The advent of X-ray free electron lasers (XFELs) producing ultrashort and intense X-ray pulses has opened new avenues for investigating atomic-level structure and dynamics. Time-resolved studies in the femto- and attosecond regimes are now possible. However, a comprehensive understanding of the complex nonlinear interactions at these extreme intensities is crucial to fully exploit XFEL capabilities. Resonant X-ray interactions, known for their high cross sections and selectivity (element, chemical site, quantum state), are widely used in techniques like XANES, various spectroscopies, RIXS, and anomalous scattering. XFELs, unlike synchrotrons, offer the generation of few-to-sub-femtosecond pulses with unprecedented intensities via the self-amplified spontaneous emission (SASE) process, enabling novel ultrafast and nonlinear techniques such as coherent core-excitation, stimulated X-ray emission, stimulated X-ray Raman scattering, and single-particle imaging experiments. At XFEL intensities, multiple X-ray photons can be absorbed within the lifetime of core-hole states, leading to X-ray pump/X-ray probe schemes and multi-core ionization experiments. Resonant excitation accompanying photoionization, including "hidden resonances," can also be exploited. The possibility of inducing multiple resonant excitations within a single pulse has been explored in the XUV regime, but less so in the X-ray regime. Multi-resonant core-level electron excitation at XFELs could enable state- and site-selective production of exotic multi-core hole states with high interaction cross sections, extending traditional XAS to multidimensional XAS and potentially enabling two-dimensional coherent X-ray spectroscopy. This paper focuses on the experimental observation and theoretical understanding of such a process, using the small molecule N2 as a well-suited system for precision measurements. In sequential core excitation, each excitation occurs at a different resonance energy due to electronic configuration rearrangement, requiring a broad incident photon bandwidth and high pulse intensity to surpass core-hole lifetimes. This contrasts with multi-excitation schemes mediated by virtual states (direct two-photon absorption). Recent calculations suggest that the shifts between subsequent resonances from different atomic sites can be small (a few eV), allowing ultrashort XFEL pulses to simultaneously cover resonance energies of sequential excitations. This is particularly relevant as the resonant pathway offers higher interaction cross sections and better selectivity compared to multi-core ionization processes.

The paper draws upon existing literature on resonant X-ray interactions, nonlinear X-ray physics with XFELs, multi-core ionization and excitation processes, and theoretical methods for calculating core-level excitation and decay spectra. Specifically, it cites studies on attosecond coherent electron motion, stimulated X-ray Raman scattering, single-particle imaging using XFELs, femtosecond electronic response to ultra-intense X-rays, double-core-hole spectroscopy, and resonance-enhanced multiphoton ionization in the X-ray regime. The literature review also acknowledges the challenges in modeling nonlinear X-ray interactions and the need for benchmark experimental data. Previous work on double core excitation in molecules and the theoretical description of double core hole states are also reviewed, emphasizing the potential of multi-dimensional XAS and its applications.

The experimental work was conducted at the European XFEL's SASE3 soft X-ray branch, using an 8 GeV electron energy to access the nitrogen K-edge energy range. X-ray pulses were delivered in trains, and electron spectra were recorded using three electron time-of-flight (TOF) analyzers at 0°, 90°, and 54.7° relative to the X-ray polarization axis. A continuous N2 flow was used. Nonlinear signatures were identified by comparing high- and low-intensity measurements, varying the X-ray focal spot size. The photon energy was calibrated using the well-known single core hole resonance. Electron TOF to kinetic energy conversion involved Jacobian corrections. Prior to the main experiment, the XFEL pulse's spectral span (bandwidth) was characterized using a grating. The average bandwidth was around 3.7 eV, and the pulse duration was estimated to be 8-10 fs. The X-ray spot size and intensity were estimated based on beamline optics. Theoretical calculations employed the restricted active space self-consistent field (RASSCF) multiconfigurational electronic structure method and its second-order perturbation theory correction (RASPT2), using the OpenMolcas software package and the ANO-L-VOZP basis set. Vibrationally resolved XANES spectra were computed using the FCclasses software, and potential energy curves (PECs) were calculated using the same electronic structure methods. Decay spectra of single and double core hole states were calculated using RASSCF/PT2 wavefunctions for bound states and the spherical continuum for ionization (SCI) model for continuum wavefunctions. SCI explicitly considers the continuum electron wavefunction by solving the radial Schrödinger equation. The partial Auger decay rates were calculated using the SCI method with strong orthogonality approximation and neglecting one-electron contributions. The calculations used a set of orthonormal molecular orbitals from the RASSI module. The angular dependence of the decay electrons is discussed, comparing results across all three analyzer angles. Partial electron yields (PEYs), total photon yields, and partial ion yields (PIYs) were extracted and analyzed.

The key finding is the experimental observation of resonant double-core excitation in N2, leading to the formation of neutral two-site double core hole (ts-DCH) states. This was achieved using intense few-femtosecond X-ray pulses from the European XFEL. The ts-DCH states were identified through their characteristic decay channels: double participator (DCH-PP) and single participator (DCH-SP) decay channels, observed in Auger electron spectra. The experimental kinetic energies of these decay electrons are in good agreement with high-level theoretical calculations. The calculated resonant core-excitation spectra for the transitions from the ground state to the single core hole (SCH) state and subsequently to the ts-DCH state, show that the resonances involved in the double excitation process fall within the bandwidth of the XFEL pulses. The experimental partial electron yield (PEY) curves for DCH-SP and SCH-P show a clear resonant behavior and a red-shift of the DCH-SP yield maximum compared to the SCH resonance, in line with the calculations. This red-shift further supports the double-excitation mechanism. The high-intensity measurements reveal the DCH-SP and DCH-PP features, while low-intensity measurements primarily show signals from SCH states. Comparison with synchrotron radiation X-ray photoelectron spectroscopy (SR XPS) data for single-photon ionizations leading to the same final states as DCH-PP and DCH-SP provides further support for the assignment of the observed peaks. Analysis of partial ion yields (PIYs) also showed evidence of nonlinear processes, such as a red-shift in the yields of multiply charged fragments, consistent with the PEY observations. However, interpretation of PIYs is complicated by the presence of alternative excitation pathways. The influence of the pulse duration (slightly longer than the core-hole lifetimes) on the results is briefly discussed.

The good agreement between experimental results and theoretical calculations strongly supports the occurrence of resonant double-core excitation. The small energy difference between the two excitation steps, as well as the broad bandwidth of the XFEL pulses, allows for the simultaneous excitation of both core electrons by a single pulse. The observed red-shift in the PEY curves for DCH transitions, compared to the SCH resonance, is consistent with the theoretical predictions. The use of high-intensity X-ray pulses is crucial for observing these nonlinear processes, as evidenced by the significant increase in the DCH-related signals at high intensity. The potential influence of competing sequential pathways, where resonant core excitation is followed by valence photoionization, is discussed. While such pathways could produce similar final states, the observed red-shift in the PEYs suggests that the double resonant pathway is dominant. The paper also addresses the influence of core-hole decay and ultrafast nuclear dynamics on the observed spectral features, noting that the larger lifetime width of the ts-DCH state is comparable to the vibrational spacing, potentially leading to strong lifetime-vibrational interference effects.

The study demonstrates the successful production of neutral two-site double core hole states in N2 via resonant double-core excitation using intense few-femtosecond X-ray pulses. The results, supported by theoretical calculations, highlight the potential of this technique for chemical analysis and the study of ultrafast dynamics. Future research directions include developing resonant core pump/resonant core probe schemes, using two-color pulses to control excitation, and extending these studies to shorter pulses to investigate coherent dynamics. Furthermore, applying advanced correlation techniques to recover high-resolution spectra from broadband pulses, and exploring the influence of spectral chirp, are also suggested.

The pulse duration in the experiment was slightly longer than the core-hole lifetimes, which could affect the interpretation of the results. The theoretical calculations were performed at the ground-state geometry and did not fully account for vibrational and lifetime broadening effects, which might contribute to slight discrepancies between experimental and theoretical values. The analysis of PIYs is complicated by the presence of alternative excitation pathways and requires coincident electron-ion detection for a more accurate interpretation.