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Physical Review A

Comment on “Energy-level shift of quantum systems via the scalar electric Aharonov-Bohm effect”

Shan Gao

Published May 29, 2025• 1 citation
Publisher

Abstract

Recently Chiao and his collaborators proposed a novel scalar electric Aharonov-Bohm (AB) effect [Phys. Rev. A 107, 042209 (2023)]. They claimed that a quantum system inside a Faraday cage with a time varying but spatially uniform scalar potential acquires an AB phase, resulting in observable energy level shifts. This comment argues that their analysis is flawed: a spatially uniform scalar potential inside the cage, despite external variations, can be gauged away without altering gauge-invariant observables, such as energy differences, thus invalidating their claim. A possible explanation of this seemingly puzzling result is also given.

DOI
10.1103/c8t7-y8h2

Related papers

Energy-level shift of quantum systems via the scalar electric Aharonov-Bohm effect

R. Y. Chiao, Harold Hart, Michael Scheibner +4 more

A version of the electric Aharonov-Bohm effect is proposed where the quantum system which picks up the Aharonov-Bohm phase is confined to a Faraday cage with a time-varying spatially uniform scalar potential. The electric and magnetic fields in this region are effectively zero for the entire period of the experiment. The observable consequence of this version of the electric Aharonov-Bohmn effect is to shift the energy levels of the quantum system rather than shift the fringes of the two-slit interference pattern. We show a strong mathematical connection between this version of the scalar electric $AB$ effect and the ac Stark effect.

arXiv:2212.03437•Physical Review A•Apr 13, 2023•12 citations

Reply to “Comment on ‘Energy-level shift of quantum systems via the scalar electric Aharonov-Bohm effect' ”

R. Y. Chiao, Harold Hart, Michael Scheibner +4 more

We respond to the points raised in the preceding Comment [] on our recent work [], which introduced a theoretical framework for the scalar electric Aharonov-Bohm effect. In this Reply, we take the opportunity to address specific criticisms, by referring to foundational aspects of gauge invariance and time-dependent quantum systems. We explain why the gauging procedure used in the Comment leads to inconsistencies and show that, when applied similarly to the well-established magnetic Aharonov-Bohm effect, it would yield demonstrably incorrect conclusions. Moreover, we highlight the consistency of our results with canonical treatments of the time-dependent Schrödinger equation, and we reaffirm the physicality of energy shifts in field-free regions, an effect observed in the magnetic and predicted in the electric Aharonov-Bohm contexts. We believe that this response contributes to a deeper understanding of the subtleties involved in gauge theory and quantum phase phenomena.

Physical Review A•May 29, 2025•0 citations

Electrodynamic Aharonov-Bohm effect

Pablo L. Saldanha

We propose an electrodynamic Aharonov-Bohm (AB) scheme where a nonzero AB phase difference appears even if the interferometer paths do not enclose a magnetic flux and are subjected to negligible scalar potential differences during the propagation of the quantum charged particle. In the proposal, the current in a solenoid outside the interferometer varies in time while the quantum particle is in a superposition state inside two Faraday cages, such that it is always subjected to negligible electromagnetic fields. At first glance, this result could challenge the topological nature of the AB effect. However, by considering the topology of the electromagnetic field configuration and the possible particle trajectories in spacetime, we demonstrate the topological nature of this situation.

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Comment on "Aharonov-Bohm Phase is Locally Generated Like All Other Quantum Phases"

Shan Gao

Marletto and Vedral [Phys. Rev. Lett. 125, 040401 (2020)] propose that the Aharonov-Bohm (AB) phase is locally mediated by entanglement between a charged particle and the quantized electromagnetic field, asserting gauge independence for non-closed paths. In this Comment, we critically analyze their model and demonstrate that the AB phase arises from the interaction with the vector potential \(\mathbf{A}\), not from entanglement, which is a byproduct of the quantum electrodynamics (QED) framework. We show that their field-based energy formulation, intended to reflect local electromagnetic interactions, is mathematically flawed due to an incorrect prefactor and yields \( +q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} \) in the Coulomb gauge, conflicting with QED's \( -q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} \). This equivalence to \( q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} \) holds only approximately in the Coulomb gauge under static conditions, failing for time-dependent fields and other gauges, undermining their claim of a gauge-independent local mechanism. Furthermore, we confirm that the AB phase is gauge-dependent for non-closed paths, contradicting their assertion. Our analysis reaffirms the conventional explanation in the semi-classical picture, where the AB phase is driven by the vector potential \(\mathbf{A}\), with entanglement playing no causal role in its generation.

quant-pharXiv:2601.17665•Physical Review Letters•Jan 27, 2026•3 citations

Topological schemes in spacetime for the electrodynamic Aharonov-Bohm effect

Pablo L. Saldanha, Herman Batelaan

Abstract We consider different schemes for the electrodynamic Aharonov-Bohm (AB) effect introduced in Saldanha P. L., Phys. Rev. A , 108 (2023) 062218, exploring the phenomenon to enhance the understanding of its topological nature in spacetime. In the treated examples, the electric current in a solenoid varies in time, changing its internal magnetic field and producing an external electric field, while a quantum charged particle is in a superposition state inside two Faraday cages in an interferometer. The Faraday cages cancel the electric field at their interiors, such that the particle is always subjected to null electromagnetic fields. We discuss how the AB phase difference depends on the topology of the electric and magnetic fields in spacetime in the different treated situations. In particular, we discuss interesting results when a conducting wire connects the two Faraday cages, with the AB phase depending on the wire position. We also show an amplification of the AB phase when the wire makes several turns around the solenoid, which could enable an experimental verification of the effect.

