Self-attention Enables Quantum Path Analysis of High-harmonic Generation and Reveals Even-order Coupling Channels

High-harmonic generation in solids offers a promising route to investigate the behaviour of electrons at incredibly short timescales, but interpreting the complex signals it produces remains a significant hurdle. Cong Zhao and Xiaozhou Zou, both from King’s College London, alongside their colleagues, now present a new analytical method that leverages the power of machine learning to unravel these intricate processes. Their approach employs a Transformer encoder, a type of neural network originally developed for natural language processing, to analyse the signals generated during high-harmonic generation. This innovative technique uses self-attention to pinpoint crucial connections between the timing of electron movements and the resulting high-frequency light emitted, revealing subtle indicators of electron interactions that conventional methods often miss. By effectively filtering and amplifying these weak signals, the team demonstrates a clearer understanding of the underlying physics, paving the way for more insightful attosecond spectroscopy and advances in nonlinear photonics.

Solid-State High Harmonic Generation and Machine Learning

Recent research converges advanced optical physics, cutting-edge machine learning techniques, and signal processing tools to understand and control matter at the atomic and electronic level. Scientists are exploring high-harmonic generation (HHG) in solids, a process where intense laser fields drive electrons to emit photons at multiples of the driving laser frequency, enabling the study of electronic structure with attosecond precision. Unlike HHG in gases, solid-state HHG is more complex due to the intricate band structure and interactions between electrons. Current investigations focus on understanding multiband effects, the role of Berry curvature, and how symmetries and conservation laws govern the process. This work also highlights the potential of machine learning to accelerate scientific discovery by uncovering hidden symmetries and conserved quantities in complex physical systems.

Transformer Encoder Decodes Solid-State Harmonics

Scientists pioneered a novel approach to analyse high-harmonic generation (HHG) signals using a Transformer encoder, originally developed for natural language processing, to dissect complex electron dynamics in solids. The study harnessed the self-attention mechanism to highlight correlations between temporal dipole dynamics and high-frequency spectral components, effectively identifying signatures of nonadiabatic band coupling often obscured by standard Fourier analysis. This method involves representing the time-dependent dipole response from simulations as a sequence of feature vectors, each encoding the local dipole signal at a specific time step, and augmenting this with positional encoding to preserve temporal order. The team constructed vectors, queries, keys, and values, through learned linear projections of the input dipole data, enabling the model to assess the relevance of each time point to all others.

Similarity between query and key vectors was calculated using a scaled dot product, then normalized with a softmax function to generate attention weights, which dynamically re-weight contributions from all positions in the input sequence. This operation allows the model to capture long-range correlations and nonlocal patterns inaccessible to conventional methods. Experiments demonstrated the self-attention mechanism’s response to signals of varying complexity, revealing that a simple sinusoidal input produces a sharply periodic attention map, while the attention map derived from complex HHG dipole simulations exhibits extended vertical and horizontal correlations, capturing both short- and long-range temporal structures. By linking self-attention weights to nonadiabatic coupling signatures, the research demonstrates that even in a minimal model system, HHG encodes rich dynamical information that can be effectively extracted using this innovative analytical tool. The workflow integrates Gabor time-frequency analysis with the Transformer output, extracting and amplifying weak coupling channels contributing to even-order harmonics and anomalous spectral features, ultimately establishing a new methodology for exploring solid-state HHG beyond traditional interband-intraband decomposition.

Self-Attention Reveals Nonadiabatic Coupling in Solids

Scientists have achieved a breakthrough in understanding high-harmonic generation (HHG) in solids by employing a novel machine learning approach based on the self-attention mechanism, originally developed for natural language processing. This work demonstrates the ability to disentangle complex electron dynamics and identify subtle signatures of nonadiabatic band coupling within the HHG spectrum. Researchers utilized a one-dimensional Kronig-Penney model, representing a periodic solid, to simulate the time-dependent Schrödinger equation and observe the emergence of even-order harmonic components, directly attributable to nonadiabatic interband coupling. The core of this achievement lies in the application of self-attention to the simulated dipole response, allowing the team to highlight temporal regions of enhanced coupling and selectively amplify even-order harmonics.

The self-attention mechanism computes correlation weights between all pairs of times in the dipole response, acting as a nonlocal filter that enhances strongly correlated segments where nonadiabatic transitions imprint phase asymmetry. By integrating this self-attention decomposition with Gabor time-frequency analysis, scientists established a clear mapping from the raw dipole signal to the underlying nonadiabatic quantum paths. Results demonstrate that this methodology not only reproduces the full HHG spectrum but also isolates coupling-induced features masked in the dominant odd-order background. The team observed that the self-attention weights directly correlate with signatures of nonadiabatic coupling, revealing rich dynamical information encoded within the HHG signal. This work establishes a new methodology for exploring solid-state HHG beyond the standard interband-intraband decomposition, positioning self-attention as a promising technique for quantum path analysis in both theoretical and experimental contexts. The findings highlight the role of nonadiabaticity as a driver of symmetry breaking in HHG, opening new avenues for understanding and controlling light-matter interactions at the attosecond timescale.

Transformer Models Reveal Electronic Dynamics in Harmonics

This work demonstrates that Transformer-based models offer a powerful new approach to analysing high-harmonic generation in solids. By reconstructing time-dependent dipole responses from simulations, the self-attention mechanism directly identifies non-adiabatic coupling effects within the harmonic spectra, revealing subtle features often obscured by traditional analysis methods. The team showed the model accurately reproduces key spectral characteristics, such as the plateau and cutoff structures, while simultaneously enhancing features indicative of strong electronic interactions. The results demonstrate that the Transformer not only replicates existing data but actively extracts hidden features of underlying electronic dynamics, specifically identifying correlations between interband polarization and intraband motion that break inversion symmetry.

This capability allows for the observation of even-order harmonics, providing direct evidence of strengthened non-adiabatic coupling. The methodology, initially applied to a simplified Kronig-Penney model, can be extended to more complex materials and experimental data, offering a versatile platform for disentangling interband and intraband contributions and mapping coupled electronic states. The authors acknowledge the current work is based on a one-dimensional model and suggest future research will focus on applying this methodology to more realistic materials and experimental datasets, potentially guiding the design of novel photonic and electronic materials.