A random wave model for the Aharonov–Bohm effect

Houston, Alexander; Gradhand, Martin; Dennis, Mark R.

doi:10.1088/1751-8121/aa660f

Cited by 4 publications

(6 citation statements)

References 40 publications

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“…It is important to know whether the VS structure survives in more general AB waves involving a single flux line. These are superpositions of AB plane waves [9][10][11] initially travelling in directions θ n :…”

Section: Discussionmentioning

confidence: 99%

“…Such superpositions will typically contain additional phase singularities away from the flux line; when α = 0, these are the much-studied wave vortices in typical complex scalar fields [12][13][14]. As α increases, these vortices move [11]. Although the pattern of phase and canonical streamlines in which they are embedded is gauge-dependent, the kinetic streamlines possess vortices at the same locations.…”

Section: Discussionmentioning

confidence: 99%

“…This possesses a vortex at the origin, even when the wave (10) does not depend on θ, for example when n = 0 and 0 < α < 1/2. Although the existence of S was not surprising, its closeness to V-always less that 1/50 of the de Broglie wavelength, was unexpected.…”

Section: Vortex-stagnation Point Pairsmentioning

confidence: 99%

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Gauge-invariant Aharonov–Bohm streamlines

Berry¹

2017

J. Phys. A: Math. Theor.

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The phase gradient of the wave describing the Aharonov–Bohm effect (AB) is proportional to the local canonical momentum. This vector field contains vortices (phase singularities), whose strengths cannot be detected in quantum mechanics because they increase (discontinuously) with the magnetic flux, violating gauge invariance. The analogous quantity which is gauge-invariant is the kinetic momentum field, proportional to the local electron velocity. Investigation of the streamlines (integral curves) of this velocity field reveals that as the flux increases from 0 to 1/2 (in quantum units), a vortex V is generated at the flux line, accompanied by a stagnation point (saddle) S that emerges from V and then collapses back into V. The VS pair is always small: the maximum distance between V and S is approximately 0.0209 de Broglie wavelengths. The VS phenomenon survives generalization to a superposition of AB waves. If the flux is confined within an impenetrable tube of radius R, S persists if R < 0.004 de Broglie wavelengths, and is swallowed by the tube for larger R. An experiment is envisaged.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Gauge-invariant Aharonov–Bohm streamlines

Berry¹

2017

J. Phys. A: Math. Theor.

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“…D v , which can be determined statistically (at least, in the mean), specifies the dislocation point (or vortex) density-the mean number of dislocation lines piercing unit area of a plane (Berry, 1978). These statistics have been thoroughly studied for quasimonochromatic paraxial waves (Baranova et al, 1981), monochromatic waves in two dimensions Freund and Freilikher, 1997;Freund and Shvartsman, 1994;Freund et al, 1993;Freund and Wilkinson, 1998;Shvartsman and Freund, 1994b), isotropic random waves (Berry and Dennis, 2000), and random waves subject to the Aharonov-Bohm effect (Houston et al, 2017) among others, with novel extensions to the statistics of knotted nodal lines by Dennis (2014, 2016).…”

Section: Flows and Vorticesmentioning

confidence: 99%

Nodal portraits of quantum billiards: Domains, lines, and statistics

Jain,

Samajdar

2017

Preprint

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We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches to counting the nodal domains-using trace formulae, graph theory, and difference equations-are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity-and far-reaching implications-of the problem. CONTENTS3. Periodic orbits of nonseparable, integrable billiards 48 C. Graph-theoretic analysis 49 D. Difference-equation formalism 50 1. Circles (and annuli/sectors thereof) 50 2. Ellipses and elliptic annuli 50 3. Confocal parabolae 51 4. Right-angled isosceles triangle 51 5. Equilateral triangle 51 E. Counting with Potts spins 52 VII. Experimental realizations 52 A. Microwave billiards: The physicist's pool table 52 B. The S-matrix and transmission measurements 54 C. The perturbing bead method 56 D. Current and vortex statistics 57 VIII. Concluding remarks 60 Acknowledgments 61 A. Isoperimetric inequalities 61 1. Rayleigh-Faber-Krahn and related inequalities 61 2. Payne-Pólya-Weinberger inequality 62 3. Szegö-Weinberger inequality 62 References 62

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“…The existence of screening among topological charges is well established in literature. [9,[17][18][19][20][21][22][23][24]. It starts to play a role when the size of the observation window is bigger than the typical inter-singularity distance, of approximately λ/2.…”

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confidence: 99%

Screening and fluctuation of the topological charge in random wave fields

Angelis

Kuipers

2018

Opt. Lett.

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Vortices, phase singularities and topological defects of any kind often reflect information that is crucial to understand physical systems in which such entities arise. With near-field experiments supported by numerical calculations, we determine the fluctuations of the topological charge for phase singularities in isotropic random waves, as a function of the size R of the observation window. We demonstrate that for 2D fields such fluctuations increase with a super-linear scaling law, consistent with a R log R behavior. Additionally, we show that such scaling remains valid in presence of anisotropy.

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A random wave model for the Aharonov–Bohm effect

Cited by 4 publications

References 40 publications

Gauge-invariant Aharonov–Bohm streamlines

Gauge-invariant Aharonov–Bohm streamlines

Nodal portraits of quantum billiards: Domains, lines, and statistics

Screening and fluctuation of the topological charge in random wave fields

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