arXiv:2411.09622•Europhysics Letters (EPL)•May 1, 2025•1 citation

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
Recently Chiao and his collaborators proposed a novel scalar electric Aharonov-Bohm (AB) effect [Phys. Rev. A 107, 042209 (2023)]. They claimed that a quantum system inside a Faraday cage with a time varying but spatially uniform scalar potential acquires an AB phase, resulting in observable energy level shifts. This comment argues that their analysis is flawed: a spatially uniform scalar potential inside the cage, despite external variations, can be gauged away without altering gauge-invariant observables, such as energy differences, thus invalidating their claim. A possible explanation of this seemingly puzzling result is also given.
A version of the electric Aharonov-Bohm effect is proposed where the quantum system which picks up the Aharonov-Bohm phase is confined to a Faraday cage with a time-varying spatially uniform scalar potential. The electric and magnetic fields in this region are effectively zero for the entire period of the experiment. The observable consequence of this version of the electric Aharonov-Bohmn effect is to shift the energy levels of the quantum system rather than shift the fringes of the two-slit interference pattern. We show a strong mathematical connection between this version of the scalar electric ABABAB effect and the ac Stark effect.
We respond to the points raised in the preceding Comment [] on our recent work [], which introduced a theoretical framework for the scalar electric Aharonov-Bohm effect. In this Reply, we take the opportunity to address specific criticisms, by referring to foundational aspects of gauge invariance and time-dependent quantum systems. We explain why the gauging procedure used in the Comment leads to inconsistencies and show that, when applied similarly to the well-established magnetic Aharonov-Bohm effect, it would yield demonstrably incorrect conclusions. Moreover, we highlight the consistency of our results with canonical treatments of the time-dependent Schrödinger equation, and we reaffirm the physicality of energy shifts in field-free regions, an effect observed in the magnetic and predicted in the electric Aharonov-Bohm contexts. We believe that this response contributes to a deeper understanding of the subtleties involved in gauge theory and quantum phase phenomena.
We propose an electrodynamic Aharonov-Bohm (AB) scheme where a nonzero AB phase difference appears even if the interferometer paths do not enclose a magnetic flux and are subjected to negligible scalar potential differences during the propagation of the quantum charged particle. In the proposal, the current in a solenoid outside the interferometer varies in time while the quantum particle is in a superposition state inside two Faraday cages, such that it is always subjected to negligible electromagnetic fields. At first glance, this result could challenge the topological nature of the AB effect. However, by considering the topology of the electromagnetic field configuration and the possible particle trajectories in spacetime, we demonstrate the topological nature of this situation.
Marletto and Vedral [Phys. Rev. Lett. 125, 040401 (2020)] propose that the Aharonov-Bohm (AB) phase is locally mediated by entanglement between a charged particle and the quantized electromagnetic field, asserting gauge independence for non-closed paths. In this Comment, we critically analyze their model and demonstrate that the AB phase arises from the interaction with the vector potential A\mathbf{A}A, not from entanglement, which is a byproduct of the quantum electrodynamics (QED) framework. We show that their field-based energy formulation, intended to reflect local electromagnetic interactions, is mathematically flawed due to an incorrect prefactor and yields +qv⋅As +q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} +qv⋅As​ in the Coulomb gauge, conflicting with QED's −qv⋅As -q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} −qv⋅As​. This equivalence to qv⋅As q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} qv⋅As​ holds only approximately in the Coulomb gauge under static conditions, failing for time-dependent fields and other gauges, undermining their claim of a gauge-independent local mechanism. Furthermore, we confirm that the AB phase is gauge-dependent for non-closed paths, contradicting their assertion. Our analysis reaffirms the conventional explanation in the semi-classical picture, where the AB phase is driven by the vector potential A\mathbf{A}A, with entanglement playing no causal role in its generation.
Abstract We consider different schemes for the electrodynamic Aharonov-Bohm (AB) effect introduced in Saldanha P. L., Phys. Rev. A , 108 (2023) 062218, exploring the phenomenon to enhance the understanding of its topological nature in spacetime. In the treated examples, the electric current in a solenoid varies in time, changing its internal magnetic field and producing an external electric field, while a quantum charged particle is in a superposition state inside two Faraday cages in an interferometer. The Faraday cages cancel the electric field at their interiors, such that the particle is always subjected to null electromagnetic fields. We discuss how the AB phase difference depends on the topology of the electric and magnetic fields in spacetime in the different treated situations. In particular, we discuss interesting results when a conducting wire connects the two Faraday cages, with the AB phase depending on the wire position. We also show an amplification of the AB phase when the wire makes several turns around the solenoid, which could enable an experimental verification of the effect.
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.