Abstract
Coherent nonlinear light–matter interaction with X-rays gives access to a regime in ultrafast spectroscopy in which atomic resolution meets femtosecond and attosecond timescales1,2. Particularly, X-ray four-wave mixing, involving several resonant transitions in a single coherent nonlinear process, has the potential to provide information on the electronic states coupling, coherent electron motion, correlation and dynamics, with state and site selectivity3,4,5. Here we demonstrate coherent, background-free four-photon interactions with core-shell electrons using single broadband X-ray pulses from a free-electron laser. The all-X-ray four-wave mixing signals, measured in gaseous neon, arise from doubly resonant nonlinear processes involving Raman transitions6, including X-ray coherent anti-Stokes electronic Raman scattering. The 2D spectral maps (photon-in/photon-out) represent a step towards multidimensional correlation spectroscopy at the atomic scale. Using a multicolour time-delayed X-ray pulse scheme, we further demonstrate the feasibility of extending the proposed methodology to the ultrafast time domain. These results reveal potential for studying localized electron dynamics in multiple systems, from biomolecules to correlated quantum materials, with applications in areas such as energy conversion, biomedical imaging and quantum information technologies.
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Data availability
The raw data used in this study are available in a collaborative manner at the PSI Public Data Repository: https://doi.org/10.16907%2F7a7bd9c2-3258-454f-96cd-4c8f9d46464c.
Code availability
Data analysis codes are available on request. The codes for the model are available in a collaborative manner on request.
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Acknowledgements
We acknowledge the technical support of S. Tiefenbacher and all of the groups involved in the operation of SwissFEL. The research leading to these results has received support from the Swiss National Science Foundation under grant agreement no. 200021-165550/1. The Maloja instrument received support from the Swiss National Science Foundation through R’Equip grant no. 206021-182988. The work of A.S.M.-C. was financed by the European Union’s Horizon 2020 programme under the Marie Skłodowska-Curie grant agreement 884104 (PSI-FELLOW-III-3i).
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G.K. initiated and led the project. G.K. and A.S.M.-C. coordinated the project. G.K. and C.B. supervised the project and managed the funding acquisition. G.K. and T.P. developed the experimental concept and T.F. and A.C. provided advice on the experimental plan. K.S., A.A.-H., A.S., Z.S., N.Y., A.S.M.-C. and G.K. prepared the experimental beamline. E.P. prepared the X-ray pulses, in particular the set-up of the two-colour mode. All authors participated in the collection and interpretation of the experimental data. A.S.M.-C. led the data analysis, with support from S.A., J.K. and S.Z. A.S.M.-C., S.A. and G.K. developed the model and set up the required computational resources for the simulations. A.S.M.-C. and G.K. prepared the initial version of the manuscript. T.P., C.O., T.F., K.S. and A.A.-H. provided feedback on the initial draft. T.P., C.O. and T.F. provided feedback to the corresponding authors, A.S.M.-C. and G.K., in preparing the submitted manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Beam propagation in the sample.
a, Calculation of the beam distribution at two positions before entering the gas cell. b,c, Propagation around the focus. c, Intensity distribution at the focus.
Extended Data Fig. 2 BoxCARS geometry.
a, Coordinate system and arrangement of beams on the mask used for the phase-matching calculation. The holes of the spatial mask have diameters of 1 mm and therefore cover a range of angles on the order 2Θmax − 2Θmin ≈ 0.04°. b, Phase-matching efficiency (sinc2 function in equation (4)) calculated for the case of ω1 = ω2 = 867 eV, a fixed θ = θ3 = θS, as a function of a varying ω3. The refractive index value is approximated to 1 and the propagation length is set as large as the gas cell, 8 mm, thus showing an unrealistic ‘worst-case scenario’ for phase mismatch. A shorter length value would increase the phase-matching energy/angle tolerance.
Extended Data Fig. 3 Spatial characterization of the XFWM signal.
