Historical research guide · 0464 · 2019
Asymmetry and non-dispersivity in the Aharonov-Bohm effect
Maria Becker · Giulio Guzzinati · Armand Béché · Johan Verbeeck · Herman Batelaan
In one minute
The Aharonov-Bohm effect is the cleanest evidence that the electromagnetic potential is physically real: an electron passing a shielded magnetic flux picks up a phase shift even though no field, and therefore no classical force, ever touches it. Maria Becker, Herman Batelaan and their colleagues in Antwerp and Nebraska ask the sharper question — is that electron really force-free? Andrei Shelankov predicted in 1998, and Michael Berry worked out in detail, that a quantum ‘force’ should nudge the beam sideways and leave a lopsided diffraction pattern behind it. The team built the experiment: a magnetised nickel nanorod strung across a five-micrometre hole inside an electron microscope, with the far-field diffraction recorded. The pattern is asymmetric, it flips when the rod’s magnetisation is reversed, and it becomes symmetric again when the rod is heated past its Curie point. The theory half of the paper shows that Zeilinger’s force-free theorem and Shelankov’s deflection are two limits of a single deflection theorem: no classical force along the beam, a quantum one across it.
Connection
Chapter 10 rests on the vector potential being physically real and something you can act with, and this is the sharpest measurement yet of what the potential does where no field exists — it steers the beam. It belongs to chapter 1 as the model of how a settled effect gets pushed further: a prediction, a purpose-built experiment, a reversal test, and a quantitative case that the obvious confound cannot explain the signal.
From the source
Electrons are affected by electromagnetic potentials in the absence of forces due to fields: outside an ideal flux tube the fields are zero, so no classical Lorentz force can act, yet the non-zero vector potential produces a local phase shift. The effect has been observed for free electrons in a series of ever more refined experiments and in conductors, and the absence of longitudinal force has been confirmed by the absence of electron time delays.Introduction, opening paragraphs
Zeilinger’s dispersivity theorem, as expounded by Peshkin, states that when an interaction is dispersionless — when the phase shift does not depend on momentum — there is no shift of the wavepacket’s position expectation value compared with free propagation. It is the formal statement of the textbook claim that the Aharonov-Bohm effect is force-free.Results, Dispersionless and quantum forces; Eqs. (6) and (7)
The paper derives one deflection theorem from which both results follow as limiting cases: apply it to a longitudinal Gaussian momentum distribution and Zeilinger’s theorem appears; apply it to a transverse Gaussian position distribution carrying a phase step and Shelankov’s deflection appears. Zeilinger’s theorem is correct but its assumptions do not cover the transverse motion of the Aharonov-Bohm system, so the usual statement that the effect is unaccompanied by forces does not hold in that direction.Results, Deflection theorem, Eq. (5); Fig. 3; Discussion
For the transverse case the average displacement comes out as y₀ plus (ħ over m β) times the square root of two over π times the sine of 2πα times t — a non-zero average deflection that oscillates with the amount of enclosed flux, with α equal to the flux in quantum units. In the de Broglie-Bohm picture the corresponding quantum potential is left-right symmetric when the rod is unmagnetised and loses that symmetry when it is magnetised.Results, Eq. (12) and the Quantum force section; Fig. 4
The experiment confirms the predicted asymmetry. A 30-micrometre nickel-coated nanorod mounted across a 5-micrometre aperture in a transmission electron microscope produces an asymmetric far-field diffraction pattern at 60 keV, matched by a path-integral simulation at a flux strength of 0.39 against the 0.41 measured independently by electron holography; the demagnetised rod gives a single symmetric peak at a fitted strength of minus 0.02. Reversing the rod’s magnetisation reverses the asymmetry, and heating the rod in ten-degree steps walks the pattern through successive reversals to symmetry above the Curie temperature.Results, Experimental asymmetry observed; Figs. 6 and 7
Fringing fields cannot account for the result: the computed phase variation across half the aperture is no more than 0.05π radians, giving an estimated deflection of about 10 micrometres against a diffraction pattern whose characteristic size is 373 micrometres. What remains to be done is the direct measurement of the deflection itself — between measurements the pattern shifts on the screen by more than the predicted deflection — and the authors suggest the effect may lead to a new detection mode for SQUID magnetometry.Results, Role of fringing fields; Discussion
The way in
https://doi.org/10.1038/s41467-019-09609-9How to cite it
Maria Becker, Giulio Guzzinati, Armand Béché, Johan Verbeeck, Herman Batelaan (2019) Asymmetry and non-dispersivity in the Aharonov-Bohm effect doi:10.1038/s41467-019-09609-9