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.

  • Review Article
  • Published:

Non-Hermitian photonics based on parity–time symmetry

Nature Photonics volume 11, pages 752–762 (2017)Cite this article

Subjects

Abstract

Nearly one century after the birth of quantum mechanics, parity–time symmetry is revolutionizing and extending quantum theories to include a unique family of non-Hermitian Hamiltonians. While conceptually striking, experimental demonstration of parity–time symmetry remains unexplored in quantum electronic systems. The flexibility of photonics allows for creating and superposing non-Hermitian eigenstates with ease using optical gain and loss, which makes it an ideal platform to explore various non-Hermitian quantum symmetry paradigms for novel device functionalities. Such explorations that employ classical photonic platforms not only deepen our understanding of fundamental quantum physics but also facilitate technological breakthroughs for photonic applications. Research into non-Hermitian photonics therefore advances and benefits both fields simultaneously.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: PT symmetry and phase transition in photonics.
Fig. 2: Experimental demonstrations of PT symmetry in different photonic platforms.
Fig. 3: Novel optical effects enabled by PT symmetry.
Fig. 4: CPA laser concept and realization.
Fig. 5: Unidirectional reflectionless resonance.
Fig. 6: Orbital angular momentum microlaser by unidirectional laser actions.

Similar content being viewed by others

References

  1. 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 

  2. Bender, C. M., Brody, D. C. & Jones, H. F. Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  3. Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).

    Article  ADS  Google Scholar 

  4. El-Ganainy, R., Makris, K. G., Christodoulides, D. N. & Musslimani, Z. H. Theory of coupled optical PT-symmetric structures. Opt. Lett. 32, 2632–2634 (2007).

    Article  ADS  MATH  Google Scholar 

  5. Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008).

    Article  ADS  Google Scholar 

  6. Musslimani, Z. H., Makris, K. G., El-Ganainy, R. & Christodoulides, D. N. Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008).

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  9. Ruschhaupt, A., Delgado, F. & Muga, J. G. Physical realization of PT-symmetric potential scattering in a planar slab waveguide. J. Phys. Math. Gen. 38, L171–L176 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Klaiman, S., Günther, U. & Moiseyev, N. Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kottos, T. Broken symmetry makes light work. Nat. Phys. 6, 166–167 (2010).

    Article  Google Scholar 

  12. Longhi, S. Quantum-optical analogies using photonic structures. Laser Photon. Rev. 3, 243–261 (2009).

    Article  Google Scholar 

  13. Lee, Y.-C., Hsieh, M.-H., Flammia, S. T. & Lee, R.-K. Local PT symmetry violates the no-signaling principle. Phys. Rev. Lett. 112, 130404 (2014).

    Article  ADS  Google Scholar 

  14. Bender, C. M. PT symmetry in quantum physics: from a mathematical curiosity to optical experiments. Europhys. News 47, 17–20 (2016).

    Article  ADS  Google Scholar 

  15. Cai, W. & Shalaev, V. Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

  16. Moiseyev, N. Non-Hermitian Quantum Mechanics (Cambridge Univ. Press, Cambridge, 2011).

  17. Mostafazadeh, A. Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205–214 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  19. Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009).

    Article  ADS  Google Scholar 

  20. Longhi, S. Optical realization of relativistic non-Hermitian quantum mechanics. Phys. Rev. Lett. 105, 013903 (2010).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Peng, B. et al. Parity–time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014).

    Article  Google Scholar 

  23. Chang, L. et al. Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators. Nat. Photon. 8, 524–529 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. 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 

  26. Lawrence, M. et al. Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces. Phys. Rev. Lett. 113, 093901 (2014).

    Article  ADS  Google Scholar 

  27. Xu, Y.-L. et al. Experimental realization of Bloch oscillations in a parity–time synthetic silicon photonic lattice. Nat. Commun. 7, 11319 (2016).

    Article  ADS  Google Scholar 

  28. Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).

    Article  ADS  Google Scholar 

  29. 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 

  30. Zhao, H. et al. Metawaveguide for asymmetric interferometric light-light switching. Phys. Rev. Lett. 117, 193901 (2016).

    Article  ADS  Google Scholar 

  31. 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 

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

    Article  ADS  Google Scholar 

  33. Longhi, S. PT-symmetric laser absorber. Phys. Rev. A 82, 031801 (2010).

    Article  ADS  Google Scholar 

  34. Chong, Y. D., Ge, L. & Stone, A. D. PT-symmetry breaking and laser-absorber modes in optical scattering systems. Phys. Rev. Lett. 106, 093902 (2011).

