Spin dynamics in Kapitza-Dirac scattering of electrons from bichromatic laser fields

Kapitza-Dirac scattering of nonrelativistic electrons from counterpropagating bichromatic laser waves of linear polarization is studied. The focus lies on the electronic spin dynamics in the Bragg regime when the laser fields possess a frequency ratio of 2. To this end, the time-dependent Pauli equation is solved numerically, both in coordinate space and momentum space. Our numerical results are corroborated by analytical derivations. We demonstrate that, for certain incident electron momenta, the scattering crucially relies on the electron spin which undergoes characteristic Rabi-like oscillations. A parameter regime is identified where the Rabi oscillations reach maximum amplitude. We also briefly discuss spin-dependent Kapitza-Dirac scattering of protons.

Figure 1

Spin-dependent Kapitza-Dirac scattering of an electron wave packet with central momentum $-2k$ and width 0.1 ${\mathrm{eV}}^{-1}c$. The spatial occupation probabilities of the upper (incident) and lower (scattered) spinor components of the electron wave function are shown in the corresponding panels. The field parameters are $\omega ={10}^3\mathrm{eV}$ and $e{a}_1=e{a}_2=2\times {10}^4\mathrm{eV}$.

Figure 2

Probabilities for the electron to be in the incident spin state and around the incident momentum $-2k$ (solid line) as well as to be in the flipped spin state with the scattered momentum $2k$ (dotted line). The interaction parameters are the same as in Fig. 1. Note that these probabilities add up to 1 only before switching on and after switching off the fields. Also shown are the total probabilities (summed over momenta) to be in the incident spin state (dashed line) and in the flipped spin state (dash-dotted line).

Figure 3

Time evolution of the occupation probabilities $|c_{-2}^{\uparrow }{\left(t\right)|}^2$ (solid line) and $|c_2^{\downarrow }{\left(t\right)|}^2$ (dashed line) for Kapitza-Dirac scattering from bichromatic counterpropagating waves with $\omega ={10}^3\mathrm{eV}$, $e{a}_1=e{a}_2=2\times {10}^4\mathrm{eV}$. The combined laser intensity is $I=6.82\times {10}^{21}\mathrm{W}/{\mathrm{cm}}^2$. The electron is incident with a longitudinal momentum of $-2k$; its transverse momentum is oriented perpendicularly to the field polarization and has an arbitrary magnitude.

Figure 4

Final occupation probabilities $|c_{-2}^{\uparrow }{\left(T\right)|}^2$ (solid line) and $|c_2^{\downarrow }{\left(T\right)|}^2$ (dashed line) for Kapitza-Dirac scattering from bichromatic counterpropagating laser waves as a function of the interaction time $T$. The electron and field parameters are chosen as in Fig. 3. Every plotted data point corresponds to a full interaction with switching on and off of $8\times {10}^2$ laser cycles each.

Figure 5

Same as Fig. 4, but with $\omega =2\times {10}^2\mathrm{eV}$ and $e{a}_1=e{a}_2=8\times {10}^3\mathrm{eV}$, resulting in a laser intensity of $I=4.36\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$. Note that the Rabi cycles are more developed, here with $C\approx 0.855$.

Figure 6

Same as Fig. 4, but with $\omega =2.5\times {10}^2\mathrm{eV}$, $e{a}_1=4\times {10}^3\mathrm{eV}$, and $e{a}_2=2\times {10}^3\mathrm{eV}$, resulting in a laser intensity of $I=6.82\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$. Here, the Rabi cycles are almost fully developed ($C\approx 1$).

Figure 7

Schematic diagram of the maximal Rabi amplitude $C$ in the $a_1^2{a}_2\text{−}\omega$ plane, indicating various parameter regimes. The position of the dyke summit is like $e^3{a}_1^2{a}_2\approx {\omega }^2{10}^6\mathrm{eV}$ (order of magnitude).

Figure 8

Resonance behavior of the Rabi oscillation of the spin-flipping Kapitza-Dirac transition from state $|-2\uparrow \rangle$ to state $|2\downarrow \rangle$. The laser parameters read $\omega ={10}^3\mathrm{eV}$ and $e{a}_1=e{a}_2=2.4\times {10}^4\mathrm{eV}$. The Rabi amplitude $C$ is shown (crosses) with a fit of Eq. (A4) over $\mathrm{\Delta }$ and ${\mathrm{\Omega }}_R$ (dashed line). Additionally, the measured frequency $\mathrm{\Omega }$ (pluses) is compared to Eq. (A3) (dash-dotted line) with the same fit parameters. The latter are marked on the corresponding axes.