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Physical Review A

Investigation of electron spin dynamic in the bichromatic Kapitza-Dirac effect via frequency ratio and amplitude of laser beams

Asma Ebadati, Mohsen Vafaee, Babak Shokri

Published Nov 27, 2019• 4 citations
Publisher

Abstract

We discuss electron diffraction from two standing waves with two different frequencies. The effects of increasing the frequency of the second laser beam and changing the laser amplitudes on the form and period of the Rabi oscillation are studied theoretically. The corresponding scattering probabilities for certain incident electron momenta are obtained by an analytical Rabi matrix and numerical solution of the Dirac equation. We show that at high intensities, $\ensuremath{\ge}{10}^{20}$ W ${\mathrm{cm}}^{\ensuremath{-}2}$, the process with an even number of photons involved in the Kapitza-Dirac effect can be used as a spin filter for free electrons. On the other hand, the process with an odd number photons and an electron with momentum along the laser polarization preserves the initial spin of the electron.

DOI
10.1103/physreva.100.052514

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We discuss electron diffraction from two standing waves with two different frequencies. The effects of increasing the frequency of the second laser beam and changing the laser amplitudes on the form and period of the Rabi oscillation are studied theoretically. The corresponding scattering probabilities for certain incident electron momenta are obtained by an analytical Rabi matrix and numerical solution of the Dirac equation. We show that at high intensities, \ensuremath≥1020\ensuremath{\ge}{10}^{20}\ensuremath≥1020 W cm\ensuremath−2{\mathrm{cm}}^{\ensuremath{-}2}cm\ensuremath−2, the process with an even number of photons involved in the Kapitza-Dirac effect can be used as a spin filter for free electrons. On the other hand, the process with an odd number photons and an electron with momentum along the laser polarization preserves the initial spin of the electron.
We theoretically demonstrate the feasibility of producing an electron spin beam splitter using Kapitza-Dirac diffraction on bichromatic standing waves which are created by the fundamental frequency and its third harmonic. The relativistic electron in the Bragg regime absorbs three photons with a frequency of $\ensuremath{\omega}$ and emits a photon with a frequency of $3\ensuremath{\omega}$; we introduce this as the four-photon Kapitza-Dirac effect. In this four-photon Kapitza-Dirac effect distinct spin effects arise in different polarizations of the third harmonic laser beam. It is shown that the shape of the Rabi oscillation between the initial and scattered states is changed and finds two unequal peaks for all polarizations of laser beams. For the case of circular polarization of the fundamental and third harmonic, despite Rabi oscillation, the spin down electron beam in 0.23 fs intervals maintains its momentum and spin. Also we find that if a suitable combination of a linearly polarized fundamental laser beam and a third harmonic with circular polarization in the condition $\stackrel{P\vec}{p}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{A}$ term is involved, it results in a spin-dependent oscillation that preserves the spin state of the initial electron.
Coherent scattering of an electron beam by the Kapitza-Dirac effect from a standing laser wave which comprises two frequency components is studied. To this end, the Schr\"odinger equation is solved numerically with a suitable ponderomotive potential. Besides, an analytical solution for electron diffraction in the asymptotic domain of large field amplitudes is obtained and a mathematical model in reduced dimensionality for the scattering amplitude in the Bragg regime is presented. We demonstrate distinct interference signatures and relative phase effects when the standing wave contains a fundamental frequency and its second harmonic. The influence of the relative field intensities on the Rabi oscillation dynamics is also discussed.
Electron spin dynamics in Kapitza-Dirac scattering from a standing laser wave of high frequency and high intensity is studied. We develop a fully relativistic quantum theory of the electron motion based on the time-dependent Dirac equation. Distinct spin dynamics, with Rabi oscillations and complete spin-flip transitions, is demonstrated for Kapitza-Dirac scattering involving three photons in a parameter regime accessible to future high-power x-ray laser sources. The Rabi frequency and, thus, the diffraction pattern is shown to depend crucially on the spin degree of freedom.
Kapitza-Dirac scattering of nonrelativistic electrons from counterpropagating bichromatic laser waves of linear polarization is studied. The focus lies on the electronic spin dynamics in the Bragg regime when the laser fields possess a frequency ratio of 2. To this end, the time-dependent Pauli equation is solved numerically, both in coordinate space and momentum space. Our numerical results are corroborated by analytical derivations. We demonstrate that, for certain incident electron momenta, the scattering crucially relies on the electron spin which undergoes characteristic Rabi-like oscillations. A parameter regime is identified where the Rabi oscillations reach maximum amplitude. We also briefly discuss spin-dependent Kapitza-Dirac scattering of protons.
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.
We demonstrate the theoretical feasibility of spin-dependent diffraction and spin polarization of an electron in two counterpropagating, circularly polarized laser beams. The spin dynamics appears in a two-photon process of the Kapitza-Dirac effect in the Bragg regime. We show the spin dependence of the diffraction process by comparison of the time evolution of spin-up and spin-down electrons in a relativistic quantum simulation. We further discuss the spin properties of the scattering by studying an analytically approximated solution of the time-evolution matrix. A classification scheme in terms of unitary or nonunitary propagation matrices is used for establishing a generalized and spin-independent description of the spin properties in the diffraction process.