The Wigner time delay for laser induced tunnel-ionization via the   electron propagator
  Recent attoclock experiments using the attsecond angular streaking technique enabled the measurement of the tunneling time delay during laser induced strong field ionization. Theoretically the tunneling time delay is commonly modelled by the Wigner time delay concept which is derived from the derivative of the electron wave function phase with respect to energy. Here, we present an alternative method for the calculation of the Wigner time delay by using the fixed energy propagator. The developed formalism is applied to the nonrelativistic as well as to the relativistic regime of the tunnel-ionization process from a zero-range potential, where in the latter regime the propagator can be given by means of the proper-time method. 
Enderalp Yakaboylu, Michael Klaiber, and Karen Z. Hatsagortsyan
quant-ph physics.atom-ph