Controlling electron spin dynamics in bichromatic Kapitza-Dirac scattering by the laser field polarization

Spin-dependent Kapitza-Dirac scattering of electron beams from counterpropagating bichromatic laser waves in various polarization geometries is studied. The corresponding scattering probabilities are obtained by analytical and numerical solutions of the time-dependent Dirac equation, assuming a field frequency ratio of 2. When the fundamental field mode is circularly polarized, we show that spin dynamics are generally suppressed at low intensities but can become distinct at high intensities. Conversely, when a linearly or elliptically polarized fundamental mode is combined with a second harmonic of circular polarization, strong spin effects arise already at low field intensities. In particular, a polarization configuration is identified which acts as a spin filter for free electrons.

Figure 1

Final occupation probabilities $|c_{-2}^{\uparrow }{\left(T\right)|}^2$ (green solid line) and $|c_2^{\downarrow }{\left(T\right)|}^2$ (black dashed line) for KD scattering from bichromatic counterpropagating laser waves of linear polarization as a function of the interaction time $T$. The field parameters are $\omega =2\times {10}^3$ eV, $e{a}_1=5.656\times {10}^4$ eV, and $e{a}_2=2\times {10}^4$ eV, corresponding to a combined laser intensity of $I=6.54\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$. The electron is incident with longitudinal momentum of $p_z=-2k$ and transverse momentum of $p_x=0$.

Figure 2

Final occupation probabilities $|c_{-2}^{\downarrow }{\left(T\right)|}^2$ (blue dash-dotted line) and $|c_2^{\uparrow }{\left(T\right)|}^2$ (red dotted line) for KD scattering from bichromatic counterpropagating laser waves with corotating circular polarization as a function of the interaction time $T$. The field parameters are $\omega =2\times {10}^3$ eV, $e{a}_1=e{a}_2=8\times {10}^4$ eV. The combined laser intensity is $I=4.36\times {10}^{23}\mathrm{W}/{\mathrm{cm}}^2$. The initial electron momentum components are the same as in Fig. 1.

Figure 3

Schematic diagram of the maximal Rabi amplitude $C$ [see Eq. (15)] in the $a_1^2{a}_2-\omega$ plane, indicating various parameter regimes, when both waves are corotating circularly polarized. The dyke summit is located approximately at $e^3{a}_1^2{a}_2\sim {\omega }^2{10}^6$ eV (order of magnitude).

Figure 4

Final occupation probabilities $|c_{-2}^{\uparrow }{\left(T\right)|}^2$ (green solid line) and $|c_2^{\downarrow }{\left(T\right)|}^2$ (black dashed line) for KD scattering from a laser field consisting of a linearly polarized wave with ${\omega }_1=2\times {10}^3$ eV, $e{a}_1=5.656\times {10}^4$ eV, and a counterpropagating circularly polarized wave with ${\omega }_2=4\times {10}^3$ eV, $e{a}_2=2\times {10}^4$ eV. The combined laser intensity is $I=6.54\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$. The electron is incident with longitudinal momentum of $p_z=-2k$ and transverse momentum of $p_x=0$.

Figure 5

Same as Fig. 4, but with $p_x=k$. Due to the nonzero transverse momentum, the electron states $|2,\uparrow \rangle$ and $|-2,\downarrow \rangle$ are also populated. The corresponding occupation probabilities are shown by the red dotted and blue dash-dotted lines, respectively.

Figure 6

Same as Fig. 5, but the fundamental laser wave is elliptically polarized with the polarization vector given by ${\vec{\epsilon }}_1=\frac{1}{\sqrt{26}}({\vec{\mathrm{e}}}_x+5i{\vec{\mathrm{e}}}_y)$. As a consequence, only the states $|-2,\uparrow \rangle$ and $|2,\downarrow \rangle$ couple with each other.

Figure 7

Scheme of the spin-polarizing interferometric beam splitter, as introduced in [13]. An incident electron beam undergoes three stages of KD scattering. As a result, the outgoing electron beam is separated into its spin components $|+\rangle$ and $|-\rangle$ along the laser magnetic field, in close analogy to the Stern-Gerlach effect.

Figure 8

Dependence on the phase parameter $\chi$ [see Eq. (17)] of the polarization degree of the outgoing electron beam with positive momentum, as obtained by the spin-polarizing interferometric beam splitter of Ref. [13]. The black crosses present the numerical data, whereas the dotted line is an analytical curve of the form $P_{0}sin(\chi +\delta )$, fitted over the parameters $P_0$ and $\delta$. The outgoing portion of the electron beam with negative momentum has the opposite degree of polarization.