- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Minimum Energy and Photon Content in PT Symmetric Metamaterials

J. B. Pendry1,* and S. A. R. Horsley2

  • *Contact author: j.pendry@imperial.ac.uk
PDF

Phys. Rev. Lett. 137, 163604 – Published 14 October, 2026

DOI: https://doi.org/10.1103/bc42-c2rg

Abstract

In the context of waves in space-time modulated materials, we ask two questions: (1) how much energy does it cost to break time reversal symmetry (T-symmetry) and transition to a PT-symmetric state? and (2) can a PT-symmetric system have a “ground state” in the sense that no photons are present? Our model system is a periodic metamaterial set in virtual motion with velocity cg to become a space-time crystal. We find that the expectation of energy content is always increased on breaking symmetry. At the same time breaking T-symmetry introduces photon pairs even when we start from a T-symmetric ground state empty of photons, except in certain pathological examples that we describe. For a range of velocities, PT symmetry is broken so that energy must be continuously invested to preserve motion, creating a trail of photon pairs. Here, energy must be continuously invested to preserve motion. We make an analogy with acoustic radiation generated from breaking the sound barrier.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (29)

  1. C. M. Bender, J. Phys. Conf. Ser. 631, 012002 (2015).
  2. C. M. Bender and D. W. Hook, Rev. Mod. Phys. 96, 045002 (2024).
  3. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
  4. C. M. Bender and S. P. Klevansky, Phys. Rev. Lett. 105, 031601 (2010).
  5. A.Fring, J. Phys. Conf. Ser. 2448, 012002 (2023).
  6. J. B.Pendry and S. A. R.Horsley, Proc. Natl. Acad. Sci. U.S.A. 120, e2302652120, (2023).
  7. E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Alù, and J. B. Pendry, Adv. Opt. Photonics 4, 014002 (2022).
  8. J. B. Pendry and S. A. R. Horsley, APL Quantum 1, 020901 (2024).
  9. J. B. Pendry, J. Phys. Condens. Matter 9, 10301 (1997).
  10. J. B. Pendry, New J. Phys. 12, 068002 (2010).
  11. A. A. Oliner and A. Hessel, IRE Trans. Microwave Theory Tech. 9, 337 (1961).
  12. E. S. Cassedy and A. A. Oliner, Proc. IEEE 51, 1342 (1963).
  13. E. S. Cassedy, Proc. IEEE 55, 1154 (1967).
  14. C. Caloz and Z.-L. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1569 (2020).
  15. C. Caloz and Z.-L. Deck-Léger, IEEE Trans. Antennas Propag. 68, 1583 (2020).
  16. J. B. Pendry, E. Galiffi, and P. A. Huidobro, J. Opt. Soc. Am. B 38, 3360 (2021).
  17. S. A. R. Horsley and J. B. Pendry, Phys. Rev. Lett. 133, 156903 (2024).
  18. P. C. W. Davies, J. Phys. A 8, 609 (1975).
  19. W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  20. J. B. Pendry, E. Galiffi, and P. A. Huidobro, Optica 9, 724 (2022).
  21. J. B. Pendry, Opt. Express 31, 452 (2023).
  22. W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).
  23. R. Schutzhold and W. G. Unruh, Phys. Rev. D 66, 044019 (2002).
  24. S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011).
  25. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, Phys. Rev. Lett. 79, 4566 (1997).
  26. C. M. de Sterke, N. G. Raphael Broderick, B. J. Eggleton, and M. J. Steel, Opt. Fiber Technol. 2, 253 (1996).
  27. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, Science 319, 1367 (2008).
  28. L. M. Procopio, R. Aguero-Santacruz, D. Bermudez, and U. Leonhardt, Nature (London) 655, 336 (2026).
  29. R. Schiek, Opt. Mater. Express 13, 1727 (2023).

Outline

Information

Phys. Rev. Lett. 137, 163604– Published 14 October, 2026
  • Received 17 April 2026
  • Revised 9 September 2026
  • Accepted 18 September 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