- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Quantum interference in two-atom resonant x-ray scattering of an intense attosecond pulse

Akilesh Venkatesh* and Phay J. Ho†

  • *Contact author: akilesh.venkatesh1@gmail.com
  • †Contact author: pho@anl.gov
PDF

Phys. Rev. A 113, 053107 – Published 11 May, 2026

DOI: https://doi.org/10.1103/97ss-g7lp

Abstract

We theoretically investigate resonant x-ray scattering from two noninteracting Ne+ ions driven by an intense attosecond pulse using a nonrelativistic, QED-based time-dependent framework. Our model includes Rabi oscillations, photoionization, Auger decay, and quantum interference among elastic scattering and resonance fluorescence pathways. We analyze how the total scattering signal depends on pulse intensity, atomic configuration, and initial electronic state. We find that the total resonant scattering yield exceeds its nonresonant counterpart; the angular dependence of the signal qualitatively resembles a two-atom structure factor; and the visibility of interference fringes is sensitive to pulse area and the initial electronic state. Only a subset of final states reached via resonance fluorescence exhibits interference, determined by the indistinguishability of photon emission pathways. Fringe visibility is maximized in the linear scattering regime, where ionization is minimal and resonance fluorescence pathways can be largely indistinguishable. These results highlight optimal conditions for applying ultrafast resonant x-ray scattering to single-particle imaging.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. W. H. Bragg and W. L. Bragg, The reflection of x-rays by crystals, Proc. R. Soc. London A 88, 428 (1913).
  2. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens, Nature (London) 400, 342 (1999).
  3. J. Miao, Computational microscopy with coherent diffractive imaging and ptychography, Nature (London) 637, 281 (2025).
  4. J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt. 21, 2758 (1982).
  5. J. Miao, D. Sayre, and H. Chapman, Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects, J. Opt. Soc. Am. A 15, 1662 (1998).
  6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, X-ray image reconstruction from a diffraction pattern alone, Phys. Rev. B 68, 140101(R) (2003).
  7. P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, A. Brachmann, P. Bucksbaum, R. Coffee, F.-J. Decker, et al., First lasing and operation of an ångstrom-wavelength free-electron laser, Nat. Photon. 4, 641 (2010).
  8. L. F. DiMauro, J. Arthur, N. Berrah, J. Bozek, J. N. Galayda, and J. Hastings, Progress report on the LCLS XFEL at SLAC, J. Phys.: Conf. Ser. 88, 012058 (2007).
  9. C. Bostedt, S. Boutet, D. M. Fritz, Z. Huang, H. J. Lee, H. T. Lemke, A. Robert, W. F. Schlotter, J. J. Turner, and G. J. Williams, Linac coherent light source: The first five years, Rev. Mod. Phys. 88, 015007 (2016).
  10. Z. Huang and K.-J. Kim, Review of x-ray free-electron laser theory, Phys. Rev. ST Accel. Beams 10, 034801 (2007).
  11. J. Rossbach, J. R. Schneider, and W. Wurth, 10 years of pioneering X-ray science at the free-electron laser FLASH at DESY, Phys. Rep. 808, 1 (2019).
  12. M. Altarelli, The European X-ray free-electron laser facility in Hamburg, Nucl. Instrum. Methods Phys. Res. B 269, 2845 (2011).
  13. J. Feldhaus, M. Krikunova, M. Meyer, T. Möller, R. Moshammer, A. Rudenko, T. Tschentscher, and J. Ullrich, AMO science at the FLASH and European XFEL free-electron laser facilities, J. Phys. B: At. Mol. Opt. Phys. 46, 164002 (2013).
  14. S. Serkez, G. Geloni, S. Tomin, G. Feng, E. V. Gryzlova, A. N. Grum-Grzhimailo, and M. Meyer, Overview of options for generating high-brightness attosecond x-ray pulses at free-electron lasers and applications at the European XFEL, J. Opt. 20, 024005 (2018).
  15. T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi, T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa, et al., A compact X-ray free-electron laser emitting in the sub-Ångström region, Nat. Photon. 6, 540 (2012).
  16. M. Yabashi, Status and perspective on the SACLA facility (Conference Presentation), in Proc. SPIE 11035, Optics Damage and Materials Processing by EUV/X-ray Radiation VII, Vol. 11035, edited by L. Juha, S. Bajt, and S. Guizard (SPIE, Philadelphia, 2019) p. 1103502.
  17. T. Schietinger, M. Pedrozzi, M. Aiba, V. Arsov, S. Bettoni, B. Beutner, M. Calvi, P. Craievich, M. Dehler, F. Frei, et al., Commissioning experience and beam physics measurements at the SwissFEL injector test facility, Phys. Rev. Accel. Beams 19, 100702 (2016).
  18. E. Prat, P. Dijkstal, E. Ferrari, and S. Reiche, Demonstration of large bandwidth hard x-ray free-electron laser pulses at SwissFEL, Phys. Rev. Lett. 124, 074801 (2020).
  19. F. Nolting, C. Bostedt, T. Schietinger, and H. Braun, The Swiss light source and SwissFEL at the Paul Scherrer Institute, Eur. Phys. J. Plus 138, 126 (2023).
  20. H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. DePonte, U. Weierstall, et al., Femtosecond x-ray protein nanocrystallography, Nature (London) 470, 73 (2011).
  21. M. M. Seibert, T. Ekeberg, F. R. Maia, M. Svenda, J. Andreasson, O. Jönsson, D. Odić, B. Iwan, A. Rocker, D. Westphal, et al., Single mimivirus particles intercepted and imaged with an x-ray laser, Nature (London) 470, 78 (2011).
  22. T. Ekeberg, M. Svenda, C. Abergel, F. R. N. C. Maia, V. Seltzer, J.-M. Claverie, M. Hantke, O. Jönsson, C. Nettelblad, G. Van Der Schot, et al., Three-dimensional reconstruction of the giant mimivirus particle with an x-ray free-electron laser, Phys. Rev. Lett. 114, 098102 (2015).
  23. T. Gorkhover, S. Schorb, R. Coffee, M. Adolph, L. Foucar, D. Rupp, A. Aquila, J. D. Bozek, S. W. Epp, B. Erk, et al., Femtosecond and nanometre visualization of structural dynamics in superheated nanoparticles, Nat. Photon. 10, 93 (2016).
  24. L. F. Gomez, K. R. Ferguson, J. P. Cryan, C. Bacellar, R. M. P. Tanyag, C. Jones, S. Schorb, D. Anielski, A. Belkacem, C. Bernando, et al., Shapes and vorticities of superfluid helium nanodroplets, Science 345, 906 (2014).
  25. K. R. Ferguson, M. Bucher, T. Gorkhover, S. Boutet, H. Fukuzawa, J. E. Koglin, Y. Kumagai, A. Lutman, A. Marinelli, M. Messerschmidt, et al., Transient lattice contraction in the solid-to-plasma transition, Sci. Adv. 2, e1500837 (2016).
  26. Projected Run 25 LCLS FEL parameters (2024), https://lcls.slac.stanford.edu/sites/default/files/2024-12/LCLS-Parameters-Run-25_0.pdf.
  27. Instrument specifications (2025), https://www.xfel.eu/facility/instruments/mid/instrument_specifications/index_eng.html.
  28. M. Yabashi, H. Tanaka, and T. Ishikawa, Overview of the SACLA facility, J. Synchrotron Radiat. 22, 477 (2015).
