- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Partial quantization of non-Hermitian Berry phases in time-varying media

Calvin M. Hooper*

  • *Contact author: ch1122@exeter.ac.uk
PDF

Phys. Rev. A 114, 043524 – Published 20 October, 2026

DOI: https://doi.org/10.1103/92c1-dd6f

Abstract

A fundamental symmetry of the non-Hermitian operators describing wave propagation in time-varying media imbues such systems with nontrivial topology. This topology may be measured directly in a wide range of experimental settings as a quantized real part of the Berry phase, contrasting unconstrained geometric gain or loss. This topological invariant is provided explicitly for practical examples, including a non-Hermitian analog of the Su-Schrieffer-Heeger model.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Alù, and J. B. Pendry, Photonics of time-varying media, Adv. Photonics 4, 014002 (2022).
  2. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, Time reversal and holography with spacetime transformations, Nat. Phys. 12, 972 (2016).
  3. S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, Optical time reversal from time-dependent epsilon-near-zero media, Phys. Rev. Lett. 120, 043902 (2018).
  4. S. A. R. Horsley and J. B. Pendry, Quantum electrodynamics of time-varying gratings, Proc. Natl. Acad. Sci. USA 120, e2302652120 (2023).
  5. V. Pacheco-Peña and N. Engheta, Spatiotemporal cascading of dielectric waveguides [Invited], Opt. Mater. Express 14, 1062 (2024).
  6. X. Wang, P. Garg, M. S. Mirmoosa, A. G. Lamprianidis, C. Rockstuhl, and V. S. Asadchy, Expanding momentum bandgaps in photonic time crystals through resonances, Nat. Photonics 19, 149 (2025).
  7. A. Ganfornina-Andrades, J. E. Vázquez-Lozano, and I. Liberal, Quantum vacuum amplification in time-varying media with arbitrary temporal profiles, Phys. Rev. Res. 6, 043320 (2024).
  8. S. A. R. Horsley, E. Galiffi, and Y.-T. Wang, Eigenpulses of dispersive time-varying media, Phys. Rev. Lett. 130, 203803 (2023).
  9. J. J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput. 2, 3 (1986).
  10. P. D. Lax, Hyperbolic Partial Differential Equations, Courant Lecture Notes (American Mathematical Society, Providence, RI, 2006).
  11. C. Singh, Student understanding of quantum mechanics, Am. J. Phys. 69, 885 (2001).
  12. K. von Klitzing, Developments in the quantum Hall effect, Philos. Trans. R. Soc. A 363, 2203 (2005).
  13. D. Tong, Lectures on the quantum Hall effect, arXiv:1606.06687.
  14. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  15. A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, Quantum many-body scars: A quasiparticle perspective, Annu. Rev. Condens. Matter Phys. 14, 443 (2023).
  16. M. Testorf, Perturbation theory as a unified approach to describe diffractive optical elements, J. Opt. Soc. Am. A 16, 1115 (1999).
  17. J. L. Remo, Second-order perturbation theory for Fabry–Perot resonators, Opt. Lett. 3, 193 (1978).
  18. C. M. Hooper, J. R. Capers, I. R. Hooper, and S. A. R. Horsley, Symmetry-protected lossless modes in dispersive time-varying media, Phys. Rev. A 111, 033507 (2025).
  19. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  20. C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947 (2007).
  21. D. Cheng, K. Wang, C. Roques-Carmes, E. Lustig, O. Y. Long, H. Wang, and S. Fan, Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions, Nature (London) 637, 52 (2025).
  22. L. Yuan, Q. Lin, M. Xiao, and S. Fan, Synthetic dimension in photonics, Optica 5, 1396 (2018).
  23. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  24. E. D. Nering, Linear Algebra and Matrix theory (Wiley, New York, 1970).
  25. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  26. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  27. K. Kawabata and S. Ryu, Non-Hermitian disordered systems, arXiv:2603.20393.
  28. C. M. Hooper, I. R. Hooper, and S. A. R. Horsley, Quasinormal modes in dispersive photonic time crystals, Phys. Rev. A 113, 043526 (2026).
