Wigner time delay for tunneling ionization via the electron propagator

Recent attoclock experiments using the attosecond angular streaking technique enabled the measurement of the tunneling time delay during laser-induced strong-field ionization. One of the theoretical models for the tunneling time delay is the Wigner time delay, which is the asymptotic time difference between the quasiclassical and the Wigner trajectories. The latter is derived from the derivative of the phase of the electron steady-state wave function with respect to energy. Here, we present an alternative method for the calculation of the Wigner trajectory by using the fixed-energy propagator. The developed formalism is applied to the nonrelativistic regime as well as to the relativistic regime of the tunnel-ionization process from a zero-range potential. Finally, it is shown that the Wigner time delay is measurable in the near-threshold-tunneling regime within the current state of the momentum spectroscopy via detecting the induced electron momentum shift in a mixture of two gas species.

Figure 1

Comparison of the Wigner (red solid line) and the classical trajectories (blue dashed line) for nonrelativistic tunnel ionization from a zero-range potential under the effect of a constant and uniform electric field; (a) the deep-tunneling regime with $E_0/E_a=1/7$ and (b) the near-threshold-tunneling regime with $E_0/E_a=1/2$. The vertical black line indicates the exit coordinate, and the applied parameter is $\kappa =1$.

Figure 2

Tunneling probability vs the kinetic momentum along the propagation direction of the crossed field at the tunnel entry (dashed line) and at the tunnel exit (solid line). In the case of tunnel ionization from a zero-range potential, the values are independent from the barrier suppression parameter $E_0/E_a$, and the transversal momentum transfer is $I_p/c$. The applied parameters are $E_0/E_a=1/7$ and $\kappa =90$.

Figure 3

Comparison of the Wigner (red solid line) and the classical trajectories (blue dashed line) for relativistic tunnel ionization from a zero-range potential under the effect of a constant and uniform crossed field; (a) the deep-tunneling regime with $E_0/E_a=1/7$ and (b) the near-threshold-tunneling regime with $E_0/E_a=1/2$. The vertical black line indicates the exit coordinate, and the applied parameter is $\kappa =90$.

Figure 4

The scaled Wigner time delay ${\tau }^w{I}_p$ vs the barrier suppression parameter $E_0/E_a$ for a certain ionization energy $I_p$. The black solid curve indicates the numerical calculation for Eq. (37), whereas the red dashed curve is for the asymptotic simple expression (38).

Figure 5

Comparison of the Wigner (red solid line), the classical (blue dashed line), and the classical with a nonzero initial momentum $p_0$ (black dotted line) trajectories for nonrelativistic tunnel ionization from a zero-range potential in the near-threshold-tunneling regime. It is shown that the Wigner trajectory coincides with the classical trajectory with a nonzero initial momentum along the tunneling direction. The vertical black line indicates the exit coordinate, and the applied parameters are $E_0/E_a=1/2$ and $\kappa =1$.