  29. K. Tono, T. Hara, M. Yabashi, and H. Tanaka, Multiple-beamline operation of SACLA, J. Synchrotron Radiat. 26, 595 (2019).
  30. R. Neutze, R. Wouts, D. Van der Spoel, E. Weckert, and J. Hajdu, Potential for biomolecular imaging with femtosecond x-ray pulses, Nature (London) 406, 752 (2000).
  31. L. Young, E. P. Kanter, B. Krässig, Y. Li, A. M. March, S. T. Pratt, R. Santra, S. H. Southworth, N. Rohringer, L. F. DiMauro, et al., Femtosecond electronic response of atoms to ultra-intense x-rays, Nature (London) 466, 56 (2010).
  32. B. Rudek, S.-K. Son, L. Foucar, S. W. Epp, B. Erk, R. Hartmann, M. Adolph, R. Andritschke, A. Aquila, N. Berrah, et al., Ultra-efficient ionization of heavy atoms by intense x-ray free-electron laser pulses, Nat. Photon. 6, 858 (2012).
  33. G. Doumy, C. Roedig, S.-K. Son, C. I. Blaga, A. D. DiChiara, R. Santra, N. Berrah, C. Bostedt, J. D. Bozek, P. H. Bucksbaum, et al., Nonlinear atomic response to intense ultrashort x rays, Phys. Rev. Lett. 106, 083002 (2011).
  34. M. Hoener, L. Fang, O. Kornilov, O. Gessner, S. T. Pratt, M. Gühr, E. P. Kanter, C. Blaga, C. Bostedt, J. D. Bozek, et al., Ultraintense x-ray induced ionization, dissociation, and frustrated absorption in molecular nitrogen, Phys. Rev. Lett. 104, 253002 (2010).
  35. S. Schorb, D. Rupp, M. L. Swiggers, R. N. Coffee, M. Messerschmidt, G. Williams, J. D. Bozek, S.-I. Wada, O. Kornilov, T. Möller, and C. Bostedt, Size-dependent ultrafast ionization dynamics of nanoscale samples in intense femtosecond x-ray free-electron-laser pulses, Phys. Rev. Lett. 108, 233401 (2012).
  36. C. Bostedt, E. Eremina, D. Rupp, M. Adolph, H. Thomas, M. Hoener, A. R. B. de Castro, J. Tiggesbäumker, K.-H. Meiwes-Broer, T. Laarmann, et al., Ultrafast x-ray scattering of xenon nanoparticles: Imaging transient states of matter, Phys. Rev. Lett. 108, 093401 (2012).
  37. P. J. Ho, C. Knight, M. Tegze, G. Faigel, C. Bostedt, and L. Young, Atomistic three-dimensional coherent x-ray imaging of nonbiological systems, Phys. Rev. A 94, 063823 (2016).
  38. E. P. Kanter, B. Krässig, Y. Li, A. M. March, P. Ho, N. Rohringer, R. Santra, S. H. Southworth, L. F. DiMauro, G. Doumy, et al., Unveiling and driving hidden resonances with high-fluence, high-intensity x-ray pulses, Phys. Rev. Lett. 107, 233001 (2011).
  39. P. J. Ho, B. J. Daurer, M. F. Hantke, J. Bielecki, A. Al Haddad, M. Bucher, G. Doumy, K. R. Ferguson, L. Flückiger, T. Gorkhover, et al., The role of transient resonances for ultra-fast imaging of single sucrose nanoclusters, Nat. Commun. 11, 167 (2020).
  40. S. Kuschel, P. J. Ho, A. Al Haddad, F. F. Zimmermann, L. Flueckiger, M. R. Ware, J. Duris, J. P. MacArthur, A. Lutman, M.-F. Lin, et al., Non-linear enhancement of ultrafast x-ray diffraction through transient resonances, Nat. Commun. 16, 847 (2025).
  41. J. Duris, S. Li, T. Driver, E. G. Champenois, J. P. MacArthur, A. A. Lutman, Z. Zhang, P. Rosenberger, J. W. Aldrich, R. Coffee, et al., Tunable isolated attosecond x-ray pulses with gigawatt peak power from a free-electron laser, Nat. Photon. 14, 30 (2020).
  42. A. Ulmer, S. Kuschel, B. Langbehn, L. Hecht, S. Dold, L. Rönnebeck, T. Driver, J. Duris, A. Kamalov, X. Li, et al., Exploring damage reduction and scattering cross section enhancement in attosecond x-ray imaging of neon near the K-edge, Bull. Am. Phys. Soc. S11-005 (2023), https://ui.adsabs.harvard.edu/abs/2023APS..DMPS11005U/abstract.
