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arXiv:2512.15935 · Physics Letters A

AC Stark effect or time-dependent Aharonov-Bohm effect for particle on a ring

Patrick Hinrichs, Douglas Singleton, Nader Inan

quant-phhep-thSubmitted Dec 17, 2025• Announced Dec 19, 2025• 0 citations
arXivHTMLPDF

Abstract

We study the effect of a time-varying solenoidal vector potential for a quantum particle confined to a ring. The setup appears to be a time-varying version of the Aharonov-Bohm effect, but since the particle moves in the presence of fields, it is not strictly an Aharonov-Bohm effect. The results are similar to the ac Stark effect, but with a time-varying electric field coming from the vector potential, rather than the scalar potential. We compare and contrast the present effect with the standard ac Stark effect. The signature of this setup is the generation of quasi-energy sidebands which are observable via spectroscopy.

DOI
10.1016/j.physleta.2025.131248
Comments
18 pages, 5 figures, revtex4, accepted for publication in Physics Letters A

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We study the effect of a time-varying solenoidal vector potential for a quantum particle confined to a ring. The setup appears to be a time-varying version of the Aharonov-Bohm effect, but since the particle moves in the presence of fields, it is not strictly an Aharonov-Bohm effect. The results are similar to the ac Stark effect, but with a time-varying electric field coming from the vector potential, rather than the scalar potential. We compare and contrast the present effect with the standard ac Stark effect. The signature of this setup is the generation of quasi-energy sidebands which are observable via spectroscopy.
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.
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.
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.
The canonical Aharonov–Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov–Bohm phase acquired by a charged particle moving in time-dependent potentials. In particular, we focus on the case of a charged particle moving in the time-varying field of a plane electromagnetic wave. We work out the Aharonov–Bohm phase using both the potential (i.e. ∮Aμdxμ\oint A_\mu \mathrm{d}x ^\mu ∮Aμ​dxμ ) and the field (i.e. 12∫Fμνdσμν\frac{1}{2}\int F_{\mu \nu } \mathrm{d}\sigma ^{\mu \nu }21​∫Fμν​dσμν ) forms of the Aharonov–Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space–time loop, amplitude of the electromagnetic wave) under which the time-varying Aharonov–Bohm effect could be observed.
Abstract In this article, we review a new approach to the scalar Aharonov-Bohm effect for both the electromagnetic and gravitational interaction. For both interactions a quantum system is placed in a time-dependent electromagnetic or gravitational potential, but with no force (spatial derivative of the potential) acting on the quantum system. Nevertheless, we show that the energy levels of the quantum system develop side bands which can be detected as the signature of this version of the scalar Aharonov-Bohm effect. We briefly look at the specific experimental setups required to detect the energy side bands.
We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.