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.
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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.
Weisskopf, V. & Wigner, E. Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. Phys. 63, 54–73 (1930).
Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).
Haroche, S. & Kleppner, D. Cavity quantum electrodynamics. Phys. Today 42, 24–30 (1989).
Pelton, M. Modified spontaneous emission in nanophotonic structures. Nat. Photonics 9, 427–435 (2015).
Noda, S., Fujita, M. & Asano, T. Spontaneous-emission control by photonic crystals and nanocavities. Nat. Photonics 1, 449–458 (2007).
Lalanne, P., Yan, W., Vynck, K., Sauvan, C. & Hugonin, J. P. Light interaction with photonic and plasmonic resonances. Laser Photonics Rev. 12, 1700113 (2018).
Liu, Y. M. & Zhang, X. Metamaterials: a new frontier of science and technology. Chem. Soc. Rev. 40, 2494–2507 (2011).
Ma, R. M. & Oulton, R. F. Applications of nanolasers. Nat. Nanotechnol. 14, 12–22 (2019).
Lodahl, P., Mahmoodian, S. & Stobbe, S. Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 87, 347 (2015).
Ding, S. Y. et al. Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials. Nat. Rev. Phys. 1, 16021 (2016).
Deng, F. G., Ren, B. C. & Li, X. H. Quantum hyperentanglement and its applications in quantum information processing. Sci. Bull. 62, 46–68 (2017).
Lodahl, P. et al. Chiral quantum optics. Nature 541, 473–480 (2017).
Rayleigh, J. W. S. The problem of the whispering gallery. Philos. Mag. 20, 1001 (1910).
Rayleigh, J. W. S. The Theory of Sound (Dover, 1945).
Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).
Bender, C. M., Boettcher, S. & Meisinger, P. N. PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999).
Guo, A. et al. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).
Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nat. Phys. 6, 192–195 (2010).
Regensburger, A. et al. Parity-time synthetic photonic lattices. Nature 488, 167–171 (2012).
Peng, B. et al. Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014).
Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time-symmetric microring lasers. Science 346, 975–978 (2014).
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).
Cao, H. & Wiersig, J. Dielectric microcavities: model systems for wave chaos and non-Hermitian physics. Rev. Mod. Phys. 87, 61 (2015).
Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics 11, 752–762 (2017).
El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).
Miri, M. A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).
Özdemir, S. K., Rotter, S., Nori, F. & Yang, L. Parity-time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).
Dembowski, C. Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett. 86, 787 (2001).
Heiss, W. D. Exceptional points of non-Hermitian operators. J. Phys. A 37, 2455–2464 (2004).
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).
Berry, M. V. Physics of non-hermitian degeneracies. Czech. J. Phys. 54, 1039–1047 (2004).
Pick, A. et al. General theory of spontaneous emission near exceptional points. Opt. Express 25, 12325–12348 (2017).
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).
Okołowicz, J., Płoszajczak, M. & Rotter, I. Dynamics of quantum systems embedded in a continuum. Phys. Rep. 374, 271 (2003).
Tureci, H. E., Ge, L., Rotter, S. & Stone, A. D. Strong interactions in multimode random lasers. Science 320, 643–646 (2008).
Ge, L., Chong, Y. D. & Stone, A. D. Steady-state ab initio laser theory: generalizations and analytic results. Phys. Rev. A 82, 063824 (2010).
Milonni, P. W. The Quantum Vacuum (Academic, 1994).
Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).
Feng, L. et al. Experimental demonstration of a unidirectional reflectionless parity–time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013).
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).
Peng, B. et al. Chiral modes and directional lasing at exceptional points. Proc. Natl Acad. Sci. USA 113, 6845–6850 (2016).
Miao, P. et al. Orbital angular momentum microlaser. Science 353, 464–467 (2016).
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).
Longhi, S. & Della Valle, G. Optical lattices with exceptional points in the continuum. Phys. Rev. A 89, 053132 (2014).
Ge, L. Non-Hermitian lattices with a flat band and polynomial power increase. Photonics Res. 6, A10–A17 (2018).
Wiersig, J. et al. Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities. Phys. Rev. A 84, 023845 (2011).
Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).
Ma, G., Xiao, M. & Chan, C. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).
Jiang, X., Li, Y., Liang, B., Cheng, J. C. & Zhang, L. Convert acoustic resonances to orbital angular momentum. Phys. Rev. Lett. 117, 034301 (2016).
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).
Zhu, X. F., Ramezani, H., Shi, C. Z., Zhu, J. & Zhang, X. PT-symmetric acoustics. Phys. Rev. X 4, 031042 (2014).
Fleury, R., Sounas, D. & Alù, A. An invisible acoustic sensor based on parity–time symmetry. Nat. Commun. 6, 5905 (2015).
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).
Padgett, M., Courtial, J. & Allen, L. Light’s orbital angular momentum. Phys. Today 57, 35–40 (2004).
Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics 3, 161–204 (2011).
Rodriguez-Fortuño, F. J. et al. Near-field interference for the unidirectional excitation of electromagnetic guided modes. Science 340, 328–330 (2013).
Petersen, J., Volz, J. & Rauschenbeutel, A. Chiral nanophotonic waveguide interface based on spin-orbit interaction of light. Science 346, 67–71 (2014).
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.
The authors declare no competing interests.
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.
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Supplementary Figs. 1–16 and Sections 1–11.
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.
The data corresponding to the graphs in Fig. 2.
The data corresponding to the graphs in Fig. 3.
The data corresponding to the graphs in Fig. 4.
The data corresponding to the graphs in Fig. 5.
The data corresponding to the graphs in Fig. 6.
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Chen, H., Liu, T., Luan, H. 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