  43. T. Richter, Interference between the resonance fluorescence fields from two independent atoms and spatial two-photon correlations, Opt. Commun. 80, 285 (1991).
  44. U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, D. J. Wineland, and M. G. Raizen, Young's interference experiment with light scattered from two atoms, Phys. Rev. Lett. 70, 2359 (1993).
  45. W. M. Itano, U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, M. G. Raizen, and D. J. Wineland, Light scattered from two atoms, in Proceedings of the International Conference on Lasers '93, Lake Tahoe, Nevada, 1993, edited by V. J. Corcoran and T. A. Goldman (STS Press, McLean, VA, 1994), pp. 412–419.
  46. W. M. Itano, J. C. Bergquist, J. J. Bollinger, D. J. Wineland, U. Eichmann, and M. G. Raizen, Complementarity and young's interference fringes from two atoms, Phys. Rev. A 57, 4176 (1998).
  47. G. S. Agarwal, J. Von Zanthier, C. Skornia, and H. Walther, Intensity-intensity correlations as a probe of interferences under conditions of noninterference in the intensity, Phys. Rev. A 65, 053826 (2002).
  48. S. Grundmann, D. Trabert, K. Fehre, N. Strenger, A. Pier, L. Kaiser, M. Kircher, M. Weller, S. Eckart, L. P. H. Schmidt, et al., Zeptosecond birth time delay in molecular photoionization, Science 370, 339 (2020).
  49. F. K. Gel'mukhanov, L. N. Mazalov, and N. A. Shklyaeva, An interference effect in x-ray fluorescence spectra, Zh. Eksp. Teor. Fiz 69, 1971 (1975) [Sov. Phys. JETP 42, 1001 (1975)].
  50. Y. Ma and M. Blume, Interference of fluorescence x rays and coherent excitation of core levels, Rev. Sci. Instrum. 66, 1543 (1995).
  51. Y. Ma, X-ray absorption, emission, and resonant inelastic scattering in solids, Phys. Rev. B 49, 5799 (1994).
  52. G. V. Smirnov and V. G. Kohn, Theory of nuclear resonant scattering of synchrotron radiation in the presence of diffusive motion of nuclei, Phys. Rev. B 52, 3356 (1995).
  53. V. G. Kohn and G. V. Smirnov, Theory of nuclear resonant scattering of synchrotron radiation in the presence of diffusive motion of nuclei. II., Phys. Rev. B 57, 5788 (1998).
  54. G. Smirnov, General properties of nuclear resonant scattering, Hyperfine Interact. 123, 31 (1999).
  55. M. A. Andreeva and B. Lindgren, Nuclear resonant spectroscopy at Bragg reflections from periodic multilayers: Basic effects and applications, Phys. Rev. B 72, 125422 (2005).
  56. R. H. Brown and R. Q. Twiss, Correlation between photons in two coherent beams of light, Nature (London) 177, 27 (1956).
  57. P. J. Ho, C. Knight, and L. Young, Fluorescence intensity correlation imaging with high spatial resolution and elemental contrast using intense x-ray pulses, Struct. Dyn. 8, 044101 (2021),.
  58. P. J. Ho, Bringing interferometric imaging into the x-ray regime, Physics 16, 66 (2023).
  59. A. Classen, K. Ayyer, H. N. Chapman, R. Röhlsberger, and J. von Zanthier, Incoherent diffractive imaging via intensity correlations of hard x rays, Phys. Rev. Lett. 119, 053401 (2017).
  60. L. M. Lohse, M. Vassholz, and T. Salditt, On incoherent diffractive imaging, Acta Crystallogr. A 77, 480 (2021).
  61. F. Trost, K. Ayyer, M. Prasciolu, H. Fleckenstein, M. Barthelmess, O. Yefanov, J. L. Dresselhaus, C. Li, S. Bajt, J. Carnis, et al., Imaging via correlation of x-ray fluorescence photons, Phys. Rev. Lett. 130, 173201 (2023).
