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Electron matter optics and the Kapitza -Dirac effect

Daniel L. Freimund

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.

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The Kapitza-Dirac effect is the diffraction of a well-collimated particle beam by a standing wave of light. Why is this interesting? Comparing this situation to the introductory physics textbook example of diffraction of a laser beam by a grating, the particle beam plays the role of the incoming wave and the standing light wave the role of the material grating, highlighting particle-wave duality. Apart from representing such a beautiful example of particle-wave duality, the diffracted particle beams are coherent. This allows the construction of matter interferometers and explains why the Kapitza-Dirac effect is one of the workhorses in the field of atom optics. Atom optics concerns the manipulation of atomic waves in ways analogous to the manipulation of light waves with optical elements. The excitement and activity in this new field of physics stems in part from the realization that the shorter de Broglie wavelengths of matter waves allow ultimate sensitivities for diffractive and interferometric experiments that in principle would far exceed their optical analogues. Not only is the Kapitza-Dirac effect an important enabling tool for this field of physics, but diffraction peaks have never been observed for electrons, for which it was originally proposed in 1933. Why has this not been observed? What is the relation between the interaction of laser light with electrons and the interaction of laser light with atoms, or in other words what is the relation between the ponderomotive potential and the lightshift potential? Would it be possible to build interferometers using the Kapitza-Dirac effect for other particles? These questions will be addressed in this paper.

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Investigation of the Kapitza-Dirac effect in the relativistic regime

Sven Ahrens

Quantum mechanical diffraction is of particular interest, because it contradicts our everyday life experience. \nThis theoretical work considers the diffraction of electrons at standing waves of light, referred \nto as the Kapitza-Dirac effect. The work focuses on a special version of a Kapitza-Dirac effect in which \nthe electron interacts with three photons. The particular property of this 3-photon Kapitza-Dirac effect \nis, that the electron spin is rotated. \nThis work considers different relativistic and non-relativistic quantum mechanical wave equations \nwhich are described in momentum space. On one hand, the quantum dynamics of the diffracted electrons \nis solved numerically in momentum space and the properties of the 3-photon Kapitza-Dirac effect \nare investigated in detail. On the other hand, the quantum dynamics is solved via time-dependent \nperturbation theory and is compared with the numerical results. \nIn contrast to the originally proposed Kapitza-Dirac effect with two interacting photons, the number \nof absorbed and emitted photons by the electron is not equal for the 3-photon Kapitza-Dirac effect. \nTherefore, the diffraction process only appears for relativistic electron momenta in laser propagation \ndirection. Furthermore, a very high field strength of the laser beam is required for driving the Kapitza- \nDirac effect with a measurable diffraction probability. The electron spin is rotated along the axis of the \nmagnetic field of the laser beam, when it undergoes the diffraction process. The rotation angle of the \nspin rotation depends on the electron momentum component in laser polarization direction. Therefore, \nthe probability for flipping the electron spin can be tuned by choosing the electron momentum in \nthe direction of the laser polarization. An experimental investigation may by established by utilizing \nfuture X-ray laser facilities.

