https://www.nature.com/articles/s41586-026-10209-z Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Advertisement Advertisement Nature * View all journals * Search * Log in * Content Explore content * About the journal * Publish with us * Subscribe * Sign up for alerts * RSS feed 1. nature 2. articles 3. article * Article * Published: 25 March 2026 Superluminal correlations in ensembles of optical phase singularities * T. Bucher ORCID: orcid.org/0000-0002-7899-5672^1^ na1, * A. Gorlach ORCID: orcid.org/0000-0002-0347-7141^1^ na1, * A. Niedermayr ORCID: orcid.org/0000-0002-7915-1201^1, * Q. Yan^1, * H. Nahari^1, * K. Wang ORCID: orcid.org/0000-0002-7857-2717^2, * R. Ruimy^1, * Y. Adiv ORCID: orcid.org/0000-0002-7451-4130^1, * M. Yannai^1, * T. L. Abudi ORCID: orcid.org/0000-0002-4432-4926^1, * E. Janzen^3, * C. Spaegele^4, * C. Roques-Carmes^5, * J. H. Edgar^3, * F. H. L. Koppens^6,7, * G. M. Vanacore ORCID: orcid.org/0000-0002-7228-7982^8, * H. H. Sheinfux ORCID: orcid.org/0000-0001-8421-5663^9, * S. Tsesses ORCID: orcid.org/0000-0003-0167-3402^1,10 & * ... * I. Kaminer ORCID: orcid.org/0000-0003-2691-1892^1,11 Show authors Nature volume 651, pages 920-926 (2026)Cite this article * 3338 Accesses * 154 Altmetric * Metrics details Subjects * Nanophotonics and plasmonics * Scanning electron microscopy * Sub-wavelength optics Abstract Phase singularities--points carrying quantized topological charge--are universal features found across diverse wave systems from superfluids and superconductors to acoustic and optical fields^1,2,3,4. Ensembles of these singularities exhibit distance correlations resembling particles in liquids^5,6,7,8, extensively studied for their role in exotic material phases^9,10,11. By contrast, the full correlations in phase space that govern the system evolution have remained unexplored and experimentally inaccessible. Here we directly measure the ultrafast dynamics of optical singularity ensembles, capturing their full phase-space correlations, presenting the joint distance-velocity distribution. Our observations show a breakdown of the particle-singularity analogy^12: phase singularities accelerate towards formally divergent velocities in the moment before annihilation^7,13,14, indicated by measurements of velocities exceeding the speed of light. These apparent superluminal velocities are paradoxically amplified by the slow group velocity of hyperbolic phonon polaritons in our material platform, hexagonal boron nitride membranes^15,16,17,18,19. We demonstrate these phenomena using combined hardware and algorithmic advances in ultrafast electron microscopy^18,20,21,22,23,24,25, achieving spatial and temporal resolutions, each an order of magnitude below the polaritonic wavelength and cycle period. Our findings deepen our understanding of phase singularities and their universality, enabling to probe topological defect dynamics at previously unattainable timescales. Access through your institution Buy or subscribe This is a preview of subscription content, access via your institution Access options Access through your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any time Learn more Subscribe to this journal Receive 51 print issues and online access $199.00 per year only $3.90 per issue Learn more Buy this article * Purchase on SpringerLink * Instant access to the full article PDF. USD 39.95 [Buy Now] Prices may be subject to local taxes which are calculated during checkout Additional access options: * Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support Fig. 1: Deep sub-wavelength and deep sub-cycle imaging of optical phase singularities in hBN, recording both phase and group dynamics. [41586_2026_10209_Fig1_HTML] Fig. 2: Deep sub-cycle annihilation of