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arXiv:1803.02748 · Physical Review A

Four-photon Kapitza-Dirac effect as an electron spin filter

Asma Ebadati, Mohsen Vafaee, Babak Shokri

Published Sep 17, 2018• 8 citations
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Abstract

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.

DOI
10.1103/physreva.98.032505

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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 \ensuremathω\ensuremath{\omega}\ensuremathω and emits a photon with a frequency of 3\ensuremathω3\ensuremath{\omega}3\ensuremathω; 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.
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.
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.
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.
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.
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.
The Kapitza-Dirac effect, which refers to electron scattering at standing light waves, is studied in the Bragg regime with counterpropagating elliptically polarized electromagnetic waves with the same intensity, wavelength, and degree of polarization for two different setups. In the first setup, where the electric-field components of the counterpropagating waves have the same sense of rotation, we find distinct spin effects. The spins of the scattered electrons and of the nonscattered electrons, respectively, precess with a frequency that is of the order of the Bragg-reflection Rabi frequency. When the electric-field components of the counterpropagating waves have an opposite sense of rotation, which is the second considered setup, the standing wave has linear polarization, and no spin effects can be observed. Our results are based on numerical solutions of the time-dependent Dirac equation and the analytical solution of a relativistic Pauli equation, which accounts for the leading relativistic effects.