Spin effects in Kapitza-Dirac scattering at light with elliptical polarization

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.

Figure 1

Schematic illustration of the two considered field configurations. Two electromagnetic plane waves with elliptical polarization traveling into opposite directions are superimposed. The sense of rotation of the electric field vector may be (top) equal (corotating setup) or (bottom) opposite (antirotating setup). The sense of rotation and the propagation direction are indicated by black arrows, red solid arrows represent electric-field components, and blue dashed arrows represent magnetic-field components.

Figure 2

Time evolution of the probabilities to find the electron, which interacts with a standing light wave of the corotating setup, in a particular quantum state. Green (gray) lines show, from top to bottom, the analytical results (32a) to (32d) based on the relativistic Pauli equation (17). Black lines represent the corresponding numerical results based on the Dirac equation (5), i.e., the probabilities to find the electron in the states ${\psi }_{-1}^{+\uparrow },{\psi }_{-1}^{+\downarrow },{\psi }_1^{+\uparrow }$, and ${\psi }_1^{+\downarrow }$, respectively, after the electromagnetic field has been turned off. Parameters are $\widehat{E}=400\mathrm{a}.\mathrm{u}.=2.06\times {10}^{14}\mathrm{V}/\mathrm{m}$ (corresponding to an intensity of $1.12\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$), $\lambda =3\mathrm{a}.\mathrm{u}.=0.159\mathrm{nm},\mathrm{\Delta }T=10\pi /\omega$, and $\eta =\pi /2$, corresponding to circular polarization.

Figure 3

Time evolution of the spin of the two diffraction modes. Green (gray) lines show, from top to bottom, the analytical results (42) and (43) based on the relativistic Pauli equation (17). Black lines show the corresponding numerical results based on the Dirac equation (5). Parameters are as in Fig. 2.

Figure 4

Time evolution of the electron's spin. Green (gray) line shows the analytical result (40) based on the relativistic Pauli equation (17). Black line shows the corresponding numerical data based on the Dirac equation (5). Parameters are as in Fig. 2.

Figure 5

Spin-oscillation frequencies for different degrees of ellipticity $\eta$ as obtained by a numerical fit procedure to data of numerical solutions of the Dirac equation compared to the predictions by the relativistic Pauli equation (17). Parameters are as in Fig. 2.

Figure 6

Rabi oscillation frequencies for different degrees of ellipticity $\eta$ for the antirotating setup as obtained by a numerical fit procedure to data of numerical solutions of the Dirac equation compared to the predictions by the relativistic Pauli equation (17). Parameters are as in Fig. 2.