singularities, showing an example of acceleration towards formally divergent velocities along a characteristic space-time trajectory. [41586_2026_10209_Fig2_HTML] Fig. 3: Distance correlations and velocity distributions of singularities. [41586_2026_10209_Fig3_HTML] Fig. 4: Full phase-space correlations of singularities. [41586_2026_10209_Fig4_HTML] Similar content being viewed by others [41566_2024] Coherently amplified ultrafast imaging using a free-electron interferometer Article 03 July 2024 [41563_2024] Solution-phase sample-averaged single-particle spectroscopy of quantum emitters with femtosecond resolution Article 08 April 2024 [42254_2021] Optical superoscillation technologies beyond the diffraction limit Article 25 October 2021 Data availability The data supporting the findings of this study are available from the corresponding authors upon reasonable request. References 1. Embon, L. et al. Imaging of super-fast dynamics and flow instabilities of superconducting vortices. Nat. Commun. 8, 85 (2017). Article ADS CAS PubMed PubMed Central Google Scholar 2. Sachkou, Y. P. et al. Coherent vortex dynamics in a strongly interacting superfluid on a silicon chip. Science 366, 1480-1485 (2019). Article ADS CAS PubMed Google Scholar 3. Bliokh, K. Y. et al. Roadmap on structured waves. J. Opt. 25, 103001 (2023). Article ADS Google Scholar 4. Dennis, M. R., King, R. P., Jack, B., O'Holleran, K. & Padgett, M. J. Isolated optical vortex knots. Nat. Phys. 6, 118-121 (2010). Article CAS Google Scholar 5. Berry, M. V. Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves. J. Phys. A Math. Gen. 11, 27-38 (1978). Article ADS MathSciNet Google Scholar 6. De Angelis, L., Alpeggiani, F., Di Falco, A. & Kuipers, L. Spatial distribution of phase singularities in optical random vector waves. Phys. Rev. Lett. 117, 093901 (2016). Article ADS PubMed Google Scholar 7. Berry, M. V. & Dennis, M. R. Phase singularities in isotropic random waves. Proc. R. Soc. Lond. A 456, 2059-2079 (2000). Article ADS MathSciNet Google Scholar 8. Hansen, J. P. & McDonald, I. R. Theory of Simple Liquids: With Applications to Soft Matter (Academic Press, 2013). 9. Toulouse, G. & Kleman, M. Principles of a classification of defects in ordered media. J. Physique Lett. 37, 149-151 (1976). Article Google Scholar 10. Blatter, G., Feigel'man, M. V., Geshkenbein, V. B., Larkin, A. I. & Vinokur, V. M. Vortices in high-temperature superconductors. Rev. Mod. Phys. 66, 1125-1388 (1994). Article ADS CAS Google Scholar 11. Drori, L. et al. Quantum vortices of strongly interacting photons. Science 381, 193-198 (2023). Article ADS CAS PubMed Google Scholar 12. Sugic, D. et al. Particle-like topologies in light. Nat. Commun. 12, 6785 (2021). Article ADS CAS PubMed PubMed Central Google Scholar 13. Indebetouw, G. Optical vortices and their propagation. J. Mod. Opt. 40, 73-87 (1993). Article ADS Google Scholar 14. Freund, I. Optical vortex trajectories. Opt. Commun. 181, 19-33 (2000). Article ADS CAS Google Scholar 15. Li, P. et al. Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing. Nat. Commun. 6, 7507 (2015). Article ADS CAS PubMed PubMed Central Google Scholar 16. Yoxall, E. et al. Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity. Nat. Photon. 9, 674-678 (2015). Article ADS CAS Google Scholar 17. Caldwell, J. D. et al. Photonics with hexagonal boron nitride. Nat. Rev. Mater. 4, 552-567 (2019). Article ADS CAS Google Scholar 18. Kurman, Y. et al. Spatiotemporal imaging of 2D polariton wave packet dynamics using free electrons. Science 372, 1181-1186 (2021). Article ADS CAS PubMed Google Scholar 19. Giles, A. J. et al. Ultralow-loss polaritons in isotopically pure boron nitride. Nat. Mater. 