To compare the signal beam with the incoming beams, we compare the images taken with a Ce-YAG screen located after the sample with and without the incoming beam open (that is, four holes or three holes opened in the spatial mask). a, Incoming beam opened in the signal direction (four holes opened in the mask). The photon energy was set at 876 eV. b, Same image as in panel a but scaled similarly as for the signal beam (see panel c) for comparison. c, XFWM signal detected with three holes opened in the spatial mask at the same photon energy, 876 eV. The signal beam profile is similar to the open incoming beam case. d, The XFWM signal vanishes when only two holes in the mask are opened, as expected from a FWM signal. The measurement in panel d was taken at 866.8 eV, around the maximum signal strength (see Fig. 1f). All measurements were done with the same gas pressure of 800 mbar. Each image is the average of 2,000 FEL shots.
Extended Data Fig. 4 Verification of the XFWM signal at 800 mbar.
Spectral maps recorded in the same experimental conditions but with different number and position of holes opened in the spatial mask. a, Standard measurement configuration, that is, three holes of the spatial mask open, only one incoming beam closed. b,c, Photon energy maps measured closing two of the incoming beams (two beams only reaching the sample). An indicative scheme of each mask and beam block configuration is shown as inset in the three 2D spectral maps. All measurements were done with the same gas pressure of 800 mbar and 25 fs r.m.s. pulse duration.
Extended Data Fig. 5 Verification of the XFWM signal at 200 mbar.
Spectral maps recorded in the same experimental conditions but with different number and position of holes opened in the spatial mask. a, Standard measurement configuration, that is, three holes of the spatial mask open, only one incoming beam closed. b,c, Photon energy maps measured closing two of the incoming beams. An indicative scheme of each mask and beam block configuration is shown as inset in the three 2D spectral maps. All measurements were done with the same gas pressure of 200 mbar and 25 fs r.m.s. pulse duration.
Extended Data Fig. 6 Pressure dependency of the XFWM spectra.
Photon energy scans for: 800 mbar (a), 400 mbar (b) and 200 mbar (c) measured with a pulse duration of 25 fs r.m.s. and plotted as spectral maps as a function of the incoming and detected photon energies.
Extended Data Fig. 7 Reference measurement, with no sample.
Photon energy map measured with three incoming beams and a gas pressure of 0.1 mbar.
Extended Data Fig. 8 Schematic energy diagrams for the XFWM processes considered in the model.
a, Two-colour XFWM (also valid for XSRS). b, Degenerate (for |f⟩ = |e⟩) or quasi-degenerate processes. c, XTG process.
Extended Data Fig. 9 Intensity scans.
a, Average spectra measured for different FEL transmissions at 800 mbar, 876.6 eV FEL photon energy and 25 fs r.m.s. pulse duration. The black lines show the region of interest taken for the fit. b, Integrated signal within the range indicated by the vertical lines in panel a, plotted as a function of intensity (FEL transmission), together with the corresponding fit and a power-three law simulation. c, Spectra measured for different FEL transmissions at 800 mbar, 866.4 eV FEL photon energy and 25 fs r.m.s. pulse duration. The black lines show the region of interest taken for the fit shown in panel d. d, Integrated signal within the range indicated by the vertical lines in panel c, plotted as a function of intensity (FEL transmission), together with the corresponding fit and a power-three law simulation. e, Spectra measured for different FEL transmissions at 200 mbar, 866.8 eV FEL photon energy and 25 fs r.m.s. pulse duration. f, Integrated signal within the range indicated by the vertical lines in panel e, plotted as a function of intensity (FEL transmission), together with the corresponding fit and a power-three law simulation. The error is calculated as the standard deviation within all shots.
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Morillo-Candas, A.S., Augustin, S., Prat, E. et al. Coherent nonlinear X-ray four-photon interaction with core-shell electrons. Nature 649, 590–596 (2026). https://doi.org/10.1038/s41586-025-09911-1
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DOI: https://doi.org/10.1038/s41586-025-09911-1
