# Approaching the Attosecond Frontier of Dynamics in Matter with the Concept of X-ray Chronoscopy
## 2 Materials and Methods
### Time-Dependent Studies of Saturable Absorption at Hard X-ray Energies  
Nonlinear spectroscopy has been studied theoretically and experimentally over decades in the optical domain [30] but has been made available recently at Angstrom wavelengths due to the sharp increase in photon flux and short pulse duration provided by the XFELs [31]. Electronic states involved in X-ray-matter interaction have characteristic lifetimes extending to the sub-femtosecond domain, which is significantly shorter than the lifetimes expected for optical light-matter interaction processes. Furthermore, photoexcitation of a single core-level electron triggers a series of Auger transitions and electronic cascades lasting for a few femtoseconds. For this reason, the ultra-short X-ray pulses are necessary to overtake the Auger decays and to initiate nonlinear phenomena in the hard X-ray regime [32]. Historically, the first experiments devoted to X-ray-matter interactions at strong X-ray fields reported processes such as hollow-atom and high charge states formation [33] or plasma creation [34] as well as processes with low cross-sections such as two-photon absorption [35], stimulated X-ray emission [36] or X-ray-laser wave mixing [37]. In the present study, we focus on the saturable absorption (SA) mechanism, a first-order nonlinear process of interaction of X-rays with matter [38; 39; 40]. We deliberately chose this process as a case study for the X-ray chronoscopy method for several reasons. First, the incidence X-ray intensity required to initiate this process was already explored at XFELs, and the available experimental conditions are sufficient to initiate the SA process at every XFEL facility. Second, SA allows for a simple description of initial and final atomic configurations, and therefore the experimental data can be evaluated with known theoretical models and by employing developed theories dedicated to the electronic structure of many-electron systems. The process of saturable absorption is the change of absorption properties of a material induced by the propagation of a strong laser beam. Such a process is a well-known phenomenon in laser technology at optical wavelengths, but it was only recently discovered in the X-ray regime due to the application of XFEL beams. Saturable absorption relates to the depletion of atomic ground-state species to the extent at which the sample becomes transparent to other photons of the same beam, i.e., there are no more absorbing states in  
Figure 2: The streaking of an X-ray-induced photoelectron pulse in a THz vector potential. The X-ray pulse temporal profile is encoded in the energy spectrum of the measured photoelectron spectrum. the system. In order to access such a regime, the X-ray ionization yields have to be higher than the decay rate transitions that allow the system to recover to the ground state. The first experiment devoted to saturable absorption and to processes leading to sample transparency was performed on the L absorption edge of aluminum [40]. The experiment was executed at FLASH XFEL [41] and with 92.5 eV photons of the pulse duration of 15 fs at X-ray fluences up to 2 J/cm\({}^{2}\). It was demonstrated that X-ray-induced transmission in the aluminum can be increased by up to 65% at the highest applied pulse fluences. From the experiment, it was concluded that the first part of the pulse ionizes L-shell electrons, and the created L-shell holes are filled at lower decay rates than ionization yields. The relatively long lifetime of the L-shell hole (40 fs) in aluminum allows for the strong depletion of L-states when using 15 fs X-ray pulse leading to the transparency increase. At hard X-ray energies, the X-ray saturable absorption was detected for metallic Fe at the K absorption edge [38]. X-rays with energies around 7100 eV were used to monitor the change of the absorption coefficient over a 30 eV energy range. It was shown that in the regime of saturable absorption, the absorption coefficient is changing for both below- and above-edge energies.