Spin Dynamics in the Kapitza-Dirac Effect

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.

Figure 1

Schematic setup. An electron with momentum $\mathbf{p}$ is incident at an angle $\vartheta$ on a linearly polarized standing laser wave with electric and magnetic components $\mathbf{E}$ and $\mathbf{B}$. The momentum $\mathbf{p}$ has components ${\mathbf{p}}_E$ and ${\mathbf{p}}_k$ along the laser’s electric field $\mathbf{E}$ and wave vector $\mathbf{k}$, respectively. The electron is initially spin-polarized along the electric field component. After Kapitza-Dirac scattering, parts of the electron wave packet may have flipped their spin orientation.

Figure 2

Rabi oscillations of the spin resolved diffraction probabilities as a function of the interaction time $T$ for a three-photon Kapitza-Dirac effect. Starting from a pure spin-up state the electron is either diffracted with probability $|c_3^{+\uparrow }(T){|}^2+|c_3^{+\downarrow }(T){|}^2$ or passes the laser beam without momentum transfer with probability $|c_0^{+\uparrow }(T){|}^2$. Laser parameters of this simulation correspond to two counterpropagating x-ray laser beams with a peak intensity of $2.0\times {10}^{23}\mathrm{W}/{\mathrm{cm}}^2$ each and a photon energy of 3.1 keV. The electron impinges at an angle of inclination of $\vartheta =0.4°$ and a momentum of $176\mathrm{keV}/c$, in order to fulfill the Bragg condition (5).

Figure 3

The spin-flip probability $P_{\mathrm{flip}}$ as a function of the electron momentum $|{\mathbf{p}}_E|$ in electric field direction. The solid black line is given by Eq. (8). White squares result from simulations with the Dirac Eq. (4). All simulation parameters except ${\mathbf{p}}_E$ are the same as those for Fig. 2.

Figure 4

Panel (a): The Rabi frequency ${\Omega }_R$ as a function of the electron momentum $|{\mathbf{p}}_E|$ in electric field direction for the Dirac equation (squares) and the Klein-Gordon equation (triangles). The solid black line is given by Eq. (9) and the dashed black line is given by Eq. (12). Panel (b): The diffraction probability after an interaction time of 0.36 fs for particles with and without spin for parameters as in Fig. 2. The light (dark) gray bars represent the spin-up (spin-down) probabilities. In the case of the Klein-Gordon equation there is no spin degree of freedom and, therefore, no dark gray bars appear. Note that the diffraction probability depends on the spin degree of freedom.