Nonlocal mechanisms of attosecond interferometry in three-dimensional systems

Attosecond interferometry (AI) is an experimental technique based on ionizing a system with an attosecond pulse train in the presence of an assisting laser. This assisting laser provides multiple pathways for the photoelectron wave packet to reach the same final state, and interference of these pathways can be used to probe properties of matter. The mechanism of AI is well-understood for isolated atoms and molecules in the gas phase, but not so much in the condensed phase, especially if the substrate under study is transparent. Then, additional pathways open up for the electron due to scattering from neighbouring atoms. We investigate to what extent these additional pathways influence the measured photoionization delay with the help of one- and three-dimensional model systems. In both cases, we find that the total delay can be expressed as the sum of a local (photoionization) delay and a non-local delay which contains the effect of electron scattering during transport. The 1D system shows that the non-local delay is an oscillatory function of the distance between the sites where ionization and scattering take place. A similar result is obtained in 3D, but the modulation depth of the non-local delay is found to strongly depend on the effective scattering cross section. We conclude that attosecond interferometry of disordered systems like liquids at low photon energies (20-30 eV) is mainly sensitive to the local delay, i.e., to changes of the photoionization dynamics induced by the immediate environment of the ionized entity, and less to electron scattering during transport through the medium.