  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.