  29. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London Ser. A 392, 45 (1984).
  30. J. Garrison and E. Wright, Complex geometrical phases for dissipative systems, Phys. Lett. A 128, 177 (1988).
  31. M. Wagner, F. Dangel, H. Cartarius, J. Main, and G. Wunner, Numerical calculation of the complex Berry phase in non-Hermitian systems, Acta Polytech. 57, 470 (2017).
  32. S. Tsubota, H. Yang, Y. Akagi, and H. Katsura, Symmetry-protected quantization of complex Berry phases in non-Hermitian many-body systems, Phys. Rev. B 105, L201113 (2022).
  33. G. Nenciu and G. Rasche, On the adiabatic theorem for nonself-adjoint Hamiltonians, J. Phys. A: Math. Gen. 25, 5741 (1992).
  34. S. Ibáñez and J. G. Muga, Adiabaticity condition for non-Hermitian Hamiltonians, Phys. Rev. A 89, 033403 (2014).
  35. E.-M. Graefe, A. A. Mailybaev, and N. Moiseyev, Breakdown of adiabatic transfer of light in waveguides in the presence of absorption, Phys. Rev. A 88, 033842 (2013).
  36. Q. Ma, J. W. You, L. Chen, Z. Gu, S. L. Qin, Z. Lan, and T. J. Cui, Floquet topological states in time-varying metasurfaces, Sci. Adv. 11, eadx9025 (2025).
  37. J. C. Serra and M. G. Silveirinha, Engineering topological phases with a traveling-wave spacetime modulation, Laser Photonics Rev. 20, e00560 (2026).
  38. E. Lustig, Y. Sharabi, and M. Segev, Topological aspects of photonic time crystals, Optica 5, 1390 (2018).
  39. X. Ni, S. Yin, H. Li, and A. Alù, Topological wave phenomena in photonic time quasicrystals, Phys. Rev. B 111, 125421 (2025).
  40. O. Segal, Y. Plotnik, E. Lustig, Y. Sharabi, M.-I. Cohen, A. Dikopoltsev, and M. Segev, Two-dimensional topological edge states in periodic space-time interfaces, Phys. Rev. Lett. 135, 163801 (2025).
  41. A. Caballero, T. F. Allard, and P. A. Huidobro, Interface states in space-time photonic crystals: Topological origin, propagation, and amplification, ACS Photonics 13, 3012 (2026).
  42. J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions (Springer International Publishing, Cham, 2016), pp. 1–53.
  43. J. Zak, Berry's phase for energy bands in solids, Phys. Rev. Lett. 62, 2747 (1989).
  44. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  45. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  46. F. Ghahari, D. Walkup, C. Gutiérrez, J. F. Rodriguez-Nieva, Y. Zhao, J. Wyrick, F. D. Natterer, W. G. Cullen, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, An on/off Berry phase switch in circular graphene resonators, Science 356, 845 (2017).
  47. O. Breach, R.-J. Slager, and F. N. Ünal, Interferometry of non-Abelian band singularities and Euler class topology, Phys. Rev. Lett. 133, 093404 (2024).
  48. C. M. Bender and S. A. Orszag, WKB theory, in Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory (Springer, New York, NY, 1999), pp. 484–543.
  49. A. Canós Valero, S. Gladyshev, D. Globosits, S. Rotter, E. A. Muljarov, and T. Weiss, Revealing the resonant physics of open photonic time crystals, Laser Photonics Rev. e71318 (2025).
  50. B. Vial and R. V. Craster, Quasinormal modes of Floquet media slabs, Phys. Rev. B 112, 214201 (2025).
  51. M. M. Sabaté, B. Vial, R. Wiltshaw, S. Guenneau, and R. V. Craster, Fabry-Pérot quasinormal modes for topological edge states, Phys. Rev. B 113, L140102 (2026).
  52. I. R. Hooper, D. B. Phillips, and S. A. R. Horsley, Harnessing the frequency eigenchannels of ultrafast time-varying media, arXiv:2508.12753.
  53. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  54. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018).
  55. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  56. C. Hooper, Time-varying adiabaticity figure generation code [Computer software], Zenodo, 2026, https://doi.org/10.5281/zenodo.23243040.

Outline

Information

Phys. Rev. A 114, 043524– Published 20 October, 2026
  • Received 12 May 2026
  • Accepted 8 October 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation