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Physical Review A

Topological quantum phase and multipole moment of neutral particles

Chia-Chu Chen

Published Mar 1, 1995• 65 citations
Publisher

Abstract

It is shown that the Aharonov-Bohm (AB) effect is a generic feature of any multipole moment moving in an electromagnetic field. Explicit examples of electric and magnetic quadrupole moments are discussed. A way to detect the AB phase of a magnetic quadrupole moment is also provided.
DOI
10.1103/physreva.51.2611

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The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two special orientations of the magnetic moment (same as in the quantum case) the motion is effectively free and similar to the AB motion of a charged particle. Thus, the AC effect is not really topological. Presence of higher multipoles totally destroys the possible AB-AC duality.
Recent works showed that the Aharonov-Bohm (AB) phase difference for a quantum charged particle can be written in terms of electric and magnetic fluxes in a space-time surface whose boundaries are the possible particle worldlines in the interferometer, relative to the possible paths. After presenting this result in a more detailed way, reinforcing its topological nature, we study the magnetic and electric versions of the AB effect in different inertial reference frames. We find a particular reference frame for a magnetic AB effect where the magnetic flux has a null contribution for the AB phase difference, which is entirely due to an electric flux. Also, we find a particular reference frame for an electric AB effect where the electric flux has a null contribution for the AB phase difference, which is entirely due to a magnetic flux. In this sense, the nomenclatures magnetic AB effect and electric AB effect lose their meaning. We have electromagnetic AB effects.
It is shown that the vector potential of the Aharonov-Bohm effect is effectively the same as vector potentials of the quantum phases of electric and magnetic dipoles such as the Aharonov-Casher effect, when the Aharonov-Bohm solenoid is a straight cylinder with symmetric cross sections. With a circular cross section, the Aharonov-Bohm vector potential is effectively the same as the Aharonov-Casher vector potential, while with a rectangular cross section, it is effectively the same as the vector potential of the Casella phase. Therefore, we may say that the quantum phases of the dipoles belong to a subclass of the Aharonov-Bohm phase, and they must share the topological origin of the Aharonov-Bohm phase, even though their topological freedom is limited to the particle trajectories. However, the dipole phases have the extra relativistic effect that a magnetic (electric) dipole moving in an electric (magnetic) field feels an additional magnetic (electric) field, in its rest frame. This effect limits further the topological freedom of the dipole phases.