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Superluminal correlations in ensembles of optical phase singularities

Nature volume 651, pages 920–926 (2026)Cite this article

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Abstract

Phase singularities—points carrying quantized topological charge—are universal features found across diverse wave systems from superfluids and superconductors to acoustic and optical fields1,2,3,4. Ensembles of these singularities exhibit distance correlations resembling particles in liquids5,6,7,8, extensively studied for their role in exotic material phases9,10,11. 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 analogy12: phase singularities accelerate towards formally divergent velocities in the moment before annihilation7,13,14, 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 membranes15,16,17,18,19. We demonstrate these phenomena using combined hardware and algorithmic advances in ultrafast electron microscopy18,20,21,22,23,24,25, 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.

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Fig. 1: Deep sub-wavelength and deep sub-cycle imaging of optical phase singularities in hBN, recording both phase and group dynamics.
Fig. 2: Deep sub-cycle annihilation of singularities, showing an example of acceleration towards formally divergent velocities along a characteristic space–time trajectory.
Fig. 3: Distance correlations and velocity distributions of singularities.
Fig. 4: Full phase-space correlations of singularities.

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Data availability

The data supporting the findings of this study are available from the corresponding authors upon reasonable request.

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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 Ciències Fotòniquses, The Barcelona Institute of Science and Technology, Castelldefels, Spain

    F. H. L. Koppens

  7. ICREA-Institució 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

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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.

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Correspondence to I. Kaminer.

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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 Si3N4 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 × 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.

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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

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