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Probing bulk topological invariants using leaky photonic lattices

Nature Physics volume 17, pages 632–638 (2021)Cite this article

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Abstract

Topological invariants characterizing filled Bloch bands underpin electronic topological insulators and analogous artificial lattices for Bose–Einstein condensates, photonics and acoustic waves. In bosonic systems, there is no Fermi exclusion principle to enforce uniform band filling, which makes measuring their bulk topological invariants challenging. Here we show how to achieve the controllable filling of bosonic bands using leaky photonic lattices. Leaky photonic lattices host transitions between bound and radiative modes at a critical energy, which plays a role analogous to the electronic Fermi level. Tuning this effective Fermi level into a bandgap results in the disorder-robust dynamical quantization of bulk topological invariants such as the Chern number. Our findings establish leaky lattices as a highly flexible platform for exploring topological and non-Hermitian wave physics.

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Fig. 1: Schematic of the method to measure topological invariants using leaky lattices.
Fig. 2: Quasi-normal mode spectrum of a one-dimensional lattice.
Fig. 3: Leaky SSH lattice.
Fig. 4: Measurement of Chern numbers using the leaky Haldane model.
Fig. 5: Array of ten slab waveguides for implementing the SSH model.

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Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code used to perform the numerical simulations within this paper is available from the corresponding author upon reasonable request.

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Acknowledgements

We thank A. Cerjan, Z. Chen, Y. Chong and M. Rechstman for illuminating discussions. D.L. was supported by the Institute for Basic Science (IBS-R024-Y1 and IBS-R024-D1). D.A.S. acknowledges funding from the Australian Research Council’s Early Career Researcher Award (DE190100430) and the Russian Science Foundation (Grant No. 20-72-00148).

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Author notes

  1. Daniel Leykam

    Present address: Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

Authors and Affiliations

  1. Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, Korea

    Daniel Leykam

  2. Basic Science Program, Korea University of Science and Technology, Daejeon, Korea

    Daniel Leykam

  3. Nonlinear Physics Centre, Australian National University, Canberra, Australian Capital Territory, Australia

    Daria A. Smirnova

  4. Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia

    Daria A. Smirnova

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D.L. and D.A.S. both performed the research and wrote the manuscript.

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Correspondence to Daniel Leykam.

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Extended data

Extended Data Fig. 1 Leaky photonic lattice design for laser-written waveguides.

(a) Schematic of a leaky photonic lattice formed by an array of 18 primary waveguides (red) weakly coupled to auxiliary arrays forming the environment. The alternating position of the environment arrays above and below the primary array can be used to minimise unwanted direct coupling between the auxiliary arrays. (b) Corresponding tight binding model. (c,d,e) Dynamics of observables in the primary array: (c) the total norm \({\mathcal{N}}={\sum }_{m}| {\psi }_{m}{| }^{2}\), (d) the mean displacement \(\Delta x={\sum }_{m}(m-x(0))| {\psi }_{m}{| }^{2}/{\mathcal{N}}\), and (e) the Zak phase ν. Error bars in (d,e) indicate one standard deviation obtained using an ensemble of 100 disorder realizations.

Extended Data Fig. 2 Leaky photonic lattice design for silicon photonics.

(a) Schematic of a dimerized array of 10 silicon waveguides with labelled dimensions. Trivial and nontrivial arrangements differ by the intersite gaps ordering: g1 > g2 (trivial), g1 < g2 (nontrivial). (b,c) Field profiles of eigenmodes from the upper (b, neff = 1.8154) and lower bands (c, neff = 1.6735 − 0.0014i) in the trivial array. (d) Power and (e,f) Zak phase obtained from the field projection operator summed over excitations of the 5th and 6th waveguides in the trivial (e, blue curves) and nontrivial (f, gold curves) arrays. For comparison, grey curves in (e,f) show the Zak phase estimated from the single 5th waveguide excitation.

Supplementary information

Supplementary Information

Discussion and Supplementary Fig 1.

Source data

Source Data Fig. 2

Source data for Fig. 2c,d.

Source Data Fig. 3

Source data for Fig. 3d,e.

Source Data Fig. 4

Source data for Fig. 4b,c.

Source Data Fig. 5

Source data for Fig. 5.

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Leykam, D., Smirnova, D.A. Probing bulk topological invariants using leaky photonic lattices. Nat. Phys. 17, 632–638 (2021). https://doi.org/10.1038/s41567-020-01144-5

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