Understanding and controlling light-induced phenomena in solids requires resolving laser-driven electron dynamics at their natural time and length scales. Current capabilities are limited by challenges in interpreting wave mixing of driving and probe pulses, low energy resolution at ultrashort timescales, and the lack of atomic-scale resolution offered by standard spectroscopic techniques. The excitation of solids by strong light fields is essential for various processes, including manipulating electronic structures and generating high harmonics (HHG), both highly promising for ultrafast optoelectronic device development. Revealing microscopic details of laser-driven electron dynamics is key to understanding strong-field-induced processes. Recent advances in generating few and sub-femtosecond x-ray pulses now enable real-time microscopic views into these dynamics, offering a crucial path to uncovering their mechanisms. When light interacts with a material, charges rearrange within the unit cell, creating microscopic electron currents. The conventional notion of optically-induced charge separation solely causing a dipole moment is insufficient at the atomic scale. Induced charge distributions possess intricate structures and diverse symmetry features. This research develops a method utilizing ultrashort nonresonant x-ray pulses to probe, in real time, charge and electron current distributions within a crystal's unit cell during interaction with an optical field. The method leverages the short wavelength of hard x-rays, providing the necessary spatial resolution to access the microscopic optical response at the atomic scale. The approach builds upon the concept of x-ray-optical wave mixing (XOWM), where simultaneous x-ray and optical field interactions with matter lead to sum and difference frequency generation, encoding the microscopic optical response. This work extends XOWM by proposing a subcycle-resolved measurement, where the x-ray pulse duration is shorter than an optical cycle, yielding significantly more information on laser-driven electron dynamics than time-unresolved XOWM.

The existing literature extensively covers high-harmonic generation (HHG) in solids and its potential for ultrafast optoelectronics. Studies like those by Ghimire et al. (2010) and Goulielmakis & Brabec (2022) have demonstrated HHG in bulk crystals and highlighted its potential. Research on the control of dielectrics with light (Schultze et al., 2012; Schiffrin et al., 2012) and sub-cycle control of terahertz high-harmonic generation (Schubert et al., 2014) demonstrate significant advances in manipulating light-matter interactions. However, a detailed understanding at the atomic scale of the underlying electron dynamics during these processes has been limited. Previous studies by Popova-Gorelova et al. (2018, 2015, 2016) and others have explored theoretical frameworks for imaging electron dynamics using ultrafast x-ray scattering and photoelectron spectroscopy, setting the stage for the current work's experimental proposal. Existing work on x-ray and optical wave mixing (Glover et al., 2012) provides a foundation for the methodology but lacks the sub-cycle temporal resolution crucial for capturing the detailed dynamics investigated here.

The study employs α-quartz as a model material, known for its interesting strong-field light-induced phenomena. The authors use the Floquet-Bloch formalism combined with density functional theory (DFT) to calculate the microscopic optical response of α-quartz to a periodic electromagnetic field (1.2 eV photon energy, 10¹² W cm⁻² intensity, polarized along (1,1,0)). The calculations reveal oscillations of electron density and electron-current density at frequencies ω, 2ω, and 3ω. The results demonstrate that nonlinear responses create complex charge rearrangements and current vortices that cannot be simply characterized by dipole moments. The interaction of the α-quartz crystal with an ultrashort x-ray pulse during its interaction with the optical field is then modeled. The total Hamiltonian considers the crystal, optical field, and x-ray pulse interactions. The interaction with the optical field is treated within the dipole approximation, neglecting the A² term. The x-ray interaction, assumed nonresonant with core-excitation energies, is described by a Hamiltonian representing nonresonant x-ray scattering, the basis for x-ray diffraction. The total x-ray scattering probability is given by the sum of quasielastic and inelastic scattering probabilities (Ptot = Pqe + P inel). The quasielastic scattering signal is shown to contain the relevant information about optically-driven dynamics. Calculations show that inelastic scattering is significantly weaker than quasielastic scattering for α-quartz. The model then analyzes x-ray scattering probability, considering a Gaussian-shaped x-ray pulse. The analysis differentiates between cases where the x-ray pulse duration is longer than one optical cycle and cases with sub-cycle resolution. With sub-cycle resolution (2.35 fs x-ray pulse), interference terms become non-negligible, making the signal time-dependent. This time dependence is crucial for phase retrieval, allowing the reconstruction of both the amplitudes and phases of the Fourier components of optically-induced charge distributions. The connection between the time-dependent antisymmetric part of the signal and the first-order oscillation of the electron-current density is established, and the connection between the centrosymmetric part and the electron density is also shown. The methodology details how the phases and amplitudes of Fourier components are extracted from the time-dependent and time-independent parts of the signal. This detailed analysis allows for the complete reconstruction of optically-induced charge distributions and the determination of electron current directions. The Kohn-Sham wave functions for the DFT calculations were obtained using the ABINIT software package.

The study's key findings center on the successful demonstration of a method to image laser-driven electron dynamics in solids at the atomic scale. The subcycle-resolved x-ray-optical wave mixing (XOWM) technique provides both the amplitudes and phases of the Fourier transform of optically induced charge distributions. The time-dependent nature of the subcycle-resolved XOWM signal, particularly its non-centrosymmetry, is crucial for extracting phase information. Analysis reveals that the antisymmetric part of the signal is directly related to the temporal oscillations of the electron-current density, while the centrosymmetric part corresponds to the electron density oscillations. The temporal evolution of these parts reveals the phases of these oscillations. The amplitudes of the Fourier components can be obtained from the time-independent part of the signal. By measuring the signal at various scattering vectors (G), the method allows the complete reconstruction of optically-induced charge distributions for all orders of the microscopic optical response. The direction of microscopic electron currents can be determined from the sign of the projections of their Fourier components. The intensity of the first- and second-order side peaks shows a non-perturbative dependence on the driving field intensity, indicating the complexity of the light-matter interaction. The response to varying photon energies and intensities of the driving field is also examined, revealing a complex dependence of certain spatial and temporal features of the optically-induced charge distributions on the driving field parameters. The analysis shows the potential to investigate systems where time-reversal symmetry is broken, extending the applicability of the technique beyond the α-quartz case.

The findings directly address the research question of developing a method for atomic-scale imaging of laser-driven electron dynamics. The successful implementation of subcycle-resolved XOWM provides a significant advancement over previous techniques, enabling a previously inaccessible level of detail in understanding light-matter interactions. The ability to reconstruct both charge distributions and electron current directions provides crucial insights into the fundamental mechanisms underlying strong-field phenomena in solids. The non-perturbative nature of the observed interactions underscores the importance of moving beyond simplified models. The sensitivity of the signal to variations in driving-field parameters highlights the potential for precise control of electron dynamics through tailored light fields. The ability to reconstruct complete charge distributions resolves the 'phase problem' inherent in crystallography, which is a significant contribution to the field. These results are relevant to a wide range of fields including ultrafast optoelectronics, nonlinear optics, and materials science, paving the way for new approaches in designing and controlling light-matter interactions at the atomic level.

This study presents a novel method, subcycle-resolved x-ray-optical wave mixing, for atomic-scale imaging of laser-driven electron dynamics in solids. This technique overcomes limitations of previous approaches by providing both the amplitude and phase information needed for complete reconstruction of optically-induced charge and current distributions. Future work could explore the application of this method to a wider range of materials, including those lacking time-reversal symmetry, and investigate the effects of different light polarizations and pulse shapes. Further research incorporating excitonic effects into the theoretical model would also enhance the accuracy and predictive power of the technique.

The current study focuses on α-quartz, and the generalizability to other materials needs further investigation. The theoretical model currently neglects excitonic effects, which could influence the accuracy of the results for certain materials. The experimental feasibility of achieving the required sub-cycle temporal resolution and signal-to-noise ratio needs to be confirmed through future experiments. The computational cost associated with the Floquet-Bloch DFT calculations may limit the scalability of the method to larger systems.