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.