Relativistic features and time delay of laser-induced tunnel ionization

The electron dynamics in the classically forbidden region during relativistic tunnel ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge-invariant total-energy operator. Introducing position-dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagation direction. This is compatible with results from a strong-field approximation calculation where the binding potential is assumed to be zero ranged. Further, the tunneling time delay, stemming from Wigner's definition, is investigated for model configurations of tunneling and compared with results obtained from the exact propagator. By adapting Wigner's time delay definition to the ionization process, the tunneling time is investigated in the deep-tunneling and in the near-threshold-tunneling regimes. It is shown that while in the deep-tunneling regime signatures of the tunneling time delay are not measurable at remote distance, they are detectable, however, in the latter regime.

Figure 1

Electron density (solid line) along the laser polarization direction at the instant of maximal field strength at the atomic core. The shaded area represents the classical forbidden area; gray dashed lines are exponential fits on the wave-function density. The electron's wave function $\Psi (x,z)$ is obtained by solving the two-dimensional Dirac equation for an electron in a soft-core potential interacting with an external laser pulse; laser parameters are the same as in [31].

Figure 2

Potential barrier $V_{\mathrm{barrier}}$ for quasistatic tunnel ionization. The laser field $E\left(t_0\right)=-{\kappa }^3/30$ is along the $x$ direction. The most probable tunneling path is indicated by the black dashed line.

Figure 3

Nonrelativistic tunneling probability versus the momentum along $z$ direction. The maximum tunneling probability occurs at $p_z=0$. The applied parameters are $E\left(t_0\right)=-{\kappa }^3/30$ and $\kappa =90$. Without loss of generality, $p_y=0$ was chosen.

Figure 4

Schematic picture of nonrelativistic [subfigures (a) and (b)] and relativistic [subfigure (c)] tunneling from a bound state into the continuum: (a) the atomic potential is approximated by a zero-range potential, (b) and (c) the Coulomb potential case. The potential barrier (solid, blue) and the energy levels (dashed, red) for $p_z=0$ (in the nonrelativistic cases) and for the most probable transversal momentum $p_z$ (relativistic case) are plotted against the longitudinal tunneling coordinate $x$. The shaded area can be interpreted as a measure for the tunneling probability. The electron wave packet is indicated in black.

Figure 5

Tunneling probability vs the kinetic momentum along the laser's propagation direction at the tunnel entry (lines with peak on the left) and tunnel exit (lines with peak on the right). Results for calculations including magnetic dipole correction are indicated in blue, while for the red lines also the leading relativistic correction to the kinetic energy are taken into account. The dashed-black lines correspond to a fully relativistic calculation, which are very close to the ones including leading relativistic corrections to the kinetic energy. The densities are normalized to the maximum density in the nonrelativistic case. A similar comparison was made in [31] for the case of a zero-range potential via SFA.

Figure 6

Kinetic momentum shift at the tunnel exit $q_z\left(x_e\right)$ versus the barrier suppression parameter $E_0/E_a$ for an electron bound by a Coulomb potential (dashed) and a zero-range atomic potential (solid).

Figure 7

Electronic density in the mixed space of position $x$ and kinetic momentum $q_z$ at the moment of maximal field strength at the atomic core (left panel). The electron's wave function has been obtained by simulating the tunnel ionization from a two-dimensional soft-core potential by solving the time-dependent Dirac equation with all numerical parameters as in Fig. 1. The right panel shows the normalized kinetic momentum distribution of the tunneled electron at the tunnel exit (indicated by the white line in the left panel) as obtained by solving the Dirac equation and by the WKB approximation.

Figure 8

Coordinate [subfigure (a)] and momentum [subfigure (b)] along the electric-field direction of the complex SFA trajectory for tunnel ionization from a zero-range potential. During the under-the-barrier motion the momentum is imaginary, whereas the coordinate is real. Tunneling finishes when the kinetic momentum becomes real.

Figure 9

Momentum distribution of the ionized electron: (a) in infinity at the detector; (b) at the tunnel exit, depending on the tunnel exit time $t_e$. The maximum tunneling probability occurs at $p_z=I_p/\left(3c\right)$. The applied parameters are $\kappa =90$, $E_0/E_a=1/30$, and $\omega =10$.

Figure 10

Comparison of the nonrelativistic (blue) and relativistic (black) complex SFA trajectories: (a) coordinates and (b) momentum components. The dashed and the solid lines correspond to the under-the-barrier motion and the motion after the tunneling, respectively. In the relativistic regime, the trajectory enters the tunneling barrier at $(0,0)$, it is complex under the barrier, and becomes real again when it leaves the tunneling barrier. The trajectory enters the barrier with the transversal momentum $-2I_p/\left(3c\right)$ and leaves the barrier with $I_p/\left(3c\right)$. The applied parameters are $\kappa =90$ and $E_0/E_a=1/30$.

Figure 11

Ionization momentum amplitude for $p_x=0$ vs the observation time $t$. The inset shows the momentum amplitude for large times, which coincides with the ionization amplitude. The applied parameters are $E_0/E_a=1/30$ and $\kappa =1$.

Figure 12

(a) Comparison of the Wigner trajectory (90) (solid red line) and the classical (dashed blue line) trajectory for tunneling through a square potential barrier (87). The applied parameters are $V_0=2{\varepsilon }_0$, ${\varepsilon }_0=I_p$, and $I_p=c^2-\sqrt{c^4-c^2{\kappa }^2}$ with $\kappa =90$ and $a=14/\kappa$. (b) Close-up of subfigure (a) with additionally showing the position of the wave packet's maximum.

Figure 13

Tunneling through a square potential with an additional magnetic field with $E_0={\kappa }^3/30$ and other parameters as in Fig. 12. Subfigures (a) and (b) compare the Wigner (red solid lines) and the classical (blue dashed lines) trajectories along the $x$ direction and in the $x$-$z$ plane, respectively.

Figure 14

Comparison of the Wigner trajectory (red solid line) and the classical trajectory (blue dashed line) for tunneling through a linear potential barrier (103) for $V_0=2{\varepsilon }_0$ and ${\varepsilon }_0=I_p$, with the numerical parameters $\kappa =90$ and $E_0/E_a=1/30$.

Figure 15

(a) Comparison of the Wigner trajectory (red solid line) and the classical trajectory (blue dashed line) for tunneling through the parabolic potential barrier (105) for $V_0=2{\varepsilon }_0$ and ${\varepsilon }_0=I_p$ with the numerical parameters $\kappa =90$ and $\beta =1/30$. (b) The coordinate $z$ as a function of $x$ for tunneling through a parabolic potential barrier in the presence of a magnetic field.

Figure 16

(a) Potential barrier (110) (red solid line) and its linear approximation (112) (blue dashed line) in the deep-tunneling regime. The blue dotted line indicates the energy level of the bound state with $\varepsilon =-I_p$. (b) The Wigner trajectory (red solid line) and the classical trajectory (blue dashed line) in the deep-tunneling regime for nonrelativistic tunnel ionization with $\kappa =90$ and $E_0/E_a=1/30$. Vertical black line indicates the exit coordinate.

Figure 17

(a) Tunneling barrier (110) (red solid line), its linear approximation (112) (blue dashed line), and the quadratic approximation (114) (black dash-dotted line) in the near-threshold-tunneling regime. The blue dotted line indicates the energy level of the bound state with $\varepsilon =-I_p$. (b) The Wigner (red solid line) and the classical trajectories (blue dashed line) in the near-threshold-tunneling regime for nonrelativistic tunnel ionization with $\kappa =90$ and $E_0/E_a=1/17$. Vertical black line indicates the exit coordinate.

Figure 18

Comparison of the Wigner (red solid line) and the classical trajectories (blue dashed line) for tunnel ionization taking into account leading effects in $1/c$ (magnetic dipole effects): (a) for the deep-tunneling regime with the most probable momentum at the exit $q_z\left(x_e\right)=0.28I_p/c$, $\kappa =90$, and $E_0/E_a=1/30$; (b) for the near-threshold-tunneling regime with $q_z\left(x_e\right)=0.12I_p/c$, $\kappa =90$, and $E_0/E_a=1/17$. Corresponding classical and Wigner trajectories $z$ as a function of $x$ are presented in subfigure (c) for parameters of subfigure (b). Vertical black lines indicate the exit coordinate.

Figure 19

Comparison of the Wigner trajectory (red solid line) and the classical trajectory (blue dashed line) for tunnel ionization taking into account kinematic relativistic effects for the deep-tunneling regime employing nonrelativistic [part (a)] and relativistic [part (b)] parameters. In both parameter regimes no nonzero time delay is detectable at remote distance. Vertical black line indicates the exit coordinate.

Figure 20

Scaled Wigner time delay ${\tau }_{\mathrm{W}}{I}_p$ as a function of $I_p$ for different values of $E_0/E_a$ for tunnel ionization taking into account relativistic kinematic effects (solid lines) and nonrelativistic tunnel ionization (dashed lines) for the zero-range potential. Note the increasing small deviations of the scaled Wigner time delay with increasing $I_p$ (relativistic effects).