High precision measurement of the topological Aharonov–Casher effect with neutrons
Introduction
First described in 1959, the Aharonov–Bohm (AB) effects [1] have spawned many hundreds of articles [2], [3]. The effects are of two versions, the scalar (or electric) and the vector (or magnetic) AB effects, based on the scalar and vector terms of the four vector describing the interaction. In the scalar AB effect, a charged particle (electron) interacts with the scalar potential in the field free (i.e. force free) region inside an electrostatic cylinder. For the vector AB effect, the electron moves through the vector potential (again in the absence of an electromagnetic field) around a magnetic flux tube. The wave function of the electron is described by the time-dependent Schrödinger equationwhere V=−eU and , with the canonical momentum given byIn these equations, U is the scalar electrostatic potential, is the classical momentum of the electron, and is the vector electromagnetic potential. The effects are based around the fact that in quantum mechanics, the potentials play a more fundamental role than the classical fields and forces they involve. These effects manifest themselves as a shift of the phase of the quantum mechanical wave function, whilst the energy remains unchanged. An interference experiment offers the possibility to measure this phase shift.
In this paper, we present previous work on the Aharonov–Casher effect and explain details of a proposed neutron interferometry experiment to measure the AC phase shift to high accuracy that will also highlight the topological nature of the effect.
Section snippets
The Aharonov–Bohm effects
The vector and scalar AB interference experiments as described in the 1959 paper are shown in Fig. 1. The vector AB effect (Fig. 1a) appears as a phase shift in the wave function of an electron as it diffracts either side of a line of magnetic flux, and is given by the path integral of the canonical momentumwhere ΦM is the total magnetic flux enclosed by the two paths. This effect had been observed in a series of investigations culminating in the unequivocal
Neutron Aharonov–Casher experiment
In 1984 Aharonov and Casher (AC) suggested the electromagnetic and quantum mechanical dual of the vector AB effect [6]. If the line of magnetic flux (Fig. 1a) is replaced with a line of magnetic dipoles (Fig. 2a), then in the reference frame of the electron, we have the theoretical and topological equivalent [7], [8] of a magnetic dipole (e.g. a neutron) diffracting around a line of electric charge (Fig. 2b). Such an experiment was performed by Cimmino et al. [9] in 1989 using a neutron
Atom Aharonov–Bohm experiments
In principle, the AC effect is much greater for atoms than for neutrons, by a factor of the order of the ratio (Bohr magneton)/(nuclear magneton) i.e. ≈mn/me, and thus much higher precision should be attainable. Recently, a series of atomic AC experiments [21], [22], [23], [24] have indeed been performed and have measured the phase shift to a few percent accuracy. However, the experimental geometry of these experiments does not represent the topological equivalent of the neutron AC experiment,
High precision Aharonov–Casher experiment
In the original AC experiment performed by Cimmino et al. [9], the observed phase shift was nearly two standard deviations above the theoretical value. Improving on this result and proving the topological nature are the reasons for performing a second high precision experiment. The AC phase shift is obtained by intergrating the canonical momentum around the interference loop, Eq. [7], so that increasing the path length increases the size of the effect. The planned experiment will use a similar
Conclusions
In this article, we have presented our plan to perform a high precision neutron interferometery Aharonov–Casher experiment, to improve on the accuracy of the previous experiment and to highlight the topological nature of the effect. AC-type phase shifts have been measured accurately in several atom interferometry experiments. However, they have failed to demonstrate the topological nature of the effect. We intend to show this aspect by placing several electrodes in the interferometer, and to
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