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.

Enhanced sensitivity at higher-order exceptional points

An Erratum to this article was published on 30 November 2017

This article has been updated

Abstract

Non-Hermitian degeneracies, also known as exceptional points, have recently emerged as a new way to engineer the response of open physical systems, that is, those that interact with the environment. They correspond to points in parameter space at which the eigenvalues of the underlying system and the corresponding eigenvectors simultaneously coalesce1,2,3. In optics, the abrupt nature of the phase transitions that are encountered around exceptional points has been shown to lead to many intriguing phenomena, such as loss-induced transparency4, unidirectional invisibility5,6, band merging7,8, topological chirality9,10 and laser mode selectivity11,12. Recently, it has been shown that the bifurcation properties of second-order non-Hermitian degeneracies can provide a means of enhancing the sensitivity (frequency shifts) of resonant optical structures to external perturbations13. Of particular interest is the use of even higher-order exceptional points (greater than second order), which in principle could further amplify the effect of perturbations, leading to even greater sensitivity. Although a growing number of theoretical studies have been devoted to such higher-order degeneracies14,15,16, their experimental demonstration in the optical domain has so far remained elusive. Here we report the observation of higher-order exceptional points in a coupled cavity arrangement—specifically, a ternary, parity–time-symmetric photonic laser molecule—with a carefully tailored gain–loss distribution. We study the system in the spectral domain and find that the frequency response associated with this system follows a cube-root dependence on induced perturbations in the refractive index. Our work paves the way for utilizing non-Hermitian degeneracies in fields including photonics, optomechanics10, microwaves9 and atomic physics17,18.

Your institute does not have access to this article

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Parity–time-symmetric coupled cavity systems that support exceptional points.
Figure 2: Bifurcations of complex eigenfrequencies around a third-order exceptional point.
Figure 3: Binary parity–time-symmetric system operating around a second-order exceptional point.
Figure 4: Response of a ternary parity–time-symmetric system biased at a third-order exceptional point.

Change history

  • 08 November 2017

    Please see accompanying Erratum (http://doi.org/10.1038/nature24024). In this Letter, the y-axis labels of the inset to Fig. 3a should have been 102 rather than 101 (top label) and 101 rather than 100 (bottom label). This error has been corrected online.

References

  1. Kato, T. Perturbation Theory for Linear Operators (Springer, 2013)

  2. Heiss, W. D. Phases of wave functions and level repulsion. Eur. Phys. J. D 7, 1–4 (1999)

    ADS  CAS  Article  Google Scholar 

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

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

    ADS  CAS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  8. 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)

    ADS  CAS  Article  Google Scholar 

  9. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016)

    ADS  CAS  Article  Google Scholar 

  10. Xu, H., Mason, D., Jiang, L. & Harris, J. G. E. Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80–83 (2016)

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  12. 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)

    ADS  CAS  Article  Google Scholar 

  13. 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)

    ADS  Article  Google Scholar 

  14. Demange, G. & Graefe, E.-M. Signatures of three coalescing eigenfunctions. J. Phys. A 45, 025303 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  15. Lin, Z., Pick, A., Loncˇar, M. & Rodriguez, A. W. Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals. Phys. Rev. Lett. 117, 107402 (2016)

    ADS  Article  Google Scholar 

  16. Jing, H., Özdemir, S¸. K., Lü, H. & Nori, F. High-order exceptional points in optomechanics. Sci. Rep. 7, 3386 (2017)

    ADS  CAS  Article  Google Scholar 

  17. Zhang, Z. et al. Observation of parity–time symmetry in optically induced atomic lattices. Phys. Rev. Lett. 117, 123601 (2016)

    ADS  Article  Google Scholar 

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

    CAS  Article  Google Scholar 

  19. Zhu, J. et al. On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator. Nat. Photon. 4, 46–49 (2010)

    ADS  CAS  Article  Google Scholar 

  20. Armani, A. M., Kulkarni, R. P., Fraser, S. E., Flagan, R. C. & Vahala, K. J. Label-free, single-molecule detection with optical microcavities. Science 317, 783–787 (2007)

    ADS  CAS  Article  Google Scholar 

  21. Arnold, S., Shopova, S. I. & Holler, S. Whispering gallery mode bio-sensor for label-free detection of single molecules: thermo-optic vs. reactive mechanism. Opt. Express 18, 281–287 (2010)

    ADS  CAS  Article  Google Scholar 

  22. Chow, W. W. et al. The ring laser gyro. Rev. Mod. Phys. 57, 61–104 (1985)

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  25. Weimann, S. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater. 16, 433–438 (2016)

    ADS  Article  Google Scholar 

  26. Ding, K., Ma, G., Xiao, M., Zhang, Z. Q. & Chan, C. T. Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization. Phys. Rev. X 6, 021007 (2016)

    Google Scholar 

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  28. Teimourpour, M. H., El-Ganainy, R., Eisfeld, A., Szameit, A. & Christodoulides, D. N. Light transport in PT-invariant photonic structures with hidden symmetries. Phys. Rev. A 90, 053817 (2014)

