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Braiding photonic topological zero modes

Nature Physics volume 16, pages 989–993 (2020)Cite this article

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Abstract

A remarkable property of quantum mechanics in two-dimensional space is its ability to support ‘anyons’, particles that are neither fermions nor bosons. Theory predicts that these exotic excitations can exist as bound states confined near topological defects, such as Majorana zero modes trapped in vortices in topological superconductors. Intriguingly, in the simplest cases the non-trivial phase that arises when such defects are ‘braided’ around one another is not intrinsically quantum mechanical; instead, it can be viewed as a manifestation of the geometric (Pancharatnam–Berry) phase in wave mechanics, which makes possible the simulation of such phenomena in classical systems. Here, we report the experimental measurement of the geometric phase owing to such a braiding process. These measurements are obtained with an interferometer constructed from highly tunable two-dimensional arrays of photonic