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Physical Review A

Quantum topological phase of an electric dipole circulating around a ferromagnetic wire

V. M. Tkachuk

Published Oct 19, 2000• 44 citations
Publisher

Abstract

The quantum topological phase of electric dipole circulating around the line of magnetic charges (monopoles) is discussed. We propose to mimic the line of magnetic monopoles using a ferromagnetic wire in which the magnetization is parallel to the wire and the magnitude of magnetization changes linearly along the wire. The phase shift in the proposed scheme is within the reach of present experimental techniques and may be observed in atomic or molecular interferometry.

DOI
10.1103/physreva.62.052112

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The quantum topological phase of electric dipole circulating around the line of magnetic charges (monopoles) is discussed. We propose to mimic the line of magnetic monopoles using a ferromagnetic wire in which the magnetization is parallel to the wire and the magnitude of magnetization changes linearly along the wire. The phase shift in the proposed scheme is within the reach of present experimental techniques and may be observed in atomic or molecular interferometry.
We have recently tested the topological phase predicted by He and McKellar and by Wilkens: this phase appears when an electric dipole propagates in a transverse magnetic field. In the present paper, we first recall the physical origin of this phase and its relations to the Aharonov-Bohm and Aharonov-Casher phases. We then explain possible detection schemes and we briefly describe the lithium atom interferometer we have used for this purpose. Finally, we analyze in great detail the phase shifts induced by electric and magnetic fields acting on such an interferometer, taking into account experimental defects. The experiment and its results are described in the following paper.
There are three topological phases related to electromagnetic interactions in quantum mechanics: 1. The Aharonov Bohm phase acquired when a charged particle encircles a magnetic field but travels through a field free region. 2. The Aharonov Casher phase acquired when a magnetic dipole encircles electric charges but travels through a charge free region. 3. The He McKellar Wilkens phase acquired when an electric dipole encircles magnetic charges but travels through a charge free region. We review the conditions under which these phases are indeed topological and their experimental realisation. Because the He McKellar Wilkens phase has been recently observed we pay particular attention to how the basic concept of "an electric dipole encircles magnetic charges" was realised experimentally, and discuss possible future experimental realisations.
We predict a quantum phase shift of a moving electric dipole in the presence of an external magnetic field at a distance. On the basis of the Lorentz-covariant field interaction approach, we show that a phase shift appears in the internal dipole state under the condition that the dipole is moving in the field-free region, which is distinct from the topological He-McKellar-Wilkens phase generated by a direct overlap of the dipole and the field. We discuss the experimental feasibility of detecting this phase with atomic interferometry and argue that detection of this phase will resolve the question of the locality in quantum electromagnetic interaction.
The smooth topology change of Berry's phase from a Dirac monopolelike configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection Ak{\mathcal{A}}_{k}Ak​ is that the geometrical center of the monopolelike configuration and the origin of the Dirac string are displaced in the parameter space. Gauss's theorem ${\ensuremath{\int}}_{S}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{S}={\ensuremath{\int}}_{V}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})dV=0$ for a volume VVV which is free of singularities shows that a combination of the monopolelike configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge ${e}_{M}=2\ensuremath{\pi}\ensuremath{\hbar}$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.