Under-the-Barrier Dynamics in Laser-Induced Relativistic Tunneling

The tunneling dynamics in relativistic strong-field ionization is investigated with the aim to develop an intuitive picture for the relativistic tunneling regime. We demonstrate that the tunneling picture applies also in the relativistic regime by introducing position dependent energy levels. The quantum dynamics in the classically forbidden region features two time scales, the typical time that characterizes the probability density’s decay of the ionizing electron under the barrier (Keldysh time) and the time interval which the electron spends inside the barrier (Eisenbud-Wigner-Smith tunneling time). In the relativistic regime, an electron momentum shift as well as a spatial shift along the laser propagation direction arise during the under-the-barrier motion which are caused by the laser magnetic field induced Lorentz force. The momentum shift is proportional to the Keldysh time, while the wave-packet’s spatial drift is proportional to the Eisenbud-Wigner-Smith time. The signature of the momentum shift is shown to be present in the ionization spectrum at the detector and, therefore, observable experimentally. In contrast, the signature of the Eisenbud-Wigner-Smith time delay disappears at far distances for pure quasistatic tunneling dynamics.

Figure 1

(a) Density plot of the asymptotic spin-averaged electron momentum distribution at the detector. The distribution’s maximum (white cross) is shifted from $p_{f,z}=0$ (nonrelativistic limit, white line) to $p_{f,z}=I_p/(3c)$. (b) The distribution of the electron’s kinetic momentum in the laser propagation direction at the tunnel exit at time $t_0$, with maximal laser electric field, in the electric dipole approximation (green, dotted line), including the magnetic dipole correction (blue, dashed line), and the magnetic dipole plus the mass-shift correction (red, dot-dashed line) and fully relativistically via the Dirac equation in the SFA (black, solid line). The applied parameters are $I_p/c^2=0.25$ and $E_0/E_a=1/30$.

Figure 2

Scheme of laser induced tunneling in a Coulomb potential with $I_p/c^2=0.25$. The potential barrier $V(x)-x{E}_0$ (black line) is displayed with $E_0/E_a=1/30$ and the relativistic and nonrelativistic modified energy levels. The dotted green line indicates the description in electric dipole approximation, whereas magnetic dipole effects are taken into account for the solid red line. The momenta $p_y$ and $p_z$ are chosen such that the WKB tunneling probability is maximal.

Figure 3

(a) The scaled time delay vs the coordinate along the laser polarization direction, via a quantum mechanical (black, solid line) and a quasiclassical description (gray, dashed line). (b) The electron wave packet trajectory in the $x\mathrm{\text{−}}z$-plane, via a quantum mechanical (black, solid line) and a quasiclassical description (gray, dashed line). In both subfigures $x/x_e<1$ corresponds to under-the-barrier motion with $x_e=I_p/E_0$ denoting the tunnel exit coordinate. The applied parameters are $I_p/c^2=0.25$, $E_0/E_a=1/30$, and the most probable initial kinetic momenta $p_{z,\mathrm{kin}}(x_i)=-2I_p/(3c)$ and $p_y=0$.

Figure 4

Numerical simulation of the electron wave function in a soft-core potential via the Dirac equation. The density plot shows the electron density at the moment when maximal laser field strength is attained. The solid black line indicates the maximum of the density in the laser propagation direction while the dashed line corresponds to the most probable trajectory resulting from the quasiclassical description. Solid green lines correspond to the border of the classical forbidden region. White arrows and the cross indicate the directions of the laser’s electromagnetic fields and its propagation direction. The inset shows a scale-up of the region close to the tunnel exit. The applied parameters are $I_p/c^2=0.25$, $E_0/E_a=1/30$, and $\gamma =0.1$.