17, 134-139 (2018). Article ADS CAS PubMed Google Scholar 20. Barwick, B., Flannigan, D. J. & Zewail, A. H. Photon-induced near-field electron microscopy. Nature 462, 902-906 (2009). Article ADS CAS PubMed Google Scholar 21. Feist, A. et al. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature 521, 200-203 (2015). Article ADS CAS PubMed Google Scholar 22. Madan, I. et al. Holographic imaging of electromagnetic fields via electron-light quantum interference. Sci. Adv. 5, eaav8358 (2019). Article ADS CAS PubMed PubMed Central Google Scholar 23. Nabben, D., Kuttruff, J., Stolz, L., Ryabov, A. & Baum, P. Attosecond electron microscopy of sub-cycle optical dynamics. Nature 619, 63-67 (2023). Article ADS CAS PubMed Google Scholar 24. Gaida, J. H. et al. Attosecond electron microscopy by free-electron homodyne detection. Nat. Photon. 18, 509-515 (2024). Article ADS CAS Google Scholar 25. Bucher, T. et al. Coherently amplified ultrafast imaging using a free-electron interferometer. Nat. Photon. 18, 809-815 (2024). Article ADS CAS Google Scholar 26. Nye, J. F. & Berry, M. V. Dislocations in wave trains. Proc. R. Soc. Lond. A 336, 165-190 (1974). Article ADS MathSciNet Google Scholar 27. Nye, J. F. The motion and structure of dislocations in wavefronts. Proc. R. Soc. Lond. A 378, 219-239 (1981). Article ADS MathSciNet Google Scholar 28. Andersen, M. F. et al. Quantized rotation of atoms from photons with orbital angular momentum. Phys. Rev. Lett. 97, 170406 (2006). Article ADS CAS PubMed Google Scholar 29. Tsesses, S. et al. Tunable photon-induced spatial modulation of free electrons. Nat. Mater. 22, 345-352 (2023). Article ADS CAS PubMed Google Scholar 30. Torok, P. & Munro, P. R. T. 2004. The use of Gauss-Laguerre vector beams in STED microscopy. Opt. Express 12, 3605-3617 (2004). Article ADS PubMed Google Scholar 31. Berry, M. et al. Roadmap on superoscillations. J. Opt. 21, 053002 (2019). Article ADS CAS Google Scholar 32. Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons. Nature 412, 313-316 (2001). Article ADS CAS PubMed Google Scholar 33. Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12, 5448-5456 (2004). Article ADS PubMed Google Scholar 34. Davis, T. J. et al. Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution. Science 368 , eaba6415 (2020). Article CAS PubMed Google Scholar 35. Ni, J. et al. Multidimensional phase singularities in nanophotonics. Science 374, eabj0039 (2021). Article CAS PubMed Google Scholar 36. Berry, M. V. & Dennis, M. R. Knotted and linked phase singularities in monochromatic waves. Proc. R. Soc. Lond. A 457, 2251-2263 (2001). Article ADS MathSciNet Google Scholar 37. Ostrovsky, E., Cohen, K., Tsesses, S., Gjonaj, B. & Bartal, G. Nanoscale control over optical singularities. Optica 5, 283-288 (2018). Article ADS CAS Google Scholar 38. Eckhardt, B., Dorr, U., Kuhl, U. & Stockmann, H. J. Correlations of electromagnetic fields in chaotic cavities. Europhys. Lett. 46 , 134-140 (1999). Article ADS CAS Google Scholar 39. Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride. Science 343, 1125-1129 (2014). Article ADS CAS PubMed Google Scholar 40. Berry, M. V. Vector fields for monochromatic waves with general dispersion: vortex and stagnation singularities. J. Phys. A Math. Theor. 58, 275201 (2025). Article MathSciNet Google Scholar 41. Vasnetsov, M., Pas' ko, V., Khoroshun, A., Slyusar, V. & Soskin, M. Observation of superluminal wave-front propagation at the shadow area behind an opaque disk. Opt. Lett. 32, 1830-1832 (2007). Article ADS CAS PubMed Google Scholar 42. Dominici, L. et al. Full-Bloch beams and ultrafast Rabi-rotating vortices. Phys. Rev. Res. 3, 013007 (2021). Article CAS Google Scholar 43. Jordan, A. N., Howell, J. C., Vamivakas, N. & Karimi, E. Superoscillations and physical applications. In Operator Theory (eds Alpay, D. et al.) https://doi.org/10.1007/ 978-3-0348-0692-3_101-1 (Springer, 2025). 44. Berry, M. V. Waves near zeros. In Proc. Conference on Coherence and Quantum Optics, CMB1, OSA Technical Digest (CD) (Optica Publishing Group, 2007). 45. Zhao, Z. et al. Experimental demonstration of 16-Gbit/s millimeter-wave communications link using thin metamaterial plates to generate data-carrying orbital-angular-momentum beams. In IEEE International Conference on Communications, 1392-1397 (IEEE, 2015). 46. Stav, T. et al. Quantum entanglement of the spin and orbital angular momentum of photons using metamaterials. Science 361, 1101-1104 (2018). Article ADS CAS PubMed Google Scholar 47. Gorlach, A. et al. Photonic quantum state tomography using free electrons. Phys. Rev. Lett. 133, 250801 (2024). Article ADS CAS PubMed Google Scholar 48. Forbes, A., Nothlawala, F. & Valles, A. Progress in quantum structured light. Nat. Photon. 19, 1291-1300 (2025). Article ADS CAS Google Scholar 49. Shpiro, T. et al. Toward photon-induced near-field electron tomography. Preprint at https://arxiv.org/abs/2510.24648 (2025). 50. Bliokh, K. Y. et al. Theory and applications of free-electron vortex states. Phys. Rep. 690, 1-70 (2017). Article ADS MathSciNet CAS Google Scholar 51. Midgley, P. A. & Dunin-Borkowski, R. E. Electron tomography and holography in materials science. Nat. Mater. 8, 271-280 (2009). Article ADS CAS PubMed Google Scholar 52. Russo, C. J. & Henderson, R. Charge accumulation in electron cryomicroscopy. Ultramicroscopy 187, 43-49 (2018). Article CAS PubMed PubMed Central Google Scholar 53. Ferrari, B. M. et al. Realization of a pre-sample photonic-based free-electron modulator in ultrafast transmission electron microscopes. ACS Photon. 12, 5864-5873 (2025). Article CAS Google Scholar 54. Wu, Y. et al. Enhanced thermal conduction by surface phonon-polaritons. Sci. Adv. 6, eabb4461 (2020). Article ADS CAS PubMed PubMed Central Google Scholar 55. Piazza, L. et al. Simultaneous observation of the quantization and the interference pattern of a plasmonic near-field. Nat. Commun. 6, 6407 (2015). Article ADS CAS PubMed PubMed Central Google Scholar 56. Wang, K. et al. Coherent interaction between free electrons and a photonic cavity. Nature 582, 50-54 (2020). Article ADS CAS PubMed Google Scholar 57. Kfir, O. et al. Controlling free electrons with optical whispering-gallery modes. Nature 582, 46-49 (2020). Article ADS CAS PubMed Google Scholar 58. Bucher, T. et al. Free-electron Ramsey-type interferometry for enhanced amplitude and phase imaging of nearfields. Sci. Adv. 9, eadi5729 (2023). Article PubMed PubMed Central Google Scholar 59. Liu, S. et al. Single crystal growth of millimeter-sized monoisotopic hexagonal boron nitride. Chem. Mater. 30, 6222-6225 (2018). Article ADS CAS Google Scholar 60. Kurman, Y. et al. Dynamics of optical vortices in van der Waals materials. Optica 10, 612-618 (2023). Article ADS CAS Google Scholar 61. Gibertini, M., Koperski, M., Morpurgo, A. F. & Novoselov, K. S. Magnetic 2D materials and heterostructures. Nat. Nanotechnol. 14, 408-419 (2019). Article ADS CAS PubMed Google Scholar 62. de Abajo, F. J. G. et al. Roadmap for photonics with 2D materials. ACS Photon. 8, 3961-4095 (2025). Article Google Scholar 63. Maleev, I. D. & Swartzlander, G. A. Jr. Composite optical vortices. J. Opt. Soc. Am. B 20, 1169-1176 (2003). Article ADS CAS Google Scholar 64. Bekshaev, A., Chernykh, A., Khoroshun, A. & Mikhaylovskaya, L. Singular skeleton evolution and topological reactions in edge-diffracted circular optical-vortex beams. Opt. Commun. 