    Article  ADS  Google Scholar 

  35. 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 

  36. Ambichl, P. et al. Breaking of PT-symmetry in bounded and unbounded scattering systems. Phys. Rev. X 3, 041030 (2013).

    Google Scholar 

  37. Ge, L., Makris, K. G., Christodoulides, D. N. & Feng, L. Scattering in PT- and RT-symmetric multimode waveguides: generalized conservation laws and spontaneous symmetry breaking beyond one dimension. Phys. Rev. A 92, 062135 (2015).

    Article  ADS  Google Scholar 

  38. Ge, L. & Feng, L. Optical-reciprocity-induced symmetry in photonic heterostructures and its manifestation in scattering PT-symmetry breaking. Phys. Rev. A 94, 043836 (2016).

    Article  ADS  Google Scholar 

  39. Schomerus, H. Quantum noise and self-sustained radiation of PT-symmetric systems. Phys. Rev. Lett. 104, 233601 (2010).

    Article  ADS  Google Scholar 

  40. Kang, M., Liu, F. & Li, J. Effective spontaneous PT-symmetry breaking in hybridized metamaterials. Phys. Rev. A 87, 053824 (2013).

    Article  ADS  Google Scholar 

  41. Ge, L. & Türeci, H. E. Antisymmetric PT-photonic structures with balanced positive- and negative-index materials. Phys. Rev. A 88, 053810 (2013).

    Article  ADS  Google Scholar 

  42. Petermann, K. Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding. IEEE J. Quant. Electron. 15, 566–570 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  43. Berry, M. V. Mode degeneracies and the Petermann excess-noise factor for unstable lasers. J. Mod. Opt. 50, 63–81 (2003).

    Article  ADS  MATH  Google Scholar 

  44. Zhao, H., Longhi, S. & Feng, L. Robust light state by quantum phase transition in non-Hermitian optical materials. Sci. Rep. 5, 17022 (2015).

    Article  ADS  Google Scholar 

  45. Longhi, S. Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009).

    Article  ADS  Google Scholar 

  46. Wimmer, M., Miri, M.-A., Christodoulides, D. & Peschel, U. Observation of Bloch oscillations in complex PT-symmetric photonic lattices. Sci. Rep. 5, 17760 (2015).

    Article  ADS  Google Scholar 

  47. Driben, R. & Malomed, B. A. Stability of solitons in parity–time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011).

    Article  ADS  Google Scholar 

  48. Wimmer, M. et al. Observation of optical solitons in PT-symmetric lattices. Nat. Commun. 6, 7782 (2015).

    Article  Google Scholar 

  49. Sarma, A. K., Miri, M.-A., Musslimani, Z. H. & Christodoulides, D. N. Continuous and discrete Schrödinger systems with parity–time-symmetric nonlinearities. Phys. Rev. E 89, 052918 (2014).

    Article  ADS  Google Scholar 

  50. Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Nonlinearly induced PT transition in photonic systems. Phys. Rev. Lett. 111, 263901 (2013).

    Article  ADS  Google Scholar 

  51. Hassan, A. U., Hodaei, H., Miri, M.-A., Khajavikhan, M. & Christodoulides, D. N. Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators. Phys. Rev. A 92, 063807 (2015).

    Article  ADS  Google Scholar 

  52. Walasik, W. & Litchinitser, N. M. Phase transition in multimode nonlinear parity–time-symmetric waveguide couplers. Sci. Rep. 6, 19826 (2016).

    Article  ADS  Google Scholar 

  53. Ge, L. & El-Ganainy, R. Nonlinear modal interactions in parity–time (PT) symmetric lasers. Sci. Rep. 6, 24889 (2016).

    Article  ADS  Google Scholar 

  54. Ge, L. Anomalous parity–time-symmetry transition away from an exceptional point. Phys. Rev. A 94, 013837 (2016).

    Article  ADS  Google Scholar 

  55. Ramezani, H., Kottos, T., El-Ganainy, R. & Christodoulides, D. N. Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A 82, 043803 (2010).

    Article  ADS  Google Scholar 

  56. Ge, L., Chong, Y. D., Rotter, S., Türeci, H. E. & Stone, A. D. Unconventional modes in lasers with spatially varying gain and loss. Phys. Rev. A 84, 023820 (2011).

    Article  ADS  Google Scholar 

  57. Ge, L. Parity–time symmetry in a flat-band system. Phys. Rev. A 92, 052103 (2015).

    Article  ADS  Google Scholar 

  58. Schomerus, H. Topologically protected midgap states in complex photonic lattices. Opt. Lett. 38, 1912–1914 (2013).

    Article  ADS  Google Scholar 

  59. Longhi, S. Convective and absolute PT-symmetry breaking in tight-binding lattices. Phys. Rev. A 88, 052102 (2013).

    Article  ADS  Google Scholar 

  60. Ge, L. & Stone, A. D. Parity–time symmetry breaking beyond one dimension: the role of degeneracy. Phys. Rev. X 4, 031011 (2014).

    Google Scholar 

  61. El-Ganainy, R., Ge, L., Khajavikhan, M. & Christodoulides, D. N. Supersymmetric laser arrays. Phys. Rev. A 92, 033818 (2015).

    Article  ADS  Google Scholar 

  62. Teimourpour, M. H., Ge, L., Christodoulides, D. N. & El-Ganainy, R. Non-Hermitian engineering of single mode two dimensional laser arrays. Sci. Rep. 6, 33253 (2016).

    Article  ADS  Google Scholar 

  63. Liertzer, M. et al. Pump-induced exceptional points in lasers. Phys. Rev. Lett. 108, 173901 (2012).

    Article  ADS  Google Scholar 

  64. Brandstetter, M. et al. Reversing the pump dependence of a laser at an exceptional point. Nat. Commun. 5, 4034 (2014).

    Article  Google Scholar 

  65. Gu, Z. et al. Experimental demonstration of PT-symmetric stripe lasers: experimental demonstration of PT-symmetric stripe lasers. Laser Photon. Rev. 10, 588–594 (2016).

    Article  Google Scholar 

  66. Chong, Y. D., Ge, L., Cao, H. & Stone, A. D. Coherent perfect absorbers: time-reversed lasers. Phys. Rev. Lett. 105, 053901 (2010).

    Article  ADS  Google Scholar 

  67. Wan, W. et al. Time-reversed lasing and interferometric control of absorption. Science 331, 889–892 (2011).

    Article  ADS  Google Scholar 

  68. Zhang, J., MacDonald, K. F. & Zheludev, N. I. Controlling light-with-light without nonlinearity. Light Sci. Appl. 1, e18 (2012).

    Article  Google Scholar 

  69. Yariv, A. Critical coupling and its control in optical waveguide-ring resonator systems. IEEE Photon. Technol. Lett. 14, 483–485 (2002).

    Article  ADS  Google Scholar 

  70. Tischler, J. R., Bradley, M. S. & Bulović, V. Critically coupled resonators in vertical geometry using a planar mirror and a 5 nm thick absorbing film. Opt. Lett. 31, 2045–2047 (2006).

    Article  ADS  Google Scholar 

  71. Hamel, W. A. & Woerdman, J. P. Nonorthogonality of the longitudinal eigenmodes of a laser. Phys. Rev. A 40, 2785–2787 (1989).

    Article  ADS  Google Scholar 

  72. Wong, Z. J. et al. Lasing and anti-lasing in a single cavity. Nat. Photon. 10, 796–801 (2016).

    Article  ADS  Google Scholar 

  73. Longhi, S. Invisibility in PT-symmetric complex crystals. J. Phys. Math. Theor. 44, 485302 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  74. Jones, H. F. Analytic results for a PT-symmetric optical structure. J. Phys. Math. Theor. 45, 135306 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. Berry, M. V. Optical lattices with PT symmetry are not transparent. J. Phys. Math. Theor. 41, 244007 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. Greenberg, M. & Orenstein, M. Irreversible coupling by use of dissipative optics. Opt. Lett. 29, 451–453 (2004).

    Article  ADS  Google Scholar 

  77. Kulishov, M., Laniel, J. M., Bélanger, N., Azaña, J. & Plant, D. V. Nonreciprocal waveguide Bragg gratings. Opt. Express 13, 3068–3078 (2005).

    Article  ADS  Google Scholar 

  78. Feng, L. et al. Nonreciprocal light propagation in a silicon photonic circuit. Science 333, 729–733 (2011).

    Article  ADS  Google Scholar 

  79. Yan, Y. & Giebink, N. C. Passive PT symmetry in organic composite films via complex refractive index modulation. Adv. Opt. Mater. 2, 423–427 (2014).

