- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Electron vortex beams in nonuniform magnetic fields

Abhijeet Melkani* and S. J. van Enk†

  • Department of Physics and Oregon Center for Optical, Molecular, and Quantum Science, University of Oregon, Eugene, Oregon 97403-1274, USA

  • *amelkani@uoregon.edu
  • †svanenk@uoregon.edu
PDF

Phys. Rev. Research 3, 033060 – Published 16 July, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.033060

Abstract

We consider the quantum theory of paraxial nonrelativistic electron beams in nonuniform magnetic fields, such as the Glaser field. We find the wave function of an electron from such a beam and show that it is a joint eigenstate of two (z-dependent) commuting gauge-independent operators. This generalized Laguerre-Gaussian vortex beam has a phase that is shown to consist of two parts, each being proportional to the eigenvalue of one of the two conserved operators and each having different symmetries. We also describe the dynamics of the angular momentum and cross-sectional area of any mode and how a varying magnetic field can split a mode into a superposition of modes. By a suitable change in the frame of reference, all of our analysis also applies to an electron in a quantum Hall system with a time-dependent magnetic field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. P. Hawkes and E. Kasper, Principles of Electron Optics, 2nd ed. (Elsevier, Amsterdam, 2018).
  2. P. Grivet, Electron Optics, 2nd ed. (Elsevier, Amsterdam, 1972).
  3. W. Glaser, Strenge berechnung magnetischer linsen der feldform H=H0/(1+(z/a))2, Z. Phys. 117, 285 (1941).
  4. W. Glaser, Grundlagen der Elektronenoptik (Springer, Berlin, 2013).
  5. K. Y. Bliokh, Y. P. Bliokh, S. Savel'ev, and F. Nori, Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices, Phys. Rev. Lett. 99, 190404 (2007).
  6. J. Verbeeck, H. Tian, and P. Schattschneider, Production and application of electron vortex beams, Nature (London) 467, 301 (2010).
  7. M. Uchida and A. Tonomura, Generation of electron beams carrying orbital angular momentum, Nature (London) 464, 737 (2010).
  8. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Electron vortex beams with high quanta of orbital angular momentum, Science 331, 192 (2011).
  9. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Overcoming the Rayleigh Criterion Limit with Optical Vortices, Phys. Rev. Lett. 97, 163903 (2006).
  10. J. Verbeeck, P. Schattschneider, S. Lazar, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, and G. Van Tendeloo, Atomic scale electron vortices for nanoresearch, Appl. Phys. Lett. 99, 203109 (2011).
  11. J. Rusz, S. Muto, J. Spiegelberg, R. Adam, K. Tatsumi, D. E. Bürgler, P. M. Oppeneer, and C. M. Schneider, Magnetic measurements with atomic-plane resolution, Nat. Commun. 7, 12672 (2016).
  12. P. Schattschneider, B. Schaffer, I. Ennen, and J. Verbeeck, Mapping spin-polarized transitions with atomic resolution, Phys. Rev. B 85, 134422 (2012).
  13. S. Lloyd, M. Babiker, and J. Yuan, Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter, Phys. Rev. Lett. 108, 074802 (2012).
  14. P. Schattschneider, S. Löffler, and J. Verbeeck, “Comment On Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter,” Phys. Rev. Lett. 110, 189501 (2013).
  15. J. Rusz, J.-C. Idrobo, and S. Bhowmick, Achieving Atomic Resolution Magnetic Dichroism by Controlling the Phase Symmetry of an Electron Probe, Phys. Rev. Lett. 113, 145501 (2014).