397, 72-83 (2017). Article ADS CAS Google Scholar 65. De Angelis, L., Alpeggiani, F., Di Falco, A. & Kuipers, L. Persistence and lifelong fidelity of phase singularities in optical random waves. Phys. Rev. Lett. 119, 203903 (2017). Article ADS PubMed Google Scholar 66. De Angelis, L. & Kuipers, L. Effective pair-interaction of phase singularities in random waves. Opt. Lett. 46, 2734-2737 (2021). Article ADS PubMed Google Scholar 67. Berry, M. V. Regular and irregular semiclassical wavefunctions. J. Phys. A Math. Gen. 10, 2083-2092 (1977). Article ADS MathSciNet Google Scholar 68. Kourkoulou, I., Landry, M. J., Nicolis, A. & Parmentier, K. Apparently superluminal superfluids. J. High Energy Phys. 2024, 80 (2024). Article MathSciNet Google Scholar 69. Seo, S. W., Ko, B., Kim, J. H. & Shin, Y. I. Observation of vortex-antivortex pairing in decaying 2D turbulence of a superfluid gas. Sci. Rep. 7, 4587 (2017). Article ADS PubMed PubMed Central Google Scholar 70. Green, S. Fluid Vortices (Springer, 2012). 71. Fiorino, M. & Elsberry, R. L. Some aspects of vortex structure related to tropical cyclone motion. J. Atmos. Sci. 46, 975-990 (1989). Article ADS Google Scholar 72. Poincare, H. Sur les courbes definies par les equations differentielles. J. Math. Pures Appl. 4, 167-244 (1885). Google Scholar 73. Vanacore, G. M. et al. Author Correction: Attosecond coherent control of free-electron wave functions using semi-infinite light fields. Nat. Commun. 10, 1069 (2019). Article ADS CAS PubMed PubMed Central Google Scholar 74. Machado, F., Rivera, N., Buljan, H., Soljacic, M. & Kaminer, I. Shaping polaritons to reshape selection rules. ACS Photon. 5, 3064-3072 (2018). Article CAS Google Scholar 75. Session, D. et al. Optical pumping of electronic quantum Hall states with vortex light. Nat. Photon. 19, 156-161 (2025). Article ADS CAS Google Scholar 76. Spektor, G. et al. and Meyer zu Heringdorf, F.J. Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices. Science 355, 1187-1191 (2017). Article ADS CAS PubMed Google Scholar 77. Lim, S. W. D., Spaegele, C. M. & Capasso, F. Multidimensional optical singularities and their applications. Preprint at https:/ /arxiv.org/abs/2406.00784 (2024). 78. Thompson, W. On Vortex Atoms by Lord Kelvin. Proc. R. Soc. Edin. 6, 94-105 (1867). Article Google Scholar 79. Zhang, Z. et al. Particlelike behavior of topological defects in linear wave packets in photonic graphene. Phys. Rev. Lett. 122, 233905 (2019). Article ADS CAS PubMed Google Scholar 80. Rumi, G. et al. Hyperuniform vortex patterns at the surface of type-II superconductors. Phys. Rev. Res. 1, 033057 (2019). Article CAS Google Scholar 81. Brandt, E. H. Electric field in superconductors with rectangular cross section. Phys. Rev. B 52, 15442 (1995). Article ADS CAS Google Scholar 82. Aharonov, Y., Popescu, S. & Rohrlich D. How can an infra-red photon behave as a gamma ray? Preprint TAUP, 1847-1890 (1991). 83. Berry, M. V. in Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (eds Anandan, J. S. & Safko, J. L.) 55-64 (World Scientific, 1994). 84. Rogers, E. T. et al. A super-oscillatory lens optical microscope for subwavelength imaging. Nat. Mater. 11, 432-435 (2012). Article ADS CAS PubMed Google Scholar Download references Acknowledgements We warmly thank M. V. Berry for stimulating and insightful discussions, and Y.-N. Li for valuable discussions that significantly contributed to this work. This research was funded by the Gordon and Betty Moore Foundation (grant no. GBMF11473). This project was also funded by the ERC COG, QinPINEM of the European Union (project no. 101125662). We acknowledge funding from the Helen Diller Quantum Center. This work is part of the SMART-electron project, which received funding from the Horizon 2020 Research and Innovation Programme of the European Union (grant agreement no. 964591). S.T. acknowledges support from the Adams fellowship of the Israeli Academy of Science and Humanities, the Yad Hanadiv foundation through the Rothschild fellowship, the VATAT-Quantum fellowship by the Israel Council for Higher Education, the Helen Diller Quantum Center post-doctoral fellowship and the Technion Viterbi fellowship. E.J. and J.H.E were supported by the Office of Naval Research (award no. N00014-20-1-2474). C.R.