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arXiv:2308.05413 · Physical Review A

Scattering of relativistic electrons and analogies with optical phenomena: A study of longitudinal and transverse shifts at step potentials

Yue Ban, Xi Chen

Published Oct 27, 2023• 4 citations
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Abstract

We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of the Dirac current, we elucidate that the Goos-H\"anchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.

DOI
10.1103/physreva.108.042218

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We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of the Dirac current, we elucidate that the Goos-H"anchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.
We study the classical optics effects known as Goos–Hänchen and Imbert–Fedorov shifts, occurring when reflecting a bounded light beam from a planar surface, by using a quantum-mechanical formalism. This new approach allows us to naturally separate the spatial shift into two parts, one independent on orbital angular momentum (OAM) and the other one showing OAM-induced spatial-versus-angular shift mixing. In addition, within this quantum-mechanical-like formalism, it becomes apparent that the angular shift is proportional to the beams angular spread, namely to the variance of the transverse components of the wave vector. Moreover, we extend our treatment to the enhancement of beam shifts via weak measurements and relate our results to the recent experiments.
A theoretical investigation is made of the spatial displacements of a Dirac electron which undergoes a series of total internal reflections from finite potential barriers of arbitrary smoothness. These displacements are analogous to the shifts observed for optical total reflection (the Goos-H\"anchen effect). For an electron, the polarization states that are invariant to a single-reflection process are identified and the change in polarization for an arbitrary incident polarized state is determined. A calculation of longitudinal and transverse shifts is made for a double-reflection process from surfaces with a specified orientation. This calculation is based on a phase-shift analysis in which a localization form-invariance argument for the wave packet is used to determine both the eigenpolarization states of the double-reflection process and the relevant phase shifts. An unpolarized beam of electrons will be spatially split into these geometry-dependent eigenpolarization states. In general, it is found that the total spatial displacement of the wave-packet center for each eigenstate consists of two parts, each of which has longitudinal and transverse components. There is a common shift (dependent on the potential) which near critical reflection is of the order of the de Broglie wavelength, and a polarization-dependent shift (independent of the potential) that causes a split that is at most of the order of a Compton wavelength. There is no splitting for reflections between parallel surfaces. In principle, a macroscopically observable common shift can be produced through multiple reflections.
The Goos-H\"anchen (GH) shift is a specifical optical phenomenon that describes a shift parallel to the reflected light inside the plane of incidence, when a finite-width light undergoes total internal reflection at the interface of medium. Although the GH shift in optics has been widely observed experimentally, its generalization remains uncovered completely in relativistic quantum mechanics for the spinor structure and the possible existence of Klein's paradox. Through approximation and angular spectrum analysis, we calculate the GH shift for Dirac fermions under relativistic conditions when they are incident obliquely on a three-dimensional infinite potential barrier. Furthermore, we find that the relativistic quantum GH shift will degenerate into the result based on Schr\"odinger equation under nonrelativistic limit, while the GH shift can be negative in Dirac's framework, which is different from the nonrelativistic case.
We describe how the notion of optical beam shifts (including the spatial and angular Goos-Hänchen shift and Imbert-Federov shift) can be understood as a classical analogue of a quantum measurement of the polarization state of a paraxial beam by its transverse amplitude distribution.Under this scheme, complex quantum weak values are interpreted as spatial and angular shifts of polarized scalar components of the reflected beam.This connection leads us to predict an extra spatial shift for beams with a radially-varying phase dependance.Contents 1. Introduction 2 2. Weak quantum measurement 4 3.The analogy between beam shifts and quantum measurement 5 4. The case of radial and azimuthally varying pointer phase 9 5. Discussion
In conventional optics for light waves, the pioneering work by Goos-Hänchen in 1947 on the lateral shift (or displacement) of a light beam along the surface of a dielectric boundary under the condition of total reflection has stimulated a large number of studies. The key physics behind the Goos-Hänchen shift is the nature of wave interference. From the perspective of wave optics, the incident beam of a finite transverse width can be viewed as composed of plane wave components, each of which has a slightly different transverse wavevector. Each wave component, after the total internal reflection, undergoes a different phase shift, and the superposition of all the reflected wave components gives rise to the lateral shift of the intensity peak in the reflected beam.
As a paraxial wave packet is reflected or refracted from a planar interface separating two material media, it experiences spatial and angular shifts of its center position with respect to predictions of the geometrical ray picture. These in-plane and out-of-plane beam shifts are known as Goos-Hänchen and Imbert-Fedorov shifts, respectively. We discover a universal link between the phase-space nonseparability of an incident wave packet of any degree of spatial coherence and the reflected beam shifts. We unveil coherence Goos-Hänchen and coherence Hall effects, absent in the fully coherent limit. While the former effect can trigger a pronounced enhancement of the spatial Goos-Hänchen shift, the latter enables control of the spatial Imbert-Fedorov shift, from complete cancellation at a certain incidence angle to dramatic enhancement of the shift to giant magnitudes for nearly incoherent incident wave packets. Our results are equally applicable to optical, x-ray, and neutron, as well as matter waves, and they showcase novel phenomena in wave-matter interactions.