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

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