Universal approach to saddle-point methods in attosecond science

Light-matter interactions within the strong-field regime, where intense laser fields can ionize a target via tunneling, give rise to fascinating phenomena such as the generation of high-order harmonic radiation (HHG) and, correspondingly, light pulses of attosecond duration. On the atomic scale, these strong-field processes are naturally described in terms of highly oscillatory time integrals which are often approximated using saddle-point methods. Those methods simultaneously simplify the calculations and let us understand the physical processes in terms of semiclassical electron trajectories, known as quantum orbits. However, applying saddle-point methods for HHG driven by polychromatic laser fields without clear dynamical symmetries has remained challenging. Here we introduce Picard–Lefschetz theory as a universal and robust link between the time integrals and the semiclassical trajectories, for arbitrary driving laser fields. The continuous deformation of the integration contour towards so-called Lefschetz thimbles allows for an exact evaluation of the integral, as well as the identification of relevant quantum orbits, independently of dynamical laser field symmetries or quantum-orbit classification heuristics. The latter is realized via the “necklace algorithm,” a solution to the open problem of determining the relevance of saddle points for a two-dimensional integral, which we introduce here. We demonstrate the versatility and rigor of Picard–Lefschetz methods by studying Stokes transitions and spectral caustics arising in HHG driven by two-color laser fields. For example, we showcase a quantum-orbit analysis of the color switchover, which links the regime of perturbative two-color fields with that of fully bichromatic driving fields. With this work, we set the foundation for a rigorous application of quantum-orbit-based approaches in attosecond science that enables the interpretation of state-of-the-art experimental setups, and guides the design of future ones.