $\Phi_1 / \Phi_0 = 1 / 10$.
  • Figure 3: The picture schematically showing the oscillatory behavior of the peak position of the interference pattern on the screen as the arrival time $T$ or equivalently the velocity $v$ of the electron beams changes.