Slalom in complex time: emergence of low-energy structures in tunnel ionization via complex time contours

TL;DR

This paper investigates how Coulomb interactions influence electron trajectories during tunnel ionization in strong fields, revealing complex time structures that explain low-energy electron phenomena.

Contribution

It introduces a complex time contour framework for including Coulomb effects in analytical R-matrix theory, explaining low-energy structures in tunnel ionization.

Findings

Complex trajectories have imaginary components affecting ionization paths.

Branch cuts in complex time are linked to electron revisits to the ionic core.

A new framework navigates these branch cuts using closest-approach times.

Abstract

The ionization of atoms by strong, low-frequency fields can generally be described well by assuming that the photoelectron is, after the ionization step, completely at the mercy of the laser field. However, certain phenomena, like the recent discovery of low-energy structures in the long-wavelength regime, require the inclusion of the Coulomb interaction with the ion once the electron is in the continuum. We explore the first-principles inclusion of this interaction, known as analytical R-matrix theory, and its consequences on the corresponding quantum orbits. We show that the trajectory must have an imaginary component, and that this causes branch cuts in the complex time plane when the real trajectory revisits the neighbourhood of the ionic core. We provide a framework for consistently navigating these branch cuts based on closest-approach times, which satisfy the equation $\mathbf{r}(t) \cdot \mathbf{v}(t) = 0$ in the complex plane. We explore the geometry of these roots and describe the geometrical structures underlying the emergence of LES in both the classical and quantum domains.