Understanding ultrafast electron and nuclear motion following light-molecule interaction is fundamental to numerous chemical reactions and biological processes. Attosecond light pulses, combined with femtosecond infrared (IR) pulses in pump-probe configurations, have enabled techniques like attosecond extreme-ultraviolet (XUV) transient absorption spectroscopy, attosecond streaking, and mass/ion spectroscopy to partially observe and manipulate molecular electronic degrees of freedom. After electron removal, the molecule is represented by a wave packet, a complex-valued wavefunction of all electrons and nuclei. Electrons move much faster than nuclei, so initially, nuclear positions can be considered fixed, and the electron wave packet (EWP) can be expressed as a superposition of electronic eigenstates oscillating at frequencies determined by energy differences between eigenstates. This time-dependent electric charge density is known as charge migration. On longer timescales, EWP evolution changes the electronic potential landscape, affecting nuclear wave packet evolution. Ultrafast electron diffraction visualizes nuclear motion, but an equivalent attosecond electron movie of charge migration is not readily available due to limitations in the time resolution of conventional pump-probe experiments. Directly obtaining experimental charge migration data has been a significant challenge. A pseudo pump-probe protocol based on electron recollision in the laser field has been suggested, where electron dynamics are triggered upon ionization (pump) and probed upon recollision through elastic scattering or harmonic spectra from electron-ion recombination. Laser-induced electron diffraction (LIED) and high harmonic spectroscopy (HHS) probe molecular structure using diffraction images and high-order harmonic generation, respectively. In HHS, temporal resolution stems from intrinsic harmonic emission frequency chirping, linking harmonic orders to defined ionization-recombination delays, achieving sub-100 attosecond resolution. Spatial resolution in HHS reaches sub-ångström levels. Previous HHS studies on charge migration, such as those with iodoacetylene, faced challenges. Reconstructing electron dynamics from harmonic spectra is an inverse scattering problem, generally solved by iterative methods. This requires a well-established scattering theory. Laser-driven electron dynamics is a single-molecule property that depends on the molecule's orientation relative to the laser polarization. Experimental harmonic signals are a coherent superposition of individual radiation, weighted by molecular angular distribution. The angular averaging compromises harmonic phases and dynamic information, making extraction of fixed-in-space single-molecule harmonic amplitudes and phases crucial. Accurate charge migration extraction requires extracting these parameters and population coefficients of different cation states that describe the single-molecule EWP. This paper addresses these challenges and films attosecond laser-driven charge migration using HHS.

The study builds upon previous research on attosecond spectroscopy and high harmonic generation (HHG). The authors mention several key developments in attosecond science, including the advent of attosecond pulses and their application in various spectroscopic techniques. The limitations of conventional pump-probe experiments in resolving attosecond charge migration are highlighted. The concept of charge migration, initially defined by Cederbaum and coworkers, is discussed, emphasizing the distinction between field-free charge migration and laser-driven charge migration observed in HHS. The authors cite prior work by Kraus et al. demonstrating charge migration in iodoacetylene using HHS, but also highlight the challenges of this approach, such as the inverse scattering problem nature of reconstruction and the influence of molecular alignment on the measured harmonics. The use of laser-induced electron diffraction (LIED) and high harmonic spectroscopy (HHS) as tools for probing molecular structure and dynamics is also reviewed. The authors note that achieving sufficient temporal and spatial resolution for observing attosecond charge migration requires overcoming several hurdles in experimental design and data analysis. They discuss the need for a refined understanding of the multichannel contributions to HHG signals and the importance of extracting single-molecule parameters rather than relying on ensemble-averaged data.

The researchers employed high harmonic spectroscopy (HHS) combined with a machine learning (ML) algorithm to reconstruct attosecond charge migration dynamics in N2 and CO2 molecules. The experimental setup utilized a commercial Ti:sapphire laser system generating 35-fs, 800-nm laser pulses. This laser was split into an alignment pulse and a probe pulse. The alignment pulse was used to induce nonadiabatic alignment of the molecules, while the probe pulse (used either as a single-color or a two-color field) generated high-order harmonics from the aligned molecules. A motorized delay line controlled the time delay between the alignment and probe pulses. The generated high-order harmonics were detected with a flat-field soft x-ray spectrometer. The one-color experiment provided data to identify the contributions of multiple orbitals in the HHG process, while the two-color experiment (with a weak second-harmonic field) provided additional data for decomposing the contributions of multiple orbitals in the HHG process. The reconstruction procedure involves two key steps: 1. Retrieving the single-molecule dipole moment from measured time-dependent HHG signals. This was done by solving a matrix equation that links the experimental signals to the single-molecule dipole using a sparse representation machine learning algorithm. 2. Deconvolving multichannel contributions from the total single-molecule dipole moment obtained in the first step. The two-color experiment was essential here because it provided more data points needed to effectively solve this inverse problem. The machine learning algorithm performed the deconvolution, extracting the time-dependent complex mixing coefficients of multiple orbitals in the molecular cation. These mixing coefficients are used to construct the wave packet of the hole dynamics of the cations, allowing for the visualization of attosecond charge migration. Theoretical calculations based on time-dependent density functional theory (TDDFT) were used to validate the reconstruction results.

The study successfully generated "movies" of attosecond charge migration in N2 and CO2 molecular ions. By using a two-color laser field and a machine learning algorithm, the researchers were able to reconstruct the time-dependent charge density distribution at 50-attosecond intervals and with angular resolution. The time resolution was further enhanced to 10 attoseconds through interpolation. Key observations include: 1. **Substantial Charge Migration:** The researchers observed significant charge migration within the first femtosecond after ionization. In CO2, the charge migration showed a swirling pattern, rather than simple linear motion along the laser polarization direction. 2. **Angular Dependence:** The charge migration dynamics were shown to be strongly dependent on the molecular alignment angle with respect to the laser polarization. In N2, this resulted in different dynamics being observed at different alignment angles. 3. **Multiple Orbital Contributions:** The harmonic spectra revealed the contributions of multiple molecular orbitals (HOMO, HOMO-1, HOMO-2) in the HHG process. The analysis of these contributions was essential for reconstructing the complex charge migration dynamics. 4. **Attosecond Time Scale:** The generated movies demonstrate the capability to track the charge migration at 10 attosecond intervals. The dynamics of the charge were shown to vary rapidly on this timescale. 5. **Validation:** The experimental results were validated through comparison with TDDFT simulations, which showed reasonable agreement with the reconstructed charge migration dynamics.

The findings address the research question by directly visualizing and quantifying attosecond charge migration in molecules. The use of a two-color laser field and machine learning analysis allows for high-resolution time-resolved imaging of this ultrafast process, overcoming previous limitations. The observation of swirling patterns in charge migration provides new insights into the complex electron dynamics in molecules, going beyond simple linear migration models. The angular dependence highlights the importance of molecular orientation in influencing the charge migration process. This research demonstrates a powerful new technique for studying ultrafast electron dynamics and has implications for various fields, including understanding and controlling photochemical reactions. The results have implications for the control and understanding of ultrafast chemical and biological processes.

This study presents a novel methodology for filming attosecond charge migration in single molecules using high harmonic spectroscopy (HHS) coupled with machine learning algorithms. The method achieved unprecedented time resolution (10 attoseconds) and provided insights into the intricate dynamics of charge migration, including swirling motion and angular dependence. This approach opens new avenues for understanding and potentially controlling ultrafast chemical processes. Future studies could explore this technique with other molecules to map diverse migration pathways and investigate the influence of nuclear motion on longer timescales. Extending the single-molecule parameter retrieval method to other ultrafast experiments could further broaden its impact.

The study's primary limitations relate to the complexity of the reconstruction process and its dependence on computational power. The accuracy of the reconstructed charge migration dynamics relies on the accuracy of the machine learning algorithms and the underlying theoretical models. The current method is computationally intensive, potentially limiting the scalability to larger molecules with a greater number of involved orbitals. While the study validates its findings through comparison with TDDFT calculations, the inherent approximations of TDDFT could introduce uncertainties. Moreover, the time window observable in the present work is limited by the capabilities of the experimental apparatus, and extending the time window would require modifications to the experimental setup or laser parameters.