  16. J. Yuan, S. M. Lloyd, and M. Babiker, Chiral-specific electron-vortex-beam spectroscopy, Phys. Rev. A 88, 031801(R) (2013).
  17. D. Pohl, S. Schneider, J. Rusz, and B. Rellinghaus, Electron vortex beams prepared by a spiral aperture with the goal to measure EMCD on ferromagnetic films via STEM, Ultramicroscopy 150, 16 (2015).
  18. P. Schattschneider, S. Löffler, M. Stöger-Pollach, and J. Verbeeck, Is magnetic chiral dichroism feasible with electron vortices? Ultramicroscopy 136, 81 (2014).
  19. T. Schachinger, S. Löffler, A. Steiger-Thirsfeld, M. Stöger-Pollach, S. Schneider, D. Pohl, B. Rellinghaus, and P. Schattschneider, EMCD with an electron vortex filter: Limitations and possibilities, Ultramicroscopy 179, 15 (2017).
  20. A. Edström, A. Lubk, and J. Rusz, Elastic Scattering of Electron Vortex Beams in Magnetic Matter, Phys. Rev. Lett. 116, 127203 (2016).
  21. A. Edström, A. Lubk, and J. Rusz, Magnetic effects in the paraxial regime of elastic electron scattering, Phys. Rev. B 94, 174414 (2016).
  22. G. M. Gallatin and B. McMorran, Propagation of vortex electron wave functions in a magnetic field, Phys. Rev. A 86, 012701 (2012).
  23. C. R. Greenshields, R. L. Stamps, and S. Franke-Arnold, Vacuum Faraday effect for electrons, New J. Phys. 14, 103040 (2012).
  24. E. Karimi, L. Marrucci, V. Grillo, and E. Santamato, Spin-to-Orbital Angular Momentum Conversion and Spin-Polarization Filtering in Electron Beams, Phys. Rev. Lett. 108, 044801 (2012).
  25. R. G. Littlejohn and S. Weigert, Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field, Phys. Rev. A 48, 924 (1993).
  26. Y. Aharonov and A. Stern, Origin of the Geometric Forces Accompanying Berry's Geometric Potentials, Phys. Rev. Lett. 69, 3593 (1992).
  27. G. Guzzinati, P. Schattschneider, K. Y. Bliokh, F. Nori, and J. Verbeeck, Observation of the Larmor and Gouy Rotations with Electron Vortex Beams, Phys. Rev. Lett. 110, 093601 (2013).
  28. A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices, Phys. Rev. Lett. 111, 173902 (2013).
  29. L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy, Ultramicroscopy 88, 85 (2001).
  30. A. Béché, R. Van Boxem, G. Van Tendeloo, and J. Verbeeck, Magnetic monopole field exposed by electrons, Nat. Phys. 10, 26 (2014).
  31. I. P. Ivanov, D. Seipt, A. Surzhykov, and S. Fritzsche, Double-slit experiment in momentum space, Europhys. Lett. 115, 41001 (2016).
  32. P. Schattschneider, T. Schachinger, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, K. Y. Bliokh, and F. Nori, Imaging the dynamics of free-electron Landau states, Nat. Commun. 5, 4586 (2014).
  33. H. Batelaan, T. J. Gay, and J. J. Schwendiman, Stern-Gerlach Effect for Electron Beams, Phys. Rev. Lett. 79, 4517 (1997).
  34. G. A. Gallup, H. Batelaan, and T. J. Gay, Quantum-Mechanical Analysis of a Longitudinal Stern-Gerlach Effect, Phys. Rev. Lett. 86, 4508 (2001).
  35. T. R. Harvey, V. Grillo, and B. J. McMorran, Stern-Gerlach-like approach to electron orbital angular momentum measurement, Phys. Rev. A 95, 021801(R) (2017).
  36. P. Schattschneider, V. Grillo, and D. Aubry, Spin polarisation with electron Bessel beams, Ultramicroscopy 176, 188 (2017).
  37. E. Karimi, V. Grillo, R. W. Boyd, and E. Santamato, Generation of a spin-polarized electron beam by multipole magnetic fields, Ultramicroscopy 138, 22 (2014).
