Self-attention enabled quantum path analysis of high-harmonic generation in solids

TL;DR

The paper tackles disentangling complex many-body contributions in solid-state high-harmonic generation by applying a Transformer-based self-attention framework to TDSE-generated signals from a 1D Kronig-Penney model. It demonstrates that self-attention highlights nonlocal temporal correlations associated with nonadiabatic band coupling and reconstructs the dipole with high fidelity, enabling amplification of weak coupling channels and revealing signatures of nonadiabatic dynamics. By coupling attention maps with Gabor time–frequency analysis, the approach exposes coupled electronic states and abrupt transitions, including enhanced even-order harmonics arising from symmetry-breaking dynamics. This physics-informed machine-learning framework offers interpretable insights into ultrafast electron dynamics in solids and holds promise for generalizing to more realistic materials and attosecond spectroscopy applications.

Abstract

High-harmonic generation (HHG) in solids provides a powerful platform to probe ultrafast electron dynamics and interband--intraband coupling. However, disentangling the complex many-body contributions in the HHG spectrum remains challenging. Here we introduce a machine-learning approach based on a Transformer encoder to analyze and reconstruct HHG signals computed from a one-dimensional Kronig--Penney model. The self-attention mechanism inherently highlights correlations between temporal dipole dynamics and high-frequency spectral components, allowing us to identify signatures of nonadiabatic band coupling that are otherwise obscured in standard Fourier analysis. By combining attention maps with Gabor time--frequency analysis, we extract and amplify weak coupling channels that contribute to even-order harmonics and anomalous spectral features. Our results demonstrate that multi-head self-attention acts as a selective filter for strong-coupling events in the time domain, enabling a physics-informed interpretation of high-dimensional quantum dynamics. This work establishes Transformer-based attention as a versatile tool for solid-state strong-field physics, opening new possibilities for interpretable machine learning in attosecond spectroscopy and nonlinear photonics.

Figures (4)

• Figure 1: (a) Illustration of the self-attention mechanism in the HHG dipole analysis. Each $d(t_i)$ is extracted from the HHG dipole in the TDSE calculation. (b) A simple sinusoid $y = \sin(\pi t)$ (top) produces an attention map (bottom) with regular red--blue bands, indicating that each time point attends to others with the same phase. (c) HHG dipole from TDSE simulations (top) has fast oscillations with a slow envelope. Its attention map (bottom) shows extended vertical and horizontal correlations, capturing both short- and long-range temporal structures. (d) The workflow of the Transformer-based analysis.
• Figure 2: Transformer reconstruction against TDSE dipole signals. (a--c) Representative HHG dipoles from TDSE simulations at different propagation times and crystal momenta: (a) 12.5 fs segment at $k=\pi/a$, (b) 75 fs signal at $k=0$, and (c) 75 fs signal at $k=\pi/a$. ((d)--(f)) Corresponding Transformer reconstructions, capturing both the oscillatory structure and envelope modulation, demonstrate retention of short-time dynamics and long-time coherence. Minor deviations appear in fine details, especially for the short segment, but the overall waveform is preserved. (g) Training curve showing the reconstruction accuracy ($1-\text{MSE}$) converging rapidly and stabilizing near unity after 300 epochs, indicating high-fidelity performance across different conditions.
• Figure 3: Comparison between original TDSE and Transformer-reconstructed HHG spectra. (a) Schematic of the simulation setup based on a one-dimensional Kronig–Penney lattice potential driven by a strong infrared laser. (b,d,e) Representative harmonic spectra at different propagation times and crystal momenta, showing that the Transformer faithfully reproduces the main features of the TDSE results, including the plateau and cutoff. The even harmonics produced by the long pulse (75 fs) indicate the nonadiabatic coupling, and the reconstructed HHG spectrum can capture and amplify this dynamic. (c) Bandgap structure indicating valence band (VB), first conduction band (CB1), and second conduction band (CB2); strong non-adiabatic coupling emerges near the Brillouin-zone edge ($k=\pi/a$), where the energy gap is small. The excellent agreement, especially at $k=\pi/a$, demonstrates that the self-attention mechanism captures spectral signatures of interband tunneling and band coupling, while only minor deviations appear at low-order harmonics.
• Figure 4: Comparison of original and Transformer-reconstructed HHG Gabor spectra.a-c present Gabor time–frequency spectra of high-harmonic emission obtained from TDSE simulations: a a 12.5 fs segment at $k=\pi/a$, b a 75 fs signal at $k=\pi/a$, and c a 75 fs signal at $k=0$. d-f show the corresponding Transformer-reconstructed spectra under the same conditions. e and f reveal additional features beyond the tenth harmonic, where elliptical regions indicate the presence of coupled electronic states and vertical blue lines correspond to abrupt non-adiabatic transitions between them. At $k=\pi/a$ (e), the ellipses are broader and more extended, reflecting stronger state mixing and enhanced non-adiabatic coupling at the Brillouin-zone edge, while at $k=0$ (f) the transitions are sharper and more localized in time. These observations demonstrate that the Transformer not only reconstructs the HHG dipoles but also captures hidden signatures of non-adiabatic dynamics in solids. The color scale denotes normalized intensity.