    ADS  Article  Google Scholar 

  29. Hodaei, H. et al. Parity–time-symmetric coupled microring lasers operating around an exceptional point. Opt. Lett. 40, 4955–4958 (2015)

    ADS  CAS  Article  Google Scholar 

  30. Chong, Y. D. & Stone, A. D. General linewidth formula for steady-state multimode lasing in arbitrary cavities. Phys. Rev. Lett. 109, 063902 (2012)

    ADS  CAS  Article  Google Scholar 

  31. Hodaei, H. et al. Design considerations for single-mode microring lasers using parity–timessymmetry. IEEE J. Sel. Top. Quant. Electron. 22, 1500307 (2016)

    Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  33. Santis, C. T., Steger, S. T., Vilenchik, Y., Vasilyev, A. & Yariv, A. High-coherence semiconductor lasers based on integral high-Q resonators in hybrid Si/III-V platforms. Proc. Natl Acad. Sci. USA 111, 2879–2884 (2014)

    ADS  CAS  Article  Google Scholar 

  34. Lee, H. et al. Chemically etched ultrahigh-Q wedge-resonator on a silicon chip. Nat. Photon. 6, 369–373 (2012)

    ADS  CAS  Article  Google Scholar 

  35. Schomerus, H. Excess quantum noise due to mode nonorthogonality in dielectric microresonator. Phys. Rev. A 79, 061801 (2009)

    ADS  Article  Google Scholar 

  36. Yoo, G., Sim, H.-S. & Schomerus, H. Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators. Phys. Rev. A 84, 063833 (2011)

    ADS  Article  Google Scholar 

  37. Ren, J. et al. Ultrasensitive micro-scale parity–time-symmetric ring laser gyroscope. Opt. Lett. 42, 1556–1559 (2017)

    ADS  CAS  Article  Google Scholar 

Download references

Acknowledgements

We acknowledge financial support from the Office of Naval Research (ONR; N00014-16-1-2640), the National Science Foundation (NSF; ECCS-1454531, DMR-1420620), the Air force Office of Scientific Research (AFOSR; FA9550-14-1-0037) and the Army Research Office (ARO; W911NF-16-1-0013). This work was also partially funded by the Qatar National Research Fund (NPRP 9-020-1-006). R.E.-G. acknowledges support from the Henes Center for Quantum Phenomena at Michigan Tech University. H.H. thanks J. Boroumand from UCF for helping with the SEM images.

Author information

Authors and Affiliations

Authors

Contributions

H.H., M.K. and D.N.C. conceived and designed the experiments, H.H. and H.G.-G. fabricated the samples, H.H. and S.W. performed the experiments, H.H., A.U.H., M.K., R.E.-G. and D.N.C. analysed the data, and H.H., M.K., D.N.C., R.E.-G. and A.U.H. wrote the paper.

Corresponding author

Correspondence to Mercedeh Khajavikhan.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks K. Bliokh, M. Rechtsman, Y.-F. Xiao and the other anonymous reviewer(s) 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.

Extended data figures and tables

Extended Data Figure 1 Fabrication steps in realizing parity–time-symmetric photonic molecules.

a, Schematic of the fabrication process. b, A microscope image of the fabricated metallic micro-heaters. c, The heaters are electrically connected to the pins of the header via wire bonding. d, The photonic molecule systems are accessible for measurement from the back side through a hole in the header.

Extended Data Figure 2 Measurement station for characterizing sensitivity at higher-order exceptional points.

A schematic of the micro-photoluminescence experimental set-up. BS, beam splitter; NIR, near-infrared; NA, numerical aperture.

Extended Data Figure 3 Pump distribution.

The intensity profile of the pump beam at the sample without (a) and with (b) the image of the knife edge. By adjusting the intensity of the pump beam and the position of the knife edge, the desired gain/loss distribution is realized.

Extended Data Figure 4 Sample imaging.

a, The intensity profile of three coupled micro-ring resonators when they all pumped equally. b, The associated heaters imaged on the measurement station using a broadband near-infrared source.

Extended Data Figure 5 Characterizing the heat induced detuning.

a, b, The change in resonance frequency (Δω) of the intentionally decoupled cavities as a function of I2 (heater power) in the binary structure (a) and the ternary structure (b). The applied perturbation varies linearly with the power of the heater, and is imposed differentially on the micro-rings with respect to their distance from the active heater. The colours of the rings are used for distinction purposes and not as a representation of gain or loss. In all cases, solid circles indicate measured data and solid lines denote linear fits to the frequency response.

Extended Data Figure 6 Effect of coupling on enhancing sensitivity.

a, b, The wavelength splitting as a function of the differential perturbation applied to the gain cavity, for parity–time systems with different coupling coefficients (solid curves depict square-root simulations) and for a single micro-ring (straight line), on linear (a) and logarithmic (b) scales. In all six cases, filled circles indicate measured values and the error bars represent the uncertainty in the measurement because of the resolution limit of the spectrometer.

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hodaei, H., Hassan, A., Wittek, S. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017). https://doi.org/10.1038/nature23280

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature23280

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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