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arXiv:2011.09005 · Physical Review A

Aharonov-Casher and shielded Aharonov-Bohm effects with a quantum electromagnetic field

Pablo L. Saldanha

Published Sep 24, 2021• 11 citations
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Abstract

We use a covariant formalism that is capable of describing the electric and magnetic versions of the Aharonov-Bohm effect, as well as the Aharonov-Casher effect, through local interactions of charges and currents with the quantum electromagnetic field. By considering that only local interactions of a quantum particle with the quantum field can affect its behavior, we show that the magnetic Aharonov-Bohm effect must be present even if the solenoid generating the magnetic field is shielded by a perfect conductor, as experimentally demonstrated.

DOI
10.1103/physreva.104.032219

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arXiv:2405.08536•Physical Review A•Jun 3, 2024•3 citations
We use a covariant formalism that is capable of describing the electric and magnetic versions of the Aharonov-Bohm effect, as well as the Aharonov-Casher effect, through local interactions of charges and currents with the quantum electromagnetic field. By considering that only local interactions of a quantum particle with the quantum field can affect its behavior, we show that the magnetic Aharonov-Bohm effect must be present even if the solenoid generating the magnetic field is shielded by a perfect conductor, as experimentally demonstrated.
Both classical and quantum wave interference phenomena rely on the notion of phase. In both classical as well as quantum electrodynamics the phases are acquired locally; however, the quantum phases have the additional property that they can be transmitted even when the classical value of the degree of freedom relevant for the transfer (such as the value of the electric field at one point) are zero. Here we use a covariant formalism that is capable of explaining the electric and magnetic versions of the Aharonov-Bohm effect, as well as the Aharonov-Casher effect, through local interactions of charges and currents with the quantum electromagnetic field. We show that Aharonov-Bohm effect, which is just another instance of interference based on a quantum phase, is present even if the solenoid generating the magnetic field is shielded by a superconductor. This is because even though the classical value of the magnetic field is zero in the superconductor (the Meissner effect), the quantum operators pertaining to the magnetic field are non-zero and are furthermore responsible for transmitting the phase information between the solenoid and the electron. The conclusion of our work is that quantum phases cannot be shielded and we believe that this, among other consequences, finally resolves the last outstanding mystery in the Aharonov-Bohm effect.
We address the question of locality versus nonlocality in the Aharonov-Bohm and the Aharonov-Casher effects. For this purpose, we investigate all possible configurations of ideal shielding of the overlap between the electromagnetic fields generated by a charge and by a magnetic flux and analyze their consequences on the Aharonov-Bohm-Casher interference. In a classical treatment of shielding, the Aharonov-Bohm-Casher effect vanishes regardless of the geometry of shielding when the local overlap of electromagnetic fields is completely eliminated. On the other hand, the result depends on the configuration of shielding if the charge quantization in the superconducting shield is taken into account. It is shown that our results are fully understood in terms of the fluctuating local-field interaction. Our analysis strongly supports the alternative view on the Aharonov-Bohm-Casher interference that the effects originate from the local action of electromagnetic fields.
A field-interaction scheme is introduced for describing the Aharonov-Bohm effect, fully consistent with the principle of relativity. Our theory is based on the fact that local field interactions are present even when a particle moves only in a field-free region. The interaction Lagrangian between a charge and a flux is uniquely constructed from three principles: Lorentz covariance, linearity in the interaction strength, and a correct stationary limit of charge. Our result resolves fundamental questions raised on the standard interpretation of the Aharonov-Bohm effect, concerning its duality with the Aharonov-Casher effect and the equivalence between the potential and the field-interaction models for describing the electromagnetic interaction. Most of all, potential is eliminated in our theory, and all kind of the force-free Aharonov-Bohm effect is understood in a unified framework of the Lorentz-covariant local interaction of electromagnetic fields.
In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable phase by enclosing an infinite solenoid, in a region where the solenoid's electric and magnetic fields are zero. Its generation seems therefore explainable only by the local action of gauge-dependent potentials, not of gauge-independent fields. This was recently challenged by Vaidman, who explained the phase by the solenoid's current interacting with the electron's field (at the solenoid). Still, his model has a residual nonlocality: it does not explain how the phase, generated at the solenoid, is detectable on the charge. In this Letter, we solve this nonlocality explicitly by quantizing the field. We show that the AB phase is mediated locally by the entanglement between the charge and the photons, like all electromagnetic phases. We also predict a gauge-invariant value for the phase difference at each point along the charge's path. We propose a realistic experiment to measure this phase difference locally, by partial quantum state tomography on the charge, without closing the interference loop.
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.