-C. was funded by a Stanford Science Fellowship. K.W. was supported by the National Natural Science Foundation of China (no. 12374321), the Shanghai Rising-Star Program (no. 22QA1410100) and the international partnership of the Chinese Academy of Sciences (111GJHZ2022024FN). H.H.S. acknowledges support from ISF (grant no. 2576/25). A.G. acknowledges financial support from the Azrieli Foundation through the Azrieli Graduate Studies Scholarship. The experiments were performed on the UTEM of the AdQuanta group of I.K., which is installed in the Electron Microscopy Center (EMC) of the Department of Materials Science and Engineering at the Technion. We thank Integrated Dynamic Electron Solutions (IDES) Inc. for the support, advice and discussions. I.K. wholeheartedly acknowledges the support of R. Magid and B. Magid, whose donation made the purchase of the UTEM possible; without their help, all the experiments presented here would not have been possible. Author information Author notes 1. These authors contributed equally: T. Bucher, A. Gorlach Authors and Affiliations 1. Andrea and Erna Viterbi Department of Electrical and Computer Engineering, Technion-Israel Institute of Technology, Haifa, Israel T. Bucher, A. Gorlach, A. Niedermayr, Q. Yan, H. Nahari, R. Ruimy, Y. Adiv, M. Yannai, T. L. Abudi, S. Tsesses & I. Kaminer 2. Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China K. Wang 3. Tim Taylor Department of Chemical Engineering, Kansas State University, Manhattan, KS, USA E. Janzen & J. H. Edgar 4. Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA C. Spaegele 5. E. L. Ginzton Laboratory, Stanford University, Stanford, CA, USA C. Roques-Carmes 6. ICFO-Institut de Ciencies Fotoniquses, The Barcelona Institute of Science and Technology, Castelldefels, Spain F. H. L. Koppens 7. ICREA-Institucio Catalana de Recerca i Estudis Avancats, Barcelona, Spain F. H. L. Koppens 8. Department of Materials Science, University of Milano-Bicocca, Milano, Italy G. M. Vanacore 9. Department of Physics, Bar-Ilan University, Ramat Gan, Israel H. H. Sheinfux 10. Department of Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA S. Tsesses 11. Faculty of Materials Science and Engineering, Technion-Israel Institute of Technology, Haifa, Israel I. Kaminer Authors 1. T. Bucher View author publications Search author on:PubMed Google Scholar 2. A. Gorlach View author publications Search author on:PubMed Google Scholar 3. A. Niedermayr View author publications Search author on:PubMed Google Scholar 4. Q. Yan View author publications Search author on:PubMed Google Scholar 5. H. Nahari View author publications Search author on:PubMed Google Scholar 6. K. Wang View author publications Search author on:PubMed Google Scholar 7. R. Ruimy View author publications Search author on:PubMed Google Scholar 8. Y. Adiv View author publications Search author on:PubMed Google Scholar 9. M. Yannai View author publications Search author on:PubMed Google Scholar 10. T. L. Abudi View author publications Search author on:PubMed Google Scholar 11. E. Janzen View author publications Search author on:PubMed Google Scholar 12. C. Spaegele View author publications Search author on:PubMed Google Scholar 13. C. Roques-Carmes View author publications Search author on:PubMed Google Scholar 14. J. H. Edgar View author publications Search author on:PubMed Google