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Bimorphic Floquet topological insulators

Nature Materials volume 21, pages 634–639 (2022)Cite this article

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Abstract

Topological theories have established a unique set of rules that govern the transport properties in a wide variety of wave-mechanical settings. In a marked departure from the established approaches that induce Floquet topological phases by specifically tailored discrete coupling protocols or helical lattice motions, we introduce a class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to reveal rich topological features in the system. In exploring a ‘chain-driven’ generalization of the archetypical Floquet honeycomb lattice, we identify a rich phase structure that can host multiple non-trivial topological phases associated simultaneously with both Chern-type and anomalous chiral states. Experiments carried out in photonic waveguide lattices reveal a strongly confined helical edge state that, owing to its origin in bulk flat bands, can be set into motion in a topologically protected fashion, or halted at will, without compromising its adherence to individual lattice sites.

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Fig. 1: A new road to topological lattices.
Fig. 2: Band structures and topological characterization.
Fig. 3: Probing the Chern edge state of a photonic chain-driven honeycomb lattice.
Fig. 4: Anomalous edge-state propagation and compact localized bulk states.

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Data availability

The experimental data that support the findings of this study are available from M.H. upon reasonable request (matthias.heinrich@uni-rostock.de).

Code availability

The MATLAB codes corresponding to the beam-propagation method and band structure algorithms are available from G.G.P. upon reasonable request (pyrialak@knights.ucf.edu).

References

  1. Maczewsky, L. J. et al. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    Article  CAS  Google Scholar 

  2. Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).

    Article  CAS  Google Scholar 

  3. McIver, J. W. et al. Light-induced anomalous Hall effect in graphene. Nat. Phys. 16, 1058–1063 (2020).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  5. Aidelsburger, M. et al. Experimental realization of strong effective magnetic fields in an optical lattice. Phys. Rev. Lett. 107, 255301 (2011).

    Article  CAS  Google Scholar 

  6. Liu, G., Hao, N., Zhu, S.-L. & Liu, W. M. Topological superfluid transition induced by a periodically driven optical lattice. Phys. Rev. A 86, 013639 (2012).

    Article  Google Scholar 

  7. Nathan, F., Abanin, D., Berg, E., Lindner, N. H. & Rudner, M. S. Anomalous Floquet insulators. Phys. Rev. B 99, 195133 (2019).

    Article  CAS  Google Scholar 

  8. Hafezi, M. et al. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    Article  CAS  Google Scholar 

  9. Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nat. Mater. 14, 290–294 (2015).

    Article  CAS  Google Scholar 

  10. Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    Article  CAS  Google Scholar 

  11. Wintersperger, K. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020).

    Article  CAS  Google Scholar 

  12. Peng, Y.-G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Commun. 7, 13368 (2016).

  13. Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet–Bloch systems. New J. Phys. 17, 125014 (2015).

    Article  Google Scholar 

  14. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    Article  Google Scholar 

  15. Roy, R. & Harper, F. Periodic table for Floquet topological insulators. Phys. Rev. B 96, 155118 (2017).

    Article  Google Scholar 

  16. Wang, C., Zhang, P., Chen, X., Yu, J. & Zhai, H. Scheme to measure the topological number of a Chern insulator from quench dynamics. Phys. Rev. Lett. 118, 185701 (2017).

    Article  Google Scholar 

  17. Tarnowski, M. et al. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat. Commun. 10, 1728 (2019).

    Article  Google Scholar 

  18. Ünal, F. N., Eckardt, A. & Slager, R.-J. Hopf characterization of two-dimensional Floquet topological insulators. Phys. Rev. Res. 1, 022003(R) (2019).

    Article  Google Scholar 

  19. Schuster, T., Gazit, S., Moore, J. E. & Yao, N. Y. Floquet Hopf insulators. Phys. Rev. Lett. 123, 266803 (2019).

    Article  CAS  Google Scholar 

  20. Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    CAS  Google Scholar 

  21. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Article  Google Scholar 

  22. Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet–Bloch systems. New J. Phys. 17, 125014 (2015).

