Caustics and catastrophes in strong-field physics -- Picard--Lefschetz theory as a universal approach to saddle-point methods in attosecond science

TL;DR

This paper introduces Picard--Lefschetz theory as a universal framework for saddle-point methods in attosecond science, enabling analysis of complex phenomena like caustics in high-order harmonic generation.

Contribution

It develops practical numerical methods based on Picard--Lefschetz theory for evaluating oscillatory integrals in strong-field physics, addressing limitations of traditional saddle-point approaches.

Findings

Methods remain valid at caustics where classical trajectories coalesce.

Systematic analysis of parameter regimes reveals new features in HHG.

Enhanced understanding of light-matter interactions at ultrafast timescales.

Abstract

Ultrashort laser pulses on the attosecond timescale are typically achieved via high-order harmonic generation (HHG), a nonlinear process in which atoms interact with intense light fields to emit a broad spectrum of harmonics. HHG is commonly described in terms of a `quantum orbits' model based on several interfering electron trajectories, thereby incorporating both quantum-mechanical effects and an intuitive picture of classical dynamics. By tuning the parameters of the driving laser field, the interplay between these trajectories can be controlled, shaping the emitted light. Mathematically, this model expresses the harmonic response as a highly oscillatory integral. Applying saddle-point methods to this integral allows it to be decomposed into contributions from distinct saddle points of the semi-classical action, thereby linking quantum dynamics to classical trajectories. However, a general framework for applying these methods across arbitrary parameters and laser configurations has been missing. In this thesis, we introduce Picard--Lefschetz theory and develop practical numerical methods for its application. These enable the evaluation of oscillatory integrals and identification of contributions from individual critical points. We apply these techniques to strong-field ionisation and HHG, focusing on caustics -- enhancement features where trajectories coalesce and standard approximations fail. Our methods remain valid in these regions, allowing systematic analysis of parameter regimes and revealing previously inaccessible features. This work improves the understanding and control of ultrafast light--matter interactions.