Scholar 15. F. H. L. Koppens View author publications Search author on:PubMed Google Scholar 16. G. M. Vanacore View author publications Search author on:PubMed Google Scholar 17. H. H. Sheinfux View author publications Search author on:PubMed Google Scholar 18. S. Tsesses View author publications Search author on:PubMed Google Scholar 19. I. Kaminer View author publications Search author on:PubMed Google Scholar Contributions H.H.S. performed the sample fabrication. E.J. and J.H.E. grew the hBN crystals. A.N., H.N., K.W., Y.A., M.Y. and T.L.A. performed the measurements. A.G., Q.Y., R.R. and T.B. developed the theory. T.B., C.S., C.R.-C. and A.G. performed the analysis. T.B. performed the algorithm development. T.B., H.H.S., K.W., A.N., S.T., Y.A., M.Y., G.M.V. and I.K. designed the experiment. F.H.L.K, G.M.V., S.T. and I.K. supervised the work. All authors contributed to the analysis, discussion and writing of this work. Corresponding author Correspondence to I. Kaminer. Ethics declarations Competing interests The authors declare no competing interests. Peer review Peer review information Nature thanks Oleg Angelsky, Francesco Caravelli, Alexander Vernon and Claudia Zenkova for their contribution to the peer review of this work. Peer reviewer reports are available. Additional information Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Extended data figures and tables Extended Data Fig. 1 The UTEM setup and the PELM integration. UTEM illustration (a) and image (b) illustrating the microscope column, electron spectrometer and detectors, optical setup, and the integration of a modified Hard X-ray Aperture (HXA) at a post-condenser lens stage (PELM). The external knob of the HXA (a and b, left side) has two rigid positioning points with 5 mm lateral travel around them for positioning the reference interaction point with respect to the electron beam path. An electron-transparent thin film sits at the place of the x-ray aperture, and light enters from the optical access port on the opposite side of the column at a 20-degree angle above the horizon (dashed red line in b). Double illumination scheme (a and b, right side) implemented on the vertical board next to the UTEM. The IR laser beam is separated into two portions using a 50:50 beam splitter. One portion is guided towards the PELM (dashed red line in b), whereas the other portion is guided towards the sample (solid red line in b). (c) Image (left) and CAD model (right) of the modified HXA aperture connected to the platelet hosting the electron-transparent light-opaque metallic thin films for electron-light interaction. The platelet is made of Aluminum alloy, whereas the clamp is made of 0.15-mm-thick Beryllium Copper. One can observe two Si-window TEM grids (Norcada Inc.), which are coated with a 25-nm-thick Aluminum film deposited via thermal evaporation on a 10-nm-thick Si[3]N[4] membrane. In each grid, nine slots are present to maximize the available points of interaction in case of local damage to one of the membranes. The platelet has also been cut at a specific angle, allowing it to host a small metallic mirror able to reflect the light down the column towards the sample position (not used in the current work). The platelet, HXA, and their integration were designed and performed in close collaboration with IDES, part of JEOL Ltd. (d) By using the pump-probe delay stage in combination with the PELM delay stage, the setup allows a very long acquisition time in high spatiotemporal resolution with a large field of view. The result is 285 frames of 1050 x 1050 pix images, a total size of ~1.5GB of data to analyze with our specialized algorithmic process. ( e) The very long acquisition time also results in sample and beam instability, which