Warm dense matter (WDM) represents a fascinating state of matter existing at the boundary between a plasma and a condensed phase. Its properties are relevant across numerous scientific disciplines, including astrophysics (modeling stellar interiors and planetary formation), planetary science (understanding the conditions within giant planets), and inertial confinement fusion (improving energy production efficiency). Despite its importance, a comprehensive understanding of WDM's electronic and ionic structures, particularly under intense laser irradiation, remains elusive. Existing theoretical models struggle to accurately capture the non-equilibrium dynamics and structural changes that occur when WDM is subjected to strong excitation pulses. This is because WDM is too hot to be adequately described by conventional condensed matter physics, yet too dense to be characterized using weakly coupled plasma physics. Experimental investigations are equally challenging, given the need for both powerful excitation sources and highly sensitive, ultrafast diagnostic tools to probe the fleeting nature of the WDM state. This research aims to address these challenges by employing an intense and ultrafast X-ray free-electron laser (XFEL) pulse to both create and characterize warm dense copper. By using X-ray absorption near-edge structure (XANES) spectroscopy, the study provides unique insights into the dynamic changes in the electronic structure of WDM and demonstrates a method with potential for X-ray pulse shaping.

Previous studies exploring XFEL-matter interactions have reported observations of saturable absorption (SA) and reverse saturable absorption (RSA), nonlinear effects leading to transient changes in a material's optical properties. SA occurs when the absorbing state is depleted faster than replenished, leading to absorption saturation. RSA, conversely, exhibits increasing absorption with increasing intensity, as the excited state has a larger absorption cross-section than the ground state. These phenomena are well-established in the optical regime, with applications in areas such as pulse shaping and mode-locking. Extending these effects into the X-ray domain is a significant objective, opening the potential for controlled X-ray pulse shaping. However, previous studies using XFELs have often relied on spectrally integrated measurements. This limits the detailed information that can be obtained about the material's electronic structure. This study moves beyond these limitations by employing spectrally resolved measurements of the transition from RSA to SA across a wide range of X-ray intensities. It builds on prior work in the optical regime and extends these findings to the X-ray domain with significantly improved spectral resolution.

The experiments were conducted at the Spectroscopy and Coherent Scattering (SCS) instrument of the European XFEL (EuXFEL). A 15-fs-long XFEL pulse, tuned to the Cu L₂ and L₃ absorption edges, was used to excite and probe a 15-eV-wide spectral window. The main excitation mechanisms are direct 2p photoionization followed by Auger decay and electron impact ionization. The XFEL intensity was varied to observe the transition from RSA to SA. The experimental setup involved focusing the XFEL beam onto a 100-nm or 500-nm thick copper foil, with the beam size characterized using a knife-edge scan. A gas attenuator was used to control the pulse energy. The resulting absorption spectra were recorded using a spectrometer and CCD camera. For each intensity level, several thousand single-shot spectra were collected and averaged to reduce noise and enhance the signal-to-noise ratio. The absorption coefficient was calculated using the Beer-Lambert law, comparing the intensity of the reference spectrum (without the sample) to the spectrum with the sample. The transmission of the XFEL pulse, both spectrally integrated and spectrally resolved, was determined by comparing the total number of photons in the reference and sample spectra. The pre-edge peak shift was extracted from the zero-crossing of the derivative of the smoothed absorption spectra. On the theoretical side, a model combining Boltzmann kinetic equations and finite-temperature DFT calculations was employed. The Boltzmann equation solver simulated the non-equilibrium evolution of the electronic configuration densities, considering photoionization, Auger decay, and electron collisions. The FEFF10 code was used for the high-temperature XANES calculations using the real-space Green's function method. The absorption spectrum was calculated by weighting the contributions from individual configurations based on their abundance from the Boltzmann simulations. The effective 3d-band temperature was incorporated to match the average 3d-band occupation from the Boltzmann model within FEFF10. The theoretical model considered various copper configurations, and their relative abundances were determined as a function of time and XFEL intensity. The model also tracked the non-equilibrium evolution of the electron energy distribution. The calculated XANES spectra were then compared to the experimental results.

The experimental results showed a clear transition from RSA to SA as the XFEL intensity was increased. Below an intensity of 10¹⁵ W cm⁻², a pre-edge peak appeared below the L₃ edge, increasing in intensity and shifting to lower energies (redshift) with increasing XFEL intensity. This redshift is attributed to the depletion of the 3d band and reduced screening effects. Above a transition intensity (I_tr ≈ 10¹⁵ W cm⁻² for the L₃ edge, and ≈ 10¹⁶ W cm⁻² for the L₂ edge), a blueshift was observed due to increased ionization and reduced screening, resulting in a decrease in overall absorption (SA). The van Hove singularity peak, characteristic of the crystalline copper structure, was observed up to intensities of 10¹⁶ W cm⁻², indicating the preservation of crystalline order on the timescale of the 15-fs pulse. At higher intensities, the peak disappeared, indicating the onset of structural changes. The theoretical calculations, combining Boltzmann kinetic equations and FEFF10, successfully reproduced the observed transition from RSA to SA and the redshift of the pre-edge peak at lower intensities. However, discrepancies exist, particularly concerning the intensity at which the transition occurs and the magnitude of the redshift. The theoretical analysis also showed the impact of the effective electronic temperature (T_eff) on the 3d density of states (DOS), explaining the observed redshift and blueshift of the pre-edge peak as a result of 3d band depletion, reduced screening, and the appearance of higher charge states.

The findings of this study demonstrate the successful observation and theoretical modeling of the transition from RSA to SA in the X-ray regime. This is a significant advance towards the goal of controlled X-ray pulse shaping, with applications in numerous areas. The combination of experimental measurements and theoretical modeling provides a robust approach to investigate the complex dynamics of electronic structure and atomic-scale changes in WDM under intense X-ray irradiation. The agreement between experimental and theoretical results, though not perfect, supports the validity of the model used and provides benchmarks for further refinement. The observed discrepancies, especially concerning the intensity of the RSA-to-SA transition and the magnitude of the redshift at lower intensities, highlight the challenges in modeling non-equilibrium electron dynamics in WDM. Future work could focus on improving the accuracy of the theoretical models, perhaps by incorporating more sophisticated treatments of electron-electron interactions and non-equilibrium electron distributions. The disappearance of the van Hove peak at high intensities indicates the potential of this technique to monitor structural transformations in WDM. The study provides a powerful tool for investigating matter under extreme conditions.

This research successfully demonstrated the transition from reverse saturable absorption (RSA) to saturable absorption (SA) in warm dense copper using an X-ray free-electron laser. Spectrally resolved X-ray absorption near-edge structure (XANES) spectroscopy, combined with a theoretical model using kinetic Boltzmann equations and finite-temperature density functional theory, provided a comprehensive understanding of the underlying physical processes. The findings have implications for X-ray pulse shaping and offer a valuable benchmark for non-equilibrium models of warm dense matter. Future studies could explore time-resolved XANES measurements to provide more detailed insight into the dynamics of warm dense matter formation and evolution. Further improvements in theoretical models and experimental techniques are also anticipated to lead to an even deeper understanding of WDM and its properties under extreme conditions.

The accuracy of the beam intensity characterization is estimated to be around 25%, which might affect the precise determination of the transition intensity from RSA to SA. The theoretical model, while successful in reproducing some aspects of the experimental data, does not perfectly match the observed transition intensity and the magnitude of the redshift at lower intensities. This indicates that further refinement of the theoretical models is needed to accurately capture all aspects of the non-equilibrium dynamics in WDM. Finally, the current study is focused on copper; extending the work to other materials would further broaden the understanding of WDM properties.