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Physical Review A

Extended theory of Kapitza-Dirac scattering

Leonard Rosenberg

Published Aug 2, 2004• 14 citations
Publisher

Abstract

The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between $S$-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.

DOI
10.1103/physreva.70.023401

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The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between SSS-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.
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.
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.
A relativistic description of the Kapitza-Dirac effect in the so-called Bragg regime with two and three interacting photons is presented by investigating both numerical and perturbative solutions of the Dirac equation in momentum space. We demonstrate that spin flips can be observed in the two-photon and the three-photon Kapitza-Dirac effects for certain parameters. During the interaction with the laser field the electron's spin is rotated, and we give explicit expressions for the rotation axis and the rotation angle. The off-resonant Kapitza-Dirac effect, that is, when the Bragg condition is not exactly fulfilled, is described by a generalized Rabi theory. We also analyze the in-field quantum dynamics as obtained from the numerical solution of the Dirac equation.
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.
Many calculations in strong-field quantum field theory are carried out by using a simple field geometry, often neglecting the spatial field envelope. In this paper, we simulate the electron-diffraction quantum dynamics of the Kapitza-Dirac effect in a Gaussian beam standing light wave. The two-dimensional simulation is computed in a relativistic framework, by solving the Dirac equation with the fast Fourier-transform split operator method. Except the numerical propagation method, our results are obtained without applying approximations and demonstrate that a spin flip in the Kapitza-Dirac effect is possible. We further discuss properties such as the validity of a plane-wave approach for the theoretical description, the influence of the longitudinal polarization component due to laser beam focusing, and higher-order diffraction peaks in Kapitza-Dirac scattering.
We present the possibility of spin-dependent Kapitza-Dirac scattering based on a two-photon interaction only. The interaction scheme is inspired from a Compton scattering process, for which we explicitly show the mathematical correspondence to the spin dynamics of an electron diffraction process in a standing light wave. The spin effect has the advantage that it already appears in a Bragg scattering setup with arbitrary low field amplitudes, for which we have estimated the diffraction count rate in a realistic experimental setup at available x-ray free-electron laser facilities.