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.

Observation of topological edge states in parity–time-symmetric quantum walks

Abstract

The study of non-Hermitian systems with parity–time (PT) symmetry is a rapidly developing frontier. Realized in recent experiments, PT-symmetric classical optical systems with balanced gain and loss hold great promise for future applications. Here we report the experimental realization of passive PT-symmetric quantum dynamics for single photons by temporally alternating photon losses in the quantum walk interferometers. The ability to impose PT symmetry allows us to realize and investigate Floquet topological phases driven by PT-symmetric quantum walks. We observe topological edge states between regions with different bulk topological properties and confirm the robustness of these edge states with respect to PT-symmetry-preserving perturbations and PT-symmetry-breaking static disorder. Our results contribute towards the realization of quantum mechanical PT-synthetic devices and suggest exciting possibilities for the exploration of the topological properties of non-Hermitian systems using discrete-time quantum walks.

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

Figure 1: Phase diagram and experimental scheme for non-unitary QWs with alternating losses.
Figure 2: Experimental PT-symmetric homogeneous QWs with the walker starting from x = 0 and the initial coin state chosen to be |L |H〉 + i|V〉 via the delayed-choice setting.
Figure 3: Experimental observation of topological edge states in non-unitary QWs with the walker starting from x = 4 and the initial coin state chosen to be |H〉 via the delayed-choice setting.
Figure 4: Long-time survived edge states in the lossy system.
Figure 5: Edge states are robust against small perturbations.
Figure 6: Edge states are robust against static disorder to one of the rotations with angles θ1 + δθ where δθ is chosen uniformly from the intervals [−0.2,0.2].

References

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

    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  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  7. 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 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015).

    Article  ADS  Google Scholar 

  12. Zeuner, J. M. et al. Observation of a topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett. 115, 040402 (2015).

    Article  ADS  Google Scholar 

  13. Weimann, S. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. 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 

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

    Article  Google Scholar 

  17. 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 

  18. 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 

  19. Hodaei, H. et al. Parity–time-symmetric microring lasers. Science 346, 975–978 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Regensburger, A. et al. Observation of defect states in PT-symmetric optical lattices. Phys. Rev. Lett. 110, 223902 (2013).

    Article  ADS  Google Scholar 

  22. Schreiber, A. et al. Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010).

    Article  ADS  Google Scholar 

  23. Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012).

    Article  ADS  Google Scholar 

  24. Ambainis, A. Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507–518 (2003).

    Article  Google Scholar 

  25. Childs, A. M. et al. Proceedings of the 35th ACM Symposium on Theory of Computing 59–68 (ACM, 2003).

    Google Scholar 

  26. Childs, A. M., Gosset, D. & Webb, Z. Universal computation by multiparticle quantum walk. Science 339, 791–794 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  27. Schreiber, A. et al. Decoherence and disorder in quantum walks: from ballistic spread to localization. Phys. Rev. Lett. 106, 180403 (2011).

    Article  ADS  Google Scholar 

  28. Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012).

    Article  ADS  Google Scholar 

  29. Crespi, A. et al. Anderson localization of entangled photons in an integrated quantum walk. Nat. Photon. 7, 322–328 (2013).

    Article  ADS  Google Scholar 

  30. Genske, M. et al. Electric quantum walks with individual atoms. Phys. Rev. Lett. 110, 190601 (2013).

    Article  ADS  Google Scholar 

  31. Xue, P. et al. Experimental quantum-walk revival with a time-dependent coin. Phys. Rev. Lett. 114, 140502 (2015).

    Article  ADS  Google Scholar 

  32. Mochizuki, K., Kim, D. & Obuse, H. Explicit definition of PT symmetry for nonunitary quantum walks with gain and loss. Phys. Rev. A 93, 062116 (2016).

    Article  ADS  Google Scholar 

  33. Jeong, Y.-C., Di Franco, C., Lim, H.-T., Kim, M. S. & Kim, Y.-H. Experimental realization of a delayed-choice quantum walk. Nat. Commun. 4, 2471 (2013).

    Article  ADS  Google Scholar 

  34. Kim, D., Mochizuki, K., Kawakami, N. & Obuse, H. Floquet Topological phases driven by PT symmetric nonunitary time evolution. Preprint at https://arxiv.org/abs/1609.09650 (2016).

  35. Kitagawa, T., Rudner, M. S., Berg, E. & Demler, E. Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010).

    Article  ADS  Google Scholar 

  36. Asbóth, J. K. & Obuse, H. Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88, 121406(R) (2013).

    Article  ADS  Google Scholar 

  37. Pasek, M. & Chong, Y. D. Network models of photonic Floquet topological insulators. Phys. Rev. B 89, 075113 (2014).

    Article  ADS  Google Scholar 

  38. Mostafazadeh, A. Pseudo-unitary operators and pseudo-unitary quantum dynamics. J. Math. Phys. 45, 932–946 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  39. Mochizuki, K. & Obuse, H. Effects of disorder on non-unitary PT symmetric quantum walks. Interdiscip. Inf. Sci. 23, 95–103 (2017).

    MathSciNet  Google Scholar 

  40. 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  Google Scholar 

  41. Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    Article  ADS  Google Scholar 

  42. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  44. Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

    Article  ADS  Google Scholar 

  45. Fleury, R., Khanikaev, A. B. & Alù, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016).

    Article  ADS  Google Scholar 

  46. Xiao, M., Zhang, Z. Q. & Chan, C. T. Surface impedance and bulk band geometric phases in one-dimensional systems. Phys. Rev. X 4, 021017 (2014).

    Google Scholar 

  47. Harari, G. et al. Topological insulators in PT-symmetric lattices. Conf. Lasers Electro-Opt. (CLEO) paper FTh3D.3 1–2 (Optical Society of America, 2015).

    Google Scholar 

  48. Esaki, K., Sato, M., Hasebe, K. & Kohmoto, M. Edge states and topological phases in non-Hermitian systems. Phys. Rev. B 84, 205128 (2011).

    Article  ADS  Google Scholar 

  49. Hu, Y. C. & Hughes, T. L. Absence of topological insulator phases in non-hermitian PT-symmetric Hamiltonians. Phys. Rev. B 84, 153101 (2011).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work has been supported by the Natural Science Foundation of China (Grant Nos. 11474049, 11674056, 11374283 and 11522545) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20160024). W.Y. acknowledges support from the National Key R&D Program (Grant No. 2016YFA0301700) and the ‘Strategic Priority Research Program(B)’ of the Chinese Academy of Sciences (Grant No. XDB01030200). N.K. and H.O. were supported by a Grant-in-Aid for Scientific Research on Innovative Areas ‘Topological Materials Science’ (Grant Nos JP15H05855, JP15K21717 and JP16H00975) and JSPS KAKENHI (Grant No. JP16K17760). B.C.S. acknowledges the 1000-Talent Plan and NSFC (Grant No. GG2340000241) for financial support.

Author information

Authors and Affiliations

Authors

Contributions

L.X., X.Z., Z.H.B. and K.K.W. performed the experiments. K.M., D.K., N.K. and H.O. performed the theoretical analysis. W.Y., H.O. and B.C.S. wrote part of the paper. P.X. designed the experiments, analysed the data and wrote most of the paper. All the authors participated in the discussions.

Corresponding author

Correspondence to P. Xue.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 1866 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, L., Zhan, X., Bian, Z. et al. Observation of topological edge states in parity–time-symmetric quantum walks. Nature Phys 13, 1117–1123 (2017). https://doi.org/10.1038/nphys4204

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys4204

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