needs to be taken into account. Extended Data Fig. 2 Correcting sample and beam drift. Left: phase reconstructions for two different times (\({t}_{1},{t}_ {2}\)), each time has a different rotation and translation, which is fixed by calculating an affine transformation from at least 10 features selected manually on each frame. Right: corresponding fixed phase reconstruction. The red circle marks the same pixel indices, which points on different coordinates on the sample for different times before the correction (left). After correction, the rectangle marks the same coordinates on the sample. Supplementary information Supplementary Information (download DOCX ) Peer Review file (download PDF ) Supplementary Video 1 (download MP4 ) Sub-cycle-resolved spatiotemporal dynamics of phase singularities Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and permissions About this article Check for updates. Verify currency and authenticity via CrossMark Cite this article Bucher, T., Gorlach, A., Niedermayr, A. et al. Superluminal correlations in ensembles of optical phase singularities. Nature 651, 920-926 (2026). https://doi.org/10.1038/s41586-026-10209-z Download citation * Received: 22 September 2025 * Accepted: 28 January 2026 * Published: 25 March 2026 * Version of record: 25 March 2026 * Issue date: 26 March 2026 * DOI: https://doi.org/10.1038/s41586-026-10209-z Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Access through your institution Buy or subscribe Advertisement Advertisement Explore content * Research articles * News * Opinion * Research Analysis * Careers * Books & Culture * Podcasts * Videos * Current issue * Browse issues * Collections * Subjects * Follow us on Facebook * Follow us on Bluesky * Follow us on X * Subscribe * Sign up for alerts * RSS feed About the journal * Journal Staff * About the Editors * Journal Information * Journal Metrics * Our publishing models * Editorial Values Statement * Editorial policies * Journalistic Principles * History of Nature * Awards * Contact * Send a news tip Publish with us * For Authors * For Referees * Language editing services * Open access funding * Submit manuscript Search Search articles by subject, keyword or author [ ] Show results from [All journals] Search Advanced search Quick links * Explore articles by subject * Find a job * Guide to authors * Editorial policies Nature (Nature) ISSN 1476-4687 (online) ISSN 0028-0836 (print) nature.com footer links About Nature Portfolio * About us * Press releases * Press office * Contact us Discover content * Journals A-Z * Articles by subject * protocols.io * Nature Index Publishing policies * Nature portfolio policies * Open access Author & Researcher services * Reprints & permissions * Research data * Language editing * Scientific editing * Nature Masterclasses * Research Solutions Libraries & institutions * Librarian service & tools * Librarian portal * Open research * Recommend to library Advertising & partnerships * Advertising * Partnerships & Services * Media kits * Branded content Professional development * Nature Awards * Nature Careers * Nature Conferences Regional websites * Nature Africa * Nature China * Nature India * Nature Japan * Nature Middle East * Privacy Policy * Use of cookies * Your privacy choices/Manage cookies * Legal notice * Accessibility statement * Terms & Conditions * Your US state privacy rights Springer Nature (c) 2026 Springer Nature Limited Close Nature Briefing Sign up for the Nature Briefing newsletter -- what matters in science, free to your inbox daily. Email address [ ] Sign up [ ] I agree my information will be processed in accordance with the Nature and Springer Nature Limited Privacy Policy. Close Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing *