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.

  • Article
  • Published:

Revealing the missing dimension at an exceptional point

Abstract

The radiation of electromagnetic and mechanical waves depends not only on the intrinsic properties of the emitter but also on the surrounding environment. This principle has laid the foundation for the development of lasers, quantum optics, sonar, musical instruments and other fields related to wave–matter interaction. In the conventional wisdom, the environment is defined exclusively by its eigenstates, and an emitter radiates into and interacts with these eigenstates. Here we show experimentally that this scenario breaks down at a non-Hermitian degeneracy known as an exceptional point. We find a chirality-reversal phenomenon in a ring cavity where the radiation field reveals the missing dimension of the Hilbert space, known as the Jordan vector. This phenomenon demonstrates that the radiation field of an emitter can become fully decoupled from the eigenstates of its environment. The generality of this striking phenomenon in wave–matter interaction is experimentally confirmed in both electromagnetic and acoustic systems. Our finding transforms the fundamental understanding of light–matter interaction and wave–matter interaction in general, and enriches the intriguing physics of exceptional points.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Chirality-reversal radiation at an exceptional point.
Fig. 2: Experimental results of chirality-reversal radiation.
Fig. 3: Chirality-reversal vortex radiation from a single emitter.
Fig. 4: Position- and frequency-dependent chirality of single-emitter radiation inside the exceptional point cavity.
Fig. 5: Experimental demonstrations of the chirality-reversal phenomenon in an acoustic wave system.
Fig. 6: Chirality evolution in the passive acoustic PT-symmetric system.

Similar content being viewed by others

Data availability

The data represented in Figs. 2–6 are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

References

  1. Weisskopf, V. & Wigner, E. Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. Phys. 63, 54–73 (1930).

    Article  ADS  MATH  Google Scholar 

  2. Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).

    Article  Google Scholar 

  3. Haroche, S. & Kleppner, D. Cavity quantum electrodynamics. Phys. Today 42, 24–30 (1989).

    Article  ADS  Google Scholar 

  4. Pelton, M. Modified spontaneous emission in nanophotonic structures. Nat. Photonics 9, 427–435 (2015).

    Article  ADS  Google Scholar 

  5. Noda, S., Fujita, M. & Asano, T. Spontaneous-emission control by photonic crystals and nanocavities. Nat. Photonics 1, 449–458 (2007).

    Article  ADS  Google Scholar 

  6. Lalanne, P., Yan, W., Vynck, K., Sauvan, C. & Hugonin, J. P. Light interaction with photonic and plasmonic resonances. Laser Photonics Rev. 12, 1700113 (2018).

    Article  ADS  Google Scholar 

  7. Liu, Y. M. & Zhang, X. Metamaterials: a new frontier of science and technology. Chem. Soc. Rev. 40, 2494–2507 (2011).

    Article  Google Scholar 

  8. Ma, R. M. & Oulton, R. F. Applications of nanolasers. Nat. Nanotechnol. 14, 12–22 (2019).

    Article  ADS  Google Scholar 

  9. Lodahl, P., Mahmoodian, S. & Stobbe, S. Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 87, 347 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  10. Ding, S. Y. et al. Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials. Nat. Rev. Phys. 1, 16021 (2016).

    Google Scholar 

  11. Deng, F. G., Ren, B. C. & Li, X. H. Quantum hyperentanglement and its applications in quantum information processing. Sci. Bull. 62, 46–68 (2017).

    Article  Google Scholar 

  12. Lodahl, P. et al. Chiral quantum optics. Nature 541, 473–480 (2017).

    Article  ADS  Google Scholar 

  13. Rayleigh, J. W. S. The problem of the whispering gallery. Philos. Mag. 20, 1001 (1910).

    Article  MATH  Google Scholar 

  14. Rayleigh, J. W. S. The Theory of Sound (Dover, 1945).

  15. Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Bender, C. M., Boettcher, S. & Meisinger, P. N. PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Guo, A. et al. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).

    Article  ADS  Google Scholar 

  18. Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nat. Phys. 6, 192–195 (2010).

    Article  Google Scholar 

  19. Regensburger, A. et al. Parity-time synthetic photonic lattices. Nature 488, 167–171 (2012).

    Article  ADS  Google Scholar 

  20. Peng, B. et al. Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014).

    Article  ADS  Google Scholar 

  21. Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time-symmetric microring lasers. Science 346, 975–978 (2014).

    Article  ADS  Google Scholar 

  22. Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity–time symmetry breaking. Science 346, 972–975 (2014).

    Article  ADS  Google Scholar 

  23. Cao, H. & Wiersig, J. Dielectric microcavities: model systems for wave chaos and non-Hermitian physics. Rev. Mod. Phys. 87, 61 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  24. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics 11, 752–762 (2017).

    Article  ADS  Google Scholar 

  25. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  26. Miri, M. A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  27. Özdemir, S. K., Rotter, S., Nori, F. & Yang, L. Parity-time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    Article  ADS  Google Scholar 

  28. Dembowski, C. Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett. 86, 787 (2001).

    Article  ADS  Google Scholar 

  29. Heiss, W. D. Exceptional points of non-Hermitian operators. J. Phys. A 37, 2455–2464 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Wiersig, J. Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection. Phys. Rev. Lett. 112, 203901 (2014).

    Article  ADS  Google Scholar 

  31. Berry, M. V. Physics of non-hermitian degeneracies. Czech. J. Phys. 54, 1039–1047 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  32. Pick, A. et al. General theory of spontaneous emission near exceptional points. Opt. Express 25, 12325–12348 (2017).

    Article  ADS  Google Scholar 

  33. Hernandez, E., Jauregui, A. & Mondragon, A. Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states. Phys. Rev. A 67, 022721 (2003).

    Article  ADS  MATH  Google Scholar 

  34. Okołowicz, J., Płoszajczak, M. & Rotter, I. Dynamics of quantum systems embedded in a continuum. Phys. Rep. 374, 271 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Tureci, H. E., Ge, L., Rotter, S. & Stone, A. D. Strong interactions in multimode random lasers. Science 320, 643–646 (2008).

    Article  ADS  Google Scholar 

  36. Ge, L., Chong, Y. D. & Stone, A. D. Steady-state ab initio laser theory: generalizations and analytic results. Phys. Rev. A 82, 063824 (2010).

    Article  ADS  Google Scholar 

  37. Milonni, P. W. The Quantum Vacuum (Academic, 1994).

  38. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).

    Article  ADS  Google Scholar 

  39. Feng, L. et al. Experimental demonstration of a unidirectional reflectionless parity–time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013).

    Article  ADS  Google Scholar 

  40. Ge, L., Chong, Y. D. & Stone, A. D. Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures. Phys. Rev. A 85, 023802 (2012).

    Article  ADS  Google Scholar 

  41. Peng, B. et al. Chiral modes and directional lasing at exceptional points. Proc. Natl Acad. Sci. USA 113, 6845–6850 (2016).

    Article  ADS  Google Scholar 

  42. Miao, P. et al. Orbital angular momentum microlaser. Science 353, 464–467 (2016).

    Article  ADS  Google Scholar 

  43. Wang, X.-Y., Chen, H.-Z., Li, Y., Li, B. & Ma, R.-M. Microscale vortex laser with controlled topological charge. Chin. Phys. B 25, 124211 (2016).

    Article  ADS  Google Scholar 

  44. Longhi, S. & Della Valle, G. Optical lattices with exceptional points in the continuum. Phys. Rev. A 89, 053132 (2014).

    Article  ADS  Google Scholar 

  45. Ge, L. Non-Hermitian lattices with a flat band and polynomial power increase. Photonics Res. 6, A10–A17 (2018).

    Article  MathSciNet  Google Scholar 

  46. Wiersig, J. et al. Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities. Phys. Rev. A 84, 023845 (2011).

    Article  ADS  Google Scholar 

  47. Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

    Article  ADS  Google Scholar 

  48. Ma, G., Xiao, M. & Chan, C. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).

    Article  Google Scholar 

  49. Jiang, X., Li, Y., Liang, B., Cheng, J. C. & Zhang, L. Convert acoustic resonances to orbital angular momentum. Phys. Rev. Lett. 117, 034301 (2016).

    Article  ADS  Google Scholar 

  50. Shi, C., Dubois, M., Wang, Y. & Zhang, X. High-speed acoustic communication by multiplexing orbital angular momentum. Proc. Natl Acad. Sci. USA 114, 7250–7253 (2017).

    Article  ADS  Google Scholar 

  51. Zhu, X. F., Ramezani, H., Shi, C. Z., Zhu, J. & Zhang, X. PT-symmetric acoustics. Phys. Rev. X 4, 031042 (2014).

    Google Scholar 

  52. Fleury, R., Sounas, D. & Alù, A. An invisible acoustic sensor based on parity–time symmetry. Nat. Commun. 6, 5905 (2015).

    Article  ADS  Google Scholar 

  53. Liu, T., Zhu, X., Chen, F., Liang, S. & Zhu, J. Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal. Phys. Rev. Lett. 120, 124502 (2018).

    Article  ADS  Google Scholar 

  54. Padgett, M., Courtial, J. & Allen, L. Light’s orbital angular momentum. Phys. Today 57, 35–40 (2004).

    Article  ADS  Google Scholar 

  55. Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics 3, 161–204 (2011).

    Article  ADS  Google Scholar 

  56. Rodriguez-Fortuño, F. J. et al. Near-field interference for the unidirectional excitation of electromagnetic guided modes. Science 340, 328–330 (2013).

    Article  ADS  Google Scholar 

  57. Petersen, J., Volz, J. & Rauschenbeutel, A. Chiral nanophotonic waveguide interface based on spin-orbit interaction of light. Science 346, 67–71 (2014).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work is supported by NSFC under project Nos. 11774014, 91950115, 11574012 and 61521004, Beijing Natural Science Foundation (Z180011) and the National Key R&D Programme of China (2018YFA0704401). J.Z. is supported by the Early Career Scheme of Hong Kong RGC (grant no. PolyU 252081/15E) and the National Natural Science Foundation of China (grant no. 11774297). L.G. is supported by NSF under grant no. PHY-1847240. L.L. is supported by the National Key R&D Programme of China under grant nos 2017YFA0303800 and 2016YFA0302400 and by NSFC under project no. 11721404. R.-J.L. is supported by NSFC under project no. 11974415. X.-F.Z. acknowledges financial support from the National Natural Science Foundation of China (grant nos 11674119, 11690030 and 11690032) and the Bird Nest Plan of HUST.

Author information

Authors and Affiliations

Authors

Contributions

R.-M.M. conceived the concept and supervised the project. H.-Z.C., X.-Y.W. and R.-M.M. performed the coupled-mode equation analysis and conducted the electromagnetic simulation. H.-Y.L., Y.-B.L., R.-J.L., L.L. and R.-M.M. carried out the microwave experiments. H.-Y.L. and R.-M.M. did the data analysis of the microwave experiment. J.Z. and R.-M.M. initiated the acoustic experiment. T.L., H.-Z.C. and X.-F.Z. performed the simulation and designed the acoustic experiment. T.L., Z.-M.G., S.-J.L. and H.G. conducted the acoustic experiments. T.L., X.Z. and R.-M.M. did the data analysis of the acoustic experiment. J.Z. supervised the acoustic experiment. L.G. performed the Green function and Jordan vector analysis. R.-M.M., H.-Z.C., L.G., T.L., J.Z., X.-Y.W. and S.Z. wrote the manuscript.

Corresponding authors

Correspondence to Li Ge, Jie Zhu or Ren-Min Ma.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Andrea Alu, Nicolas Bachelard, Romain Fleury and Stefan Rotter for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–16 and Sections 1–11.

Supplementary Video

The three-dimensional radiation fields of the coalesced eigenstates of a ring cavity operating close to an exceptional point, and a dipole emitter inside a ring cavity operating close to an exceptional point.

Source data

Source Data Fig. 2

The data corresponding to the graphs in Fig. 2.

Source Data Fig. 3

The data corresponding to the graphs in Fig. 3.

Source Data Fig. 4

The data corresponding to the graphs in Fig. 4.

Source Data Fig. 5

The data corresponding to the graphs in Fig. 5.

Source Data Fig. 6

The data corresponding to the graphs in Fig. 6.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, HZ., Liu, T., Luan, HY. et al. Revealing the missing dimension at an exceptional point. Nat. Phys. 16, 571–578 (2020). https://doi.org/10.1038/s41567-020-0807-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-020-0807-y

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing