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Optica

Forces in Aharonov–Bohm optical setting

Sergey V. Sukhov, Veerachart Kajorndejnukul, J. Broky, Aristide C. Dogariu

Published Dec 3, 2014• 10 citations
Publisher

Abstract

The Aharonov-Bohm effect is usually associated with a path-dependent phase accumulated by a charged matter wave and determined by an effective vector potential. In a more general geometrical framework, such phase alterations have been the hallmark of a host of related phenomena in many different fields. However, besides phase changes, it was suggested that for finite wave-packets there is also an additional deflection leading to observable changes in the wave's canonical momentum. In this paper, we create an optical scattering situation that permits observing nonconservative reaction forces, which result from the conservation of canonical momentum. We demonstrate experimentally, for the first time, the presence of such mechanical forces with the magnitude and direction determined by the phase dislocation of the vortex state. Our experimental results offer insights into essentially untested phenomena, where forces act on vortex fields, and allow examining the role of conservation laws and symmetries in complex interacting systems. (C) 2014 Optical Society of America

DOI
10.1364/optica.1.000383

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The Aharonov-Bohm effect is usually associated with a path-dependent phase accumulated by a charged matter wave and determined by an effective vector potential. In a more general geometrical framework, such phase alterations have been the hallmark of a host of related phenomena in many different fields. However, besides phase changes, it was suggested that for finite wave-packets there is also an additional deflection leading to observable changes in the wave's canonical momentum. In this paper, we create an optical scattering situation that permits observing nonconservative reaction forces, which result from the conservation of canonical momentum. We demonstrate experimentally, for the first time, the presence of such mechanical forces with the magnitude and direction determined by the phase dislocation of the vortex state. Our experimental results offer insights into essentially untested phenomena, where forces act on vortex fields, and allow examining the role of conservation laws and symmetries in complex interacting systems. (C) 2014 Optical Society of America
Recent experiments with the Aharonov-Bohm geometry have shown that in addition to an electron-interference fringe shift, there is also a lateral displacement of the electron diffraction envelope. In this paper, we derive an electron displacement force based on a second-order expansion of the magnetic vector potential. The analysis illustrates the conservation of canonical angular momentum, where the mechanical angular momentum and field angular momentum sum to a constant of the motion; the azimuthal force required to change the mechanical momentum is thus supplied by changes in field momentum associated with the second-order vector potential term. Our results are consistent with all known Aharonov-Bohm experiments, including interference fringe shifts, lateral displacements, and the absence of longitudinal time delays.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.
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 Newtonian mechanics a force changes momentum, for example it may change its magnitude only or its direction only or both. In the case of the Bohm Aharonov effect for a particle with rest mass there exists a potential which does not change momentum, but introduces a phase shift. Thus exp(ipx) for say an electron becomes exp(ip(x-xo)). Given that the form exp(-iEt+px) holds for both particles with rest mass and photons one might expect to find a Bohm Aharonov effect for photons. In (1), it is stated that a photonic Bohm Aharonov effect may be “induced by a dynamic modulation of a material’s permittivity”. In this note we argue that the refraction of light seems to have the same form as a Bohm Aharonov effect in the sense that if one begins and ends in the same medium, the particle has the same energy E, momentum p, speed of light c and makes the same angle with the normal. In a Newtonian sense there has been no change to momentum, but the ray is shifted in the x direction from one that does not refract. It seems that this may be described by exp(i py*y) exp(i px(x-xo)) i.e. a phase shifted form suggesting that for the x direction one has a Bohm Aharonov effect. We try to link this phase shift in the x direction to more general ideas. We argue as in (2) that a lack of information related to speed (i.e. the index of refraction) in the x direction leads to a conservation law related to speed in the x direction. This speed, however, is a hypothetical one such that it acts as a hypotenuse with c/n(i) being a projection. This differs from the actual x component of velocity which is a projection of c/n(i). Thus this actual x speed is not conserved meaning that light moving in a new medium will struck the outer bound at a different x than it would have it had not moved in a new medium. Thus the beam coming out of the new medium has the same E and p vector, but a shifted x i.e. there is a phase shift of the Bohm Aharonov type we argue even though we do not formally deal with a potential.
We provide arguments which demonstrate that the Berry phase effects involving particle interference in the presence of external electric and magnetic fields—namely the Aharonov–Bohm (AB), Aharonov–Casher (AC) and the Röntgen (R) effects—cannot be unambiguously explained (as has been controversially argued) through the use of any classical mechanical force. In fact, in the three scenarios which we consider, the only means to reclaim the standard quantum mechanical results is to use the expressions for the force that fail to satisfy the necessary conditions of gauge invariance. Our arguments explain the foundation of the controversy and confirm the standard idea that the Berry phase effects in question are non-local phenomena which only arise in the application of quantum theory. We lay special emphasis on the use of the concept of a canonical force, this being the quantity which ensures the agreement between the classical and quantum formulations. It is shown that if the Aharonov–Bohm effect is interpreted as being due to a force, through the use of the Bohmian formulation of quantum mechanics, then this force is equal in magnitude to the canonical force.