    Article  Google Scholar 

  80. Wiersig, J. Chiral and nonorthogonal eigenstate pairs in open quantum systems with weak backscattering between counterpropagating traveling waves. Phys. Rev. A 89, 012119 (2014).

    Article  ADS  Google Scholar 

  81. Jia, Y., Yan, Y., Kesava, S. V., Gomez, E. D. & Giebink, N. C. Passive parity–time symmetry in organic thin film waveguides. ACS Photon. 2, 319–325 (2015).

    Article  Google Scholar 

  82. Hahn, C. et al. Observation of exceptional points in reconfigurable non-Hermitian vector-field holographic lattices. Nat. Commun. 7, 12201 (2016).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  84. Horsley, S. A. R., Artoni, M. & La Rocca, G. C. Spatial Kramers–Kronig relations and the reflection of waves. Nat. Photon. 9, 436–439 (2015).

    Article  ADS  Google Scholar 

  85. Longhi, S. Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures. Opt. Lett. 40, 5694–5697 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  87. Kharitonov, S. & Brès, C.-S. Isolator-free unidirectional thulium-doped fiber laser. Light Sci. Appl. 4, e340 (2015).

    Article  Google Scholar 

  88. Hopkins, B., Poddubny, A. N., Miroshnichenko, A. E. & Kivshar, Y. S. Circular dichroism induced by Fano resonances in planar chiral oligomers: circular dichroism induced by Fano resonances in planar chiral oligomers. Laser Photon. Rev. 10, 137–146 (2016).

    Article  Google Scholar 

  89. Zhong, Q., Ahmed, A., Dadap, J. I., Osgood, R. M. Jr & El-Ganainy, R. Parametric amplification in quasi-PT symmetric coupled waveguide structures. New J. Phys. 18, 125006 (2016).

    Article  ADS  Google Scholar 

  90. Fleury, R., Sounas, D. L. & Alù, A. Negative refraction and planar focusing based on parity–time symmetric metasurfaces. Phys. Rev. Lett. 113, 023903 (2014).

    Article  ADS  Google Scholar 

  91. Peng, P. et al. Anti-parity–time symmetry with flying atoms. Nat. Phys. 12, 1139–1145 (2016).

    Article  Google Scholar 

  92. Malzard, S., Poli, C. & Schomerus, H. Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity–time symmetry. Phys. Rev. Lett. 115, 200402 (2015).

    Article  ADS  Google Scholar 

  93. Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    Article  ADS  Google Scholar 

  94. Miri, M.-A., Heinrich, M., El-Ganainy, R. & Christodoulides, D. N. Supersymmetric optical structures. Phys. Rev. Lett. 110, 233902 (2013).

    Article  ADS  Google Scholar 

  95. Heinrich, M. et al. Supersymmetric mode converters. Nat. Commun. 5, 3698 (2014).

    Article  Google Scholar 

  96. Ge, L. Symmetry-protected zero-mode laser with a tunable spatial profile. Phys. Rev. A 95, 023812 (2017).

    Article  ADS  Google Scholar 

  97. Longhi, S., Gatti, D. & Della Valle, G. Robust light transport in non-Hermitian photonic lattices. Sci. Rep. 5, 13376 (2015).

    Article  ADS  Google Scholar 

  98. Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos, T. Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101 (2011).

    Article  ADS  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

Download references

Acknowledgements

L.F. acknowledges support from the Army Research Office (W911NF-15-1-0152), the Army Research Office Young Investigator Research Program (W911NF-16-1-0403) and the National Science Foundation (DMR-1506884 and ECCS-1507312). R.E. acknowledges support from the National Science Foundation (ECCS-1545804). L.G. acknowledges support from the National Science Foundation (DMR-1506987).

Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Liang Feng

  2. Department of Physics and Henes Center for Quantum Phenomena, Michigan Technological University, Houghton, MI, USA

    Ramy El-Ganainy

  3. Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

    Ramy El-Ganainy

  4. Department of Engineering Science and Physics, College of Staten Island, City University of New York, Staten Island, NY, USA

    Li Ge

  5. The Graduate Center, City University of New York, New York, NY, USA

    Li Ge

Authors
  1. Liang Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramy El-Ganainy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

L.F. led the project. All authors contributed significantly to the preparation of the manuscript.

Corresponding authors

Correspondence to Liang Feng or Li Ge.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nature Photon 11, 752–762 (2017). https://doi.org/10.1038/s41566-017-0031-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-017-0031-1

This article is cited by

Search

Advanced 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