•Jan 1, 2012•7 citations
The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.
The Kapitza-Dirac effect is the diffraction of a well-collimated particle beam by a standing wave of light. Why is this interesting? Comparing this situation to the introductory physics textbook example of diffraction of a laser beam by a grating, the particle beam plays the role of the incoming wave and the standing light wave the role of the material grating, highlighting particle-wave duality. Apart from representing such a beautiful example of particle-wave duality, the diffracted particle beams are coherent. This allows the construction of matter interferometers and explains why the Kapitza-Dirac effect is one of the workhorses in the field of atom optics. Atom optics concerns the manipulation of atomic waves in ways analogous to the manipulation of light waves with optical elements. The excitement and activity in this new field of physics stems in part from the realization that the shorter de Broglie wavelengths of matter waves allow ultimate sensitivities for diffractive and interferometric experiments that in principle would far exceed their optical analogues. Not only is the Kapitza-Dirac effect an important enabling tool for this field of physics, but diffraction peaks have never been observed for electrons, for which it was originally proposed in 1933. Why has this not been observed? What is the relation between the interaction of laser light with electrons and the interaction of laser light with atoms, or in other words what is the relation between the ponderomotive potential and the lightshift potential? Would it be possible to build interferometers using the Kapitza-Dirac effect for other particles? These questions will be addressed in this paper.
Abstract In their famous 1927 experiment, Davisson and Germer observed1 the diffraction of electrons by a periodic material structure, so showing that electrons can behave like waves. Shortly afterwards, Kapitza2 and Dirac3 predicted that electrons should also be diffracted by a standing light wave4. This Kapitza–Dirac effect is analogous to the diffraction of light by a grating, but with the roles of the wave and matter reversed. The electron and the light grating interact extremely weakly, via the ‘ponderomotive potential’5, so attempts to measure the Kapitza–Dirac effect had to wait for the development of the laser. The idea6 that the underlying interaction with light is resonantly enhanced for electrons in an atom led to the observation7 that atoms could be diffracted by a standing wave of light. Deflection of electrons by high-intensity laser light, which is also a consequence of the Kapitza–Dirac effect, has also been demonstrated8. But the coherent interference that characterizes wave diffraction has not hitherto been observed9, 10. Here we report the diffraction of free electrons from a standing light wave—a realization of the Kapitza–Dirac effect as originally proposed.
The Bragg diffraction of electrons off of a standing electromagnetic wave, first predicted by Kapitza and Dirac, is discussed. The original derivation of Kapitza and Dirac is reviewed. This is followed by a more general derivation, which uses quantum mechanical, Born approximation, scattering theory. A review of the recent experiment results is given. Because the Kapitza-Dirac effect reverses the usual roles of “particles” and “waves” in Bragg diffraction, the discussion emphasizes the physical insight that this phenomenon gives into light and the particle-wave duality of quantum mechanics.
We report on the possibility of diffracting electrons from light waves traveling inside a dielectric medium. We show that, in the frame of reference which moves with the group velocity of light, the traveling wave acts as a stationary diffraction grating from which electrons can diffract, similar to the conventional Kapitza–Dirac effect. To characterize the Kapitza–Dirac effect with traveling light waves , we make use of the Hamiltonian Analogy between electron optics and quantum mechanics and apply the Helmholtz–Kirchhoff theory of diffraction. Introduction and background. The Kapitza-Dirac effect, i.e. the diffraction of electrons from a periodic light wave, has been predicted in the 1930's. Only now, with the advent of sufficiently intense lasers, it became possible to observe the effect experimentally. Commonly a standing wave is used to create the diffraction grating, but also other configurations, involving counter propagating waves of different frequencies have been considered in the past. In this paper, we address the question whether it is possible to observe the effect with a single light beam, travelling with a reduced group velocity within a dielectric medium. Main results. We confirm that the Kapitza-Dirac effect can be observed with a single beam of light, provided the longitudinal velocity of the electrons is commensurate with the reduced group velocity. Using Lorentz transformations between a laboratory and reference frame associated with the travelling light field, we show that in this frame of reference the light acts as a stationary diffraction grating for electrons. In contrast to a standing wave, a single, travelling light wave can give rise to inelastic diffraction when electrons exchanges energy with the light field. Wider implications. Diffraction phenomena occur for all kinds of waves and represent a signature of fundamental interference processes. Our extension of the standard Kapitza-Dirac effect to the case of a single light beam enlarges the scope to observe such effects, and might lead to simplified schemes in interferometry and microscopy. Figure. Geometry of the electron diffraction in the laboratory (a) and wave (b) frames of reference. Electrons overtake the slowed light and experience diffraction for specific angles (c). The "butterfly wings" represent the regions where the Kapitza-Dirac effect with traveling waves occurs (c).
Quantum mechanical diffraction is of particular interest, because it contradicts our everyday life experience. \nThis theoretical work considers the diffraction of electrons at standing waves of light, referred \nto as the Kapitza-Dirac effect. The work focuses on a special version of a Kapitza-Dirac effect in which \nthe electron interacts with three photons. The particular property of this 3-photon Kapitza-Dirac effect \nis, that the electron spin is rotated. \nThis work considers different relativistic and non-relativistic quantum mechanical wave equations \nwhich are described in momentum space. On one hand, the quantum dynamics of the diffracted electrons \nis solved numerically in momentum space and the properties of the 3-photon Kapitza-Dirac effect \nare investigated in detail. On the other hand, the quantum dynamics is solved via time-dependent \nperturbation theory and is compared with the numerical results. \nIn contrast to the originally proposed Kapitza-Dirac effect with two interacting photons, the number \nof absorbed and emitted photons by the electron is not equal for the 3-photon Kapitza-Dirac effect. \nTherefore, the diffraction process only appears for relativistic electron momenta in laser propagation \ndirection. Furthermore, a very high field strength of the laser beam is required for driving the Kapitza- \nDirac effect with a measurable diffraction probability. The electron spin is rotated along the axis of the \nmagnetic field of the laser beam, when it undergoes the diffraction process. The rotation angle of the \nspin rotation depends on the electron momentum component in laser polarization direction. Therefore, \nthe probability for flipping the electron spin can be tuned by choosing the electron momentum in \nthe direction of the laser polarization. An experimental investigation may by established by utilizing \nfuture X-ray laser facilities.