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arXiv:2408.06162 · Axioms

A Kuramoto Model for the Bound State Aharonov–Bohm Effect

Alviu Rey Nasir, José Luís da Silva, Jingle B. Magallanes, Herry Pribawanto Suryawan, Roshin Marielle Nasir-Britos

Published Nov 27, 2024• 0 citations
arXivHTMLPDF

Abstract

The Aharonov–Bohm effect can be described as a phase difference in interfering charged particles that travel through two distinct pathways oppositely surrounding a perpendicularly-positioned solenoid. The magnetic field emanates from the solenoid but does not intersect the pathways. On the other hand, the Kuramoto model can be used to identify the synchronization conditions that lead to a particular phase difference by treating the phases as coupled oscillators. Starting with the overall wave function expression for the electron in an Aharonov–Bohm potential, we derive a version of the Kuramoto model describing the phase dynamics of the bound state of the quantum mechanical system. We show that the resulting synchronization condition of the model coincides with the allowable values of the flux parameter for our case to achieve an Aharonov–Bohm effect.

DOI
10.3390/axioms13120828

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The Aharonov–Bohm effect can be described as a phase difference in interfering charged particles that travel through two distinct pathways oppositely surrounding a perpendicularly-positioned solenoid. The magnetic field emanates from the solenoid but does not intersect the pathways. On the other hand, the Kuramoto model can be used to identify the synchronization conditions that lead to a particular phase difference by treating the phases as coupled oscillators. Starting with the overall wave function expression for the electron in an Aharonov–Bohm potential, we derive a version of the Kuramoto model describing the phase dynamics of the bound state of the quantum mechanical system. We show that the resulting synchronization condition of the model coincides with the allowable values of the flux parameter for our case to achieve an Aharonov–Bohm effect.
The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
In the usual Aharonov-Bohm effect, representing an especial case of the Berry phase phenomenon, classical magnetic field within long and thin solenoid (or classical vector potential of the electromagnetic field without this solenoid) causes phase difference and interference shape translation of the quantum propagating (external) electron. In this work we consider a variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. In this way we obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept. As it is well-known in the usual experimental arrangement corresponding to the AharonovBohm effect [1]-[4] (representing an especial case of the Berry phase phenomenon [4], [5]) behind a diaphragm with two slits there is a long and very thin solenoid (placed perpendicularly, i.e. in the z-axis direction, to the x-0-y plane of the propagation of the (external, without solenoid
In the Aharonov–Bohm (AB) effect, electrons form an interference pattern whose bright and dark fringes shift position when the magnetic flux confined within a nearby solenoid is changed, even though the electrons travel only through regions where the magnetic field is zero. Operationally, the experiment compares electron interference patterns obtained under different solenoid current (or enclosed flux) conditions, revealing a measurable phase difference—observed as fringe displacement—between the two electron paths despite the absence of any force acting on the electrons along those paths. Each electron arrives as a dot, but many dots build up the interference pattern. The effect tracks the enclosed flux (current level), with on/off as an illustrative extreme. What is measured is phase difference / fringe displacement, i.e., a relative effect between the two paths. This Aharonov-Bohm effect is famous in modern physics and widely regarded as proof that electromagnetic potentials possess physical reality even in regions where fields vanish. This paper reinterprets the AB effect by reframing it using the SP3 (space-phase) theory.
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.
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).