  62. R. G. Radloff, F. F. Zimmermann, S. Li, S. Kuschel, A. Ulmer, Y. Sun, T. Sato, P. Sun, J. Haber, D. Zhu, et al., Fluorescence intensity correlations enable 3D imaging without sample rotations, arXiv:2510.24386.
  63. A. Venkatesh and P. J. Ho, Effect of Rabi dynamics in resonant x-ray scattering of intense attosecond pulses, Phys. Rev. A 111, L021101 (2025).
  64. A. Venkatesh and P. J. Ho, Theory of resonant x-ray scattering with ultrafast intense pulses, Phys. Rev. A 111, 023101 (2025).
  65. A. Venkatesh and F. Robicheaux, Simulation of nonlinear Compton scattering from bound electrons, Phys. Rev. A 101, 013409 (2020).
  66. R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 1983).
  67. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, New York, 1967).
  68. S. M. Cavaletto, C. Buth, Z. Harman, E. P. Kanter, S. H. Southworth, L. Young, and C. H. Keitel, Resonance fluorescence in ultrafast and intense x-ray free-electron-laser pulses, Phys. Rev. A 86, 033402 (2012).
  69. J. Eberly, Area theorem rederived, Opt. Express 2, 173 (1998).
  70. P. J. Ho and C. Knight, Large-scale atomistic calculations of clusters in intense x-ray pulses, J. Phys. B 50, 104003 (2017).
  71. M. Macovei, J. Evers, G.-X. Li, and C. H. Keitel, Strong-field spatial interference in a tailored electromagnetic bath, Phys. Rev. Lett. 98, 043602 (2007).
  72. F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, Creation and measurement of a coherent superposition of quantum states, Phys. Rev. Lett. 91, 213001 (2003).
  73. F. Vewinger, M. Heinz, U. Schneider, C. Barthel, and K. Bergmann, Amplitude and phase control of a coherent superposition of degenerate states. II. Experiment, Phys. Rev. A 75, 043407 (2007).
  74. J. Yan, W. Qin, Y. Chen, W. Decking, P. Dijkstal, M. Guetg, I. Inoue, N. Kujala, S. Liu, T. Long, et al., Terawatt-attosecond hard X-ray free-electron laser at high repetition rate, Nat. Photon. 18, 1293 (2024).
  75. A. Venkatesh and P. J. Ho, Dataset of resonant x-ray scattering of two neon ions exposed to intense attosecond x-ray pulses, Zenodo (2025), doi:10.5281/zenodo.15604377.
  76. K. Li, M. Labeye, P. J. Ho, M. B. Gaarde, and L. Young, Resonant propagation of x rays from the linear to the nonlinear regime, Phys. Rev. A 102, 053113 (2020).
  77. Y. Li, C. Gao, W. Dong, J. Zeng, Z. Zhao, and J. Yuan, Coherence and resonance effects in the ultra-intense laser-induced ultrafast response of complex atoms, Sci. Rep. 6, 18529 (2016).
  78. D. A. Lidar, Lecture notes on the theory of open quantum systems, arXiv:1902.00967.
  79. D. Manzano, A short introduction to the Lindblad master equation, AIP Adv. 10, 025106 (2020).
  80. F. Campaioli, J. H. Cole, and H. Hapuarachchi, Quantum master equations: Tips and tricks for quantum optics, quantum computing, and beyond, PRX Quantum 5, 020202 (2024).
  81. P. J. Ho, C. Bostedt, S. Schorb, and L. Young, Theoretical tracking of resonance-enhanced multiple ionization pathways in x-ray free-electron laser pulses, Phys. Rev. Lett. 113, 253001 (2014).

Outline

Information

Phys. Rev. A 113, 053107– Published 11 May, 2026
  • Received 6 June 2025
  • Revised 14 December 2025
  • Accepted 16 April 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation