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Journal of Physics A Mathematical and General

The Aharonov-Bohm effect in double- and single-slit diffraction

D Shapiro, Walter C. Henneberger

Published Sep 7, 1989• 11 citations
Publisher

Abstract

Single- and double-slit diffraction patterns are obtained by means of path integrals. The slit function is taken to be unity. This permits the study of the single-slit case as a limit of the double-slit case when the slit width is equal to the slit separation. Introduction of a whisker of flux at the midpoint of the space between slits gives the Aharonov-Bohm shift. In the case of a wide single slit, one obtains conditions approaching those postulated in calculations similar to the scattering calculation of Aharanov and Bohm (1959).

DOI
10.1088/0305-4470/22/17/024

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Single- and double-slit diffraction patterns are obtained by means of path integrals. The slit function is taken to be unity. This permits the study of the single-slit case as a limit of the double-slit case when the slit width is equal to the slit separation. Introduction of a whisker of flux at the midpoint of the space between slits gives the Aharonov-Bohm shift. In the case of a wide single slit, one obtains conditions approaching those postulated in calculations similar to the scattering calculation of Aharanov and Bohm (1959).
We consider a two-slit diffraction experiment with a magnetic flux confined to an inaccessible region between two rectangular slits. We then obtain a leading-order analytic form for the asymmetry in the resulting diffraction pattern. The corrections to the expression are bounded and disappear in the limit of long wavelengths and/or infinite source-slit-screen spacing. Using the analytic form obtained, we obtain a nonzero value for the asymmetry in the number of electrons scattered to the left and to the right but a zero value for their average displacement.
In a "thought experiment," now a classic in physics pedagogy, Feynman visualizes Young's double-slit interference experiment with electrons in magnetic field. He shows that the addition of an Aharonov-Bohm phase is equivalent to shifting the zero-field wave interference pattern by an angle expected from the Lorentz force calculation for classical particles. We have performed this experiment with one slit, instead of two, where ballistic electrons within two-dimensional electron gas diffract through a small orifice formed by a quantum point contact (QPC). As the QPC width is comparable to the electron wavelength, the observed intensity profile is further modulated by the transverse waveguide modes present at the injector QPC. Our experiments open the way to realizing diffraction-based ideas in mesoscopic physics.
In a recent paper [D. H. Kobe, V. C. Aguilera-Navarro, and R. M. Ricotta, Phys. Rev. A 45, 6192 (1992)], Kobe et al. have calculated the expectation values of position and momentum in the Aharonov-Bohm effect and have found that both are shifted by the Aharonov-Bohm potentials. Although these shifts appear to contradict certain ``no-shift'' theorems proved by various authors, we show that this apparent contradiction is the result of different assumptions. Specifically, one of the assumptions on which the no-shift theorems are based does not hold for the parameters used by Kobe et al.; that is, with their parameters, there is no reason to expect the no-shift theorems to hold. On the other hand, with the parameters of actual two-slit experiments, all of the conditions of the no-shift theorems do hold and the no-shift theorems do apply.
The Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.
Since the Aharonov-Bohm effect is the purely quantum effect that has no analogues in classical physics, its persistence in the quasiclassical limit seems to be hardly possible. Nevertheless, we show that the scattering Aharonov-Bohm effect does persist in the quasiclassical limit owing to the diffraction, i.e. the Fraunhofer diffraction in the case when space outside the enclosed magnetic flux is Euclidean, and the Fresnel diffraction in the case when the outer space is conical. Hence, the enclosed magnetic flux can serve as a gate for the propagation of short-wavelength, almost classical, particles. In the case of conical space, this quasiclassical effect which is in principle detectable depends on the particle spin.