A universal approach to saddle-point methods in attosecond science

TL;DR

The paper addresses the challenge of applying saddle-point methods to attosecond processes under arbitrary driving fields. It introduces Picard-Lefschetz theory as a universal contour-deformation framework, enabling exact evaluation of highly oscillatory integrals via Lefschetz thimbles and robust identification of quantum orbits. Two numerical tools—the downwards flow and the necklace algorithm—are developed to decompose one- and two-dimensional integrals and determine saddle relevance, which are then applied to HHG driven by two-colour fields. The approach reveals spectral caustics, Stokes transitions, and the colour switchover phenomenon, while providing a rigorous basis for tracking individual quantum orbits across parameter spaces. This work lays the foundation for rigorous, trajectory-based semi-classical analysis in attosecond science and guides future experimental and theoretical developments.

Abstract

Light-matter interactions within the strong-field regime, where intense laser fields can ionise a target via tunnelling, give rise to fascinating phenomena such as the generation of high-order harmonic radiation (HHG). On the atomic scale, these strong-field processes are described in terms of highly-oscillatory time integrals which are often approximated using saddle-point methods. These methods simultaneously simplify the calculations and let us understand the physical processes in terms of semi-classical electron trajectories, or 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 semi-classical trajectories. The continuous deformation of the integration contour towards so-called Lefschetz thimbles allows an exact evaluation of the integral, as well as the identification of relevant quantum orbits, for arbitrary driving fields. The latter is realised via the ``necklace algorithm'', a novel 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 rigour of Picard-Lefschetz methods by studying Stokes transitions and spectral caustics arising in HHG driven by two-colour laser fields. For example, we showcase a quantum-orbit analysis of the colour switchover, which links the regime of perturbative two-colour 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.

Figures (20)

• Figure 1: Bottom row: Contour plots of $\mathrm{Im}(S_\mathrm{ATI}(t))$ for the complex $t$ plane for the two driving fields shown on top (panels (a) and (b)) and drift momenta $p=0$ and $p=1.2\au$ respectively. Steepest-descent and steepest-ascent contour lines (black and grey lines, respectively) are attached to saddle points Eq. \ref{['eq:Satidrv']} (black markers, labelled in (d) for convenience), with the resulting integration path drawn as a heavy black line. The electric field in (b) is composed of the two constituent fields of frequency $\omega$ (red dashed) $2 \omega$ (blue dotted) with phase shift $\varphi = 0.5$ and amplitude ratio $E_2/E_1 = 0.15$ acc. to Eq. \ref{['eq:tc-field']}.
• Figure 2: Typical structure of the complex saddle point solutions for HHG in the complex plane of ionisation and recombination times. For the monochromatic driving field shown in (a) the solutions across a range of harmonic orders (colour bar) follow lines in the complex planes (b) and (c) and can be classified as 'short' (S) and 'long' (L) trajectories. For the electric field shown in (d), a two-colour field with $\varphi = 0.75$ and $E_2/E_1 = 0.44$, as per Eq. \ref{['eq:tc-field']}, the solutions still trace lines in the complex plane and form several ionisation windows (labelled A,B,C,D)), but their structure is more intricate, hindering a classification.
• Figure 3: Fundamental idea of Picard-Lefschetz theory, shown on the toy model function $\phi(z) = z^2$: The integrand $\mathrm{e}^{\mathrm{i} z^2}$ is highly oscillatory when evaluated along the real axis (top left panel). The continuation of $z$ into the complex plane (bottom left panel) shows, that the oscillations (along the light blue line) vanish if we evaluate the integrand along a different contour (dark blue line). The contour that localises the integrand by minimising the oscillations follows steepest-descent paths of $\mathrm{Im}(z^2)$ (contour plot in the bottom right panel) and is identified by deforming the original integration domain according to the downwards flow (red arrows in the bottom right panel) and leads across the saddle point at $z=0+0\mathrm{i}$, where $\phi'(z)=0$. Often, the integrand along the new contour has Gaussian shape (top right panel) and can be calculated analytically.
• Figure 4: Flowing the integration contour (dark blue) acc. to the downwards flow Eq. \ref{['eq:downwards-flow']} for the same scenario as in Fig. \ref{['fig:ATI-landscapes']}(d) for discretised flow steps $i$ to minimise the oscillations of the integrand. The integrand evaluated along the contour is shown in the top rows.
• Figure 5: For (a) one- and (b) two-dimensional path integrals the downwards flow (directions indicated by red arrows) transform the original, real-valued integration domain (light blue) into the complex domain, ultimately towards the steepest descent contours ("thimbles", grey) attached to the critical points (cross markers).