Recent works showed that the Aharonov-Bohm (AB) phase difference for a quantum charged particle can be written in terms of electric and magnetic fluxes in a space-time surface whose boundaries are the possible particle worldlines in the interferometer, relative to the possible paths. After presenting this result in a more detailed way, reinforcing its topological nature, we study the magnetic and electric versions of the AB effect in different inertial reference frames. We find a particular reference frame for a magnetic AB effect where the magnetic flux has a null contribution for the AB phase difference, which is entirely due to an electric flux. Also, we find a particular reference frame for an electric AB effect where the electric flux has a null contribution for the AB phase difference, which is entirely due to a magnetic flux. In this sense, the nomenclatures magnetic AB effect and electric AB effect lose their meaning. We have electromagnetic AB effects.
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.
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.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux,
ΔφAB=T1∫0TeΦ(t)dt, with the total phase shift, including kinetic contributions, equaling
eΦ(0). For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius
R, the AB phase shift is also proportional to the time average of the enclosed flux
Φenc(R,t), with the total phase shift depending only on the initial enclosed flux
eΦenc(R,0); for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential
A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics
A is in some sense more ``fundamental'' than
B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant
B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the
B field in the region inaccessible to the particle.
We discuss the consequences of the Aharonov-Bohm (AB) effect in setups involving several charged particles, wherein none of the charged particles encloses a closed loop around the magnetic flux. We show that in such setups, the AB phase is encoded either in the relative phase of a bipartite or multipartite entangled photons states, or alternatively, gives rise to an overall AB phase that can be measured relative to another reference system. These setups involve processes of annihilation or creation of electron-hole pairs. We discuss the relevance of such effects in ``vacuum birefringence" in QED, and comment on their connection to other known effects.