    Article  Google Scholar 

  23. Carpentier, D. et al. Topological index for periodically driven time-reversal invariant 2D systems. Phys. Rev. Lett. 114, 106806 (2015).

    Article  Google Scholar 

  24. Noh, J. et al. Experimental observation of optical Weyl points and Fermi arc-like surface states. Nat. Phys. 13, 611–617 (2017).

    Article  CAS  Google Scholar 

  25. Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461–465 (2018).

    Article  Google Scholar 

  26. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    Article  CAS  Google Scholar 

  27. Maczewsky, L. J. et al. Fermionic time-reversal symmetry in a photonic topological insulator. Nat. Mater. 19, 855–860 (2020).

    Article  CAS  Google Scholar 

  28. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  Google Scholar 

  29. Mukherjee, S. & Rechtsman, M. C. Observation of Floquet solitons in a topological bandgap. Science 368, 856–859 (2020).

    Article  CAS  Google Scholar 

  30. Maczewsky, L. J. et al. Nonlinearity-induced photonic topological insulator. Science 370, 701–704 (2020).

    Article  CAS  Google Scholar 

  31. Kirsch, M. et al. Nonlinear second-order photonic topological insulators. Nat. Phys. https://doi.org/10.1038/s41567-021-01275-3 (2021).

  32. Pyrialakos, G. et al. Symmetry-controlled edge states in the type-II phase of Dirac photonic lattices. Nat. Commun. 11, 2074 (2020).

    Article  CAS  Google Scholar 

  33. Ramachandran, A., Andreanov, A. & Flach, S. Chiral flat bands: existence, engineering, and stability. Phys. Rev. B 96, 161104(R) (2017).

    Article  Google Scholar 

  34. Xia, S. et al. Unconventional flatband line states in photonic Lieb lattices. Phys. Rev. Lett. 121, 263902 (2018).

    Article  Google Scholar 

  35. Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).

  36. Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).

    Article  Google Scholar 

  37. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    Article  CAS  Google Scholar 

  38. Hadad, Y. et al. Self-induced topological protection in nonlinear circuit arrays. Nat. Electron. 1, 178–182 (2018).

    Article  Google Scholar 

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

    Article  Google Scholar 

  40. Szameit, A. et al. Quasi-incoherent propagation in waveguide arrays. Appl. Phys. Lett. 90, 241113 (2007).

    Article  Google Scholar 

Download references

Acknowledgements

We thank C. Otto for preparing the high-quality fused silica samples used for the inscription of all photonic structures employed in the experiments presented here. G.G.P. acknowledges the support of the Bodossaki Foundation. This work was partially supported by Defense Advanced Research Projects Agency (DARPA; D18AP00058), Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI; N00014-16-1-2640, N00014-18-1-2347, N00014-19-1-2052, N00014-20-1-2522, N00014-20-1-2789), Air Force Office of Scientific Research (AFOSR MURI; FA9550-20-1-0322, FA9550-21-1-0202), the National Science Foundation (DMR-1420620, EECS-1711230, ECCS-1454531, DMR-1420620, ECCS-1757025, CBET-1805200, ECCS-2000538, ECCS-2011171), Mathematics and Physical Sciences (MPS) Simons collaboration (Simons grant 733682), W. M. Keck Foundation, US–Israel Binational Science Foundation (BSF 2016381) and the US Air Force Research Laboratory (FA9550-14-1-0037, FA9550-20-1-0322, FA9550-21-1-0202, FA86511820019). We furthermore acknowledge funding from the Deutsche Forschungsgemeinschaft (SCHE 612/6-1, SZ 276/12-1, BL 574/13-1, SZ 276/15-1, SZ 276/20-1) and the Alfried Krupp von Bohlen and Halbach Foundation.

