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

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 $\omega$ and emits a photon with a frequency of $3\omega$; 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 $\vec{p}·\vec{A}$ term is involved, it results in a spin-dependent oscillation that preserves the spin state of the initial electron.

Figure 1

Sketch of dominant pathway of four-photon KD effect in bichromatic standing waves. The dispersion relation of energy and momentum is in the Bragg regime and each wiggly arrow shows a photon.

Figure 2

Temporal evolution of the occupation probability in four-photon KD effect with linear fundamental beam and counterpropagating linear third harmonic. The overall laser intensity is $8.8\times {10}^{22}\mathrm{W}c{m}^{-2}$ with wavelength $\lambda =0.6$ nm; field parameters for beam $\omega$ and $3\omega$ are $e{A}_1=6\times {10}^4$ eV and $e{A}_2=2\times {10}^4$ eV, respectively. The electron enters the laser fields with $p_z=-3k$ along the laser field direction.

Figure 3

Rabi oscillation in both linear polarization setups for vector potential mentioned in Eq. (13). All other laser and electron parameters are the same as in Fig. 2.

Figure 4

Temporal evolution of the occupation probability in four-photon KD effect with corotating circular bichromatic waves. The combined laser intensity is $19\times {10}^{23}\mathrm{W}c{m}^{-2}$ with wavelength $\lambda =0.6$ nm; field amplitude for both beams is $e{A}_1=e{A}_2=9\times {10}^4$ eV. The electron enters the laser fields with $p_z=-3k$ along the laser direction. Upper panel of the figure shows that an electron with spin up does not scatter in this setup.

Figure 5

Temporal evolution of the occupation probability in four-photon KD effect with a hybrid setup that the fundamental laser beam is linear and the third harmonic is circularly polarized. The other simulation parameters are the same as in Fig. 2. The electron just has momentum $-3k$ along the laser propagation.

Figure 6

Temporal evolution of the occupation probability in four-photon KD effect from a linear polarized fundamental laser beam and third harmonic with circularly polarization. The fundamental laser wavelength $\lambda =0.6\mathrm{nm}$ and field amplitudes are the same as in Fig. 2. The incident electron spin up with $p_x=2\hslash k$, in upper panel, after interaction is diffracted to $+3k$ momentum and maintains its spin state. For spin down electron, in lower panel, the four allowed modes are populated.