Spin-Polarizing Interferometric Beam Splitter for Free Electrons

A spin-polarizing electron beam splitter is described that relies on an arrangement of linearly polarized laser waves of nonrelativistic intensity. An incident electron beam is first coherently scattered off a bichromatic laser field, splitting the beam into two portions, with electron spin and momentum being entangled. Afterwards, the partial beams are coherently superposed in an interferometric setup formed by standing laser waves. As a result, the outgoing electron beam is separated into its spin components along the laser magnetic field, which is shown by both analytical and numerical solutions of Pauli’s equation. The proposed laser field configuration thus exerts the same effect on free electrons as an ordinary Stern-Gerlach magnet does on atoms.

Figure 1

Scheme of the spin-polarizing interferometric beam splitter. An incident electron beam is first coherently Bragg scattered off a bichromatic laser field with frequencies $\omega$ (red) and $2\omega$ (blue), splitting the beam into two portions. Afterwards, the latter are coherently superposed via scattering from monochromatic standing laser waves. Because of quantum interference, the outgoing electron beam is separated into its spin components along the laser magnetic field. Further details of the beam geometry are specified in the text.

Figure 2

Spatial probability density of an electron wave packet with initial central momentum $p_z=400\mathrm{eV}/c$, width $0.11\mu \mathrm{m}$, and spin along the positive $z$ axis. The projections on $|\uparrow ⟩$ in the upper and $|\downarrow ⟩$ in the lower panel are shown. At the bottom, the interaction periods with the lasers are marked. The phase parameter $\chi =-\pi /10$ was used.

Figure 3

Same as in Fig. 2, but with the wave function projected on the ${\sigma }_y$ eigenstates $|+⟩=(|\uparrow ⟩+i|\downarrow ⟩)/\sqrt{2}$ in the upper panel and $|-⟩=(|\uparrow ⟩-i|\downarrow ⟩)/\sqrt{2}$ in the lower panel.