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Author notes

  1. These authors contributed equally: Georgios G. Pyrialakos, Julius Beck.

Authors and Affiliations

  1. College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, USA

    Georgios G. Pyrialakos & Demetrios N. Christodoulides

  2. Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Georgios G. Pyrialakos & Nikolaos V. Kantartzis

  3. Institute for Physics, University of Rostock, Rostock, Germany

    Julius Beck, Matthias Heinrich, Lukas J. Maczewsky & Alexander Szameit

  4. Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA

    Mercedeh Khajavikhan

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  1. Georgios G. Pyrialakos
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Contributions

G.G.P. initiated the idea, formulated the index-modulated lattice and performed the theoretical calculations and simulations. J.B. developed the experimental implementation, fabricated the samples and conducted the measurements. J.B., L.J.M. and M.H. evaluated the measurements and interpreted the data. M.K., N.V.K., A.S. and D.N.C. supervised the efforts of their respective groups. All authors discussed the results and cowrote the manuscript.

Corresponding author

Correspondence to Demetrios N. Christodoulides.

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The authors declare no competing interests.

Peer review

Peer review information

Nature Materials thanks Alexander Khanikaev, Sunil Mittal and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Comparison of staggered and flat-phased broad excitations.

(a-d): Staggered excitations of the primary waveguides of edge unit cells successfully populate the topological Chern mode near the edge of the Brillouin zone. (e-h) Absent the appropriate phase modulation, the injected wave packets instead represent a superposition of the bulk bands near the center of the Brillouin zone. As a result, the light diffracts freely across the entire lattice instead of being captured in the helical Chern channel.

Extended Data Fig. 2 Honeycomb helical FTI.

A driven honeycomb lattice with sinusoidally modulated time-periodic coupling terms exhibits a secondary topological phase in response to an increase of the driving period. At the critical driving period \(T_{{{\mathrm{C}}}} = \pi /3\) the gap at the Floquet zone collapses and reopens with a topologically non-trivial winding number. This corresponds to an anomalous phase with a trivial Chern number, signifying a topological phase transition.

Extended Data Fig. 3 Chained honeycomb FTI.

Above its critical modulation period \(T_{{{\mathrm{C}}}}^\prime = \pi\), the chain-driven honeycomb lattice enters a secondary topological phase characterized by the band diagram shown on the right. In this configuration, the bands manifest a non-trivial topological structure characterized by higher order Chern invariants (\({{{\mathcal{C}}}} = 2\)), and, in turn, the Chern gaps host an increased number of unidirectional edge states. Details on the modal amplitudes and propagation dynamics of the Chern states in the \({{{\mathcal{C}}}} = 2\) phase are provided in Supplementary Fig. 5.

Extended Data Fig. 4 Losses in index-modulated waveguides.

Waveguide fluorescence40 characterization confirms that even coarsely discretized index modulation (twelve constant-index segments approximating the ideal cosine Floquet cycle) only introduces excess losses of (0.096 ± 0.010)dB/cm relative to a straight constant-index waveguide. With half of the lattice sites being modulated, the mean excess losses of the bimorphic FTI are (0.048 ± 0.005)dB/cm, substantially below the value of 1.7 dB/cm reported for bending losses in conventional FTIs based on helically modulated waveguides4. Note that the apparent oscillations in the normalized intensity of the modulated waveguide are due to the different concentration of color centers formed at different writing speeds, resulting in a modulation of the fluorescence efficiency between the individual segments of each modulation period. Due to the large number of periods, this oscillation does not notably impact the measurement of the loss coefficient γ. The fluorescence itself is a feature of femtosecond laser-written waveguides in fused silica, and does not pose a substantial source of propagation losses40.

Supplementary information

Supplementary Information

Supplementary Figs. 1–6 and Discussion.

Reporting Summary

Supplementary Video 1

Simulation of propagation dynamics of two isolated compact localized states in a ribbon lattice. Power is injected into the sites indicated by labels A and B. At the 00:06 time mark, the Floquet modulation is switched off in the vicinity of wave packet B (shaded region). As a consequence, wave packet B couples to a static compact localized state and remains stationary, while wave packet A finds a route around the shaded region. At the 00:20 time mark, the Floquet modulation is restored in the entire lattice, and both wave packets resume propagation to, finally, become spatially interchanged.

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Pyrialakos, G.G., Beck, J., Heinrich, M. et al. Bimorphic Floquet topological insulators. Nat. Mater. 21, 634–639 (2022). https://doi.org/10.1038/s41563-022-01238-w

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