  38. V. Grillo, L. Marrucci, E. Karimi, R. Zanella, and E. Santamato, Quantum simulation of a spin polarization device in an electron microscope, New J. Phys. 15, 093026 (2013).
  39. D. Stoler, Operator methods in physical optics, J. Opt. Soc. Am. 71, 334 (1981).
  40. S. J. van Enk and G. Nienhuis, Eigenfunction description of laser beams and orbital angular momentum of light, Opt. Commun. 94, 147 (1992).
  41. M. Szilágyi, Electron and Ion Optics (Springer, New York, 1998).
  42. Y. Kitadono, M. Wakamatsu, L. Zou, and P. Zhang, Role of guiding center in Landau level system and mechanical and pseudo orbital angular momenta, Int. J. Mod. Phys. A 35, 2050096 (2020).
  43. S. J. van Enk, Angular momentum in the fractional quantum Hall effect, Am. J. Phys. 88, 286 (2020).
  44. M. Wakamatsu, Y. Kitadono, L. Zou, and P. Zhang, The physics of helical electron beam in a uniform magnetic field as a testing ground of gauge principle, Phys. Lett. A 384, 126415 (2020).
  45. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, and M. O. Scully, Photons & Atoms: Introduction to Quantum Electrodynamics (Wiley-VCH, Weinheim, 1992).
  46. C. R. Greenshields, R. L. Stamps, S. Franke-Arnold, and S. M. Barnett, Is the Angular Momentum of an Electron Conserved in a Uniform Magnetic Field? Phys. Rev. Lett. 113, 240404 (2014).
  47. H. R. Lewis and W. B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 (1969).
  48. P. G. L. Leach and K. Andriopoulos, The Ermakov equation: A commentary, Appl. Anal. Discr. Math. 2, 146 (2008).
  49. E. Pinney, The nonlinear differential equation y′′+p(x)y+cy−3=0, Proc. Am. Math. Soc. 1, 681 (1950).
  50. L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 1999), Vol. 39, pp. 291–372.
  51. S. Menouar, M. Maamache, and J. R. Choi, The time-dependent coupled oscillator model for the motion of a charged particle in the presence of a time-varying magnetic field, Phys. Scr. 82, 065004 (2010).
  52. K. Y. Bliokh, P. Schattschneider, J. Verbeeck, and F. Nori, Electron Vortex Beams in a Magnetic Field: A New Twist on Landau Levels and Aharonov-Bohm States, Phys. Rev. X 2, 041011 (2012).
  53. H. R. Lewis, Class of exact invariants for classical and quantum time-dependent harmonic oscillators, J. Math. Phys. 9, 1976 (1968).
  54. C. J. Eliezer and A. Gray, A note on the time-dependent harmonic oscillator, SIAM J. Appl. Math. 30, 463 (1976).
  55. L. Landau, Diamagnetismus der metalle, Z. Phys. 64, 629 (1930).
  56. K. Y. Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. Béché, R. Juchtmans, M. A. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, Theory and applications of free-electron vortex states, Phys. Rep. 690, 1 (2017).
  57. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature (London) 496, 196 (2013).
  58. Y. Lumer, M. A. Bandres, M. Heinrich, L. J. Maczewsky, H. Herzig-Sheinfux, A. Szameit, and M. Segev, Light guiding by artificial gauge fields, Nat. Photon. 13, 339 (2019).
  59. H. Abbaszadeh, A. Souslov, J. Paulose, H. Schomerus, and V. Vitelli, Sonic Landau Levels and Synthetic Gauge Fields in Mechanical Metamaterials, Phys. Rev. Lett. 119, 195502 (2017).
  60. J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011).
  61. R. Jagannathan, Quantum theory of electron lenses based on the Dirac equation, Phys. Rev. A 42, 6674 (1990).

Outline

Information

Phys. Rev. Research 3, 033060– Published 16 July, 2021
  • Received 25 November 2020
  • Accepted 28 June 2021
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation