Core-level nonlinear spectroscopy triggered by stochastic X-ray pulses

Stochastic processes are highly relevant in research fields as different as neuroscience, economy, ecology, chemistry, and fundamental physics. However, due to their intrinsic unpredictability, stochastic mechanisms are very challenging for any kind of investigations and practical applications. Here we report the deliberate use of stochastic X-ray pulses in two-dimensional spectroscopy to the simultaneous mapping of unoccupied and occupied electronic states of atoms in a regime where the opacity and transparency properties of matter are subject to the incident intensity and photon energy. A readily transferable matrix formalism is presented to extract the electronic states from a dataset measured with the monitored input from a stochastic excitation source. The presented formalism enables investigations of the response of the electronic structure to irradiation with intense X-ray pulses while the time structure of the incident pulses is preserved.

3 during the synchrotron beamtime. A Pilatus 100K detector was used for the detection of the XES signal. The different source size, background conditions and detector pixel size, resulting in different energy resolutions for the detected XES signals make a direct comparison of the datasets tedious.
Nonetheless the lineshape of the K and K emission lines is well reproduced as well as the absorption edge and its width. This demonstrates a sufficient resolution for recording the RXES maps.
However, in the XAS curve extracted from the reconstructed RXES map the pre-edge feature, which arises from 1s to 3d and valence band transitions, of the synchrotron data (where the total incident intensity was about ten orders more than 10 orders of magnitude lower) is not reproduced. Possible reasons for this present limitation might be the measurement precision of the transmissive and the von Hamos spectrometer (elaborate data treatment for the transmissive spectrometer to attenuate background contributions; single-shot counting statistics for the emission spectrometer). The experimental noise required and the proneness if matrix inversion algorithms to numerical errors required using numerical optimization which in turn may affect the accuracy of the final result, hence in a larger discrepancy in the pre-edge region. This could not be mediated by using a larger number of incident pulses, since the reconstructed maps are consistent with each other when varying the number of pulses for a given incident intensity (see also Supplementary Figure 2). In future the exclusive use of direct photon detecting sensors or a different tuning of the SASE beam (less spectral modes, increased jitter in energy) might help remedying the current limitations.  towards studying X-ray interaction with matter in the vicinity of a core ionization threshold while using X-ray pulses with a similar duration than the excited electronic states and rearrangement processes.  Basics of the mathematical reconstruction procedure on the simplest example of two different incident energies, two pulses and one emission energy (k=2, m=2, n=1): In the experiment for each pulse Ixg and IAg and IBg with g=1,2 are measured for the emission energy Ex and the incident energies EA and EB. Note that here only the number of photons within a given incident energy bin are considered as the intensity elements. The aim of the reconstruction methodology is to retrieve the response of the system (a, b), which corresponds to the RXES map.
Since two incident energies are considered, at least two pulses need to be considered, marked by the indexes 1 and 2, to solve the system.
At the same time it is worthwhile to stress that the mathematical procedure remains fully valid when applying higher total incidence X-ray intensities in units of W/cm 2 (equivalently the total or integral number of photons per unit time and per unit area) leading to possibly to a different intensity response

Supplementary Note 1: Crystal-field multiplet calculations
Calculations of the K spectrum of Fe2O3 were performed using the crystal field multiplet model through the CTM4XAS interface [1]. A ligand-field splitting of 1.45 eV was used and the d-d interactions were reduced to 70% of their original values [2]. A Gaussian broadening of 2.0 eV was applied to all transitions. Different Lorentzian broadenings were used for the K1,3 (0.8 eV FWHM) and K' (2.0 eV FWHM) to account for state-dependent lifetimes [3].

Supplementary Note 2: Time-dependent rate equations
The rate equations for two-photon absorption (TPA) at an incident photon energy below the ionization threshold as well as for one photon absorption (OPA) at energies above the ionization threshold were   For the temporal spiky pulse structure, the degree of second order of coherence, g (2) was calculated with the following equation: Supplementary Figure 8 Degree of second order coherence for randomly generated incident X-ray pulses. From g (2) the coherence time of simulated pulse was estimated to be 0.5 fs.

Supplementary Note 3: Monte Carlo simulation of the vacancy transfer
Monte Carlo simulations of all possible fluorescence and Auger decay cascades were realized starting from one electron vacancy in the K-shell of a metallic Fe atom with the initial electronic configuration 1s 2 2s 2 2p 6 3s 2 3p 6 3d 6 4s 2 . The creation of an unoccupied state in the 1s 2 level is the most likely event in the experiment realized. In the simulation the type of decay, fluorescence or Auger, is first randomly selected using the corresponding Auger yield [8]. In case of a fluorescence decay, the vacancy in the electronic configuration is removed and another one is created on an energy level drawn using X-ray transition probabilities [9] as weighting factors. In case of an Auger decay, the vacancy is removed and two are created at energy levels taken from an Auger decay path chosen randomly among the allowed ones. At every electron vacancy removal and creation a decay schedule is updated using reported lifetime broadenings [8]. The decay schedule is used to process the transfer of the vacancies decays in the correct time sequence and allows determining the temporal evolution of energy levels' occupancies. Once an electron vacancy has been processed and the decay schedule has been updated, the simulation treats the next scheduled electron vacancy decay in the described way and the algorithm continues iteratively. If the algorithm encounters an electron vacancy which occupies a zero-Auger yield energy level or if higher energy levels are empty of electrons, the electron vacancy is left at its position and the next scheduled one is processed. The simulation stops when the decay schedule contains no more electron vacancies which can be processed.

Supplementary Note 4: Xraypac simulations
The field-dependent X-ray-induced dynamics simulations were performed using the Xmdyn module in Xraypac [10][11][12][13][14]. As Xmdyn simulations can only start from the standard atomic configuration, atomic iron was used as the model system. The incident energy was chosen 2 eV above the calculated ionization edge for iron. This is below the ionization energy for Fe+ and therefore, once ionized, the system cannot have further 1s-ionisation. The simulations were performed at different incident intensities and between 200 and 20000 independent runs were computed for each intensity in order to obtain proper statistics. More runs are required to obtain converged results for the lowest intensities since excitations events are less likely. A constant intensity shift and cross-section scaling was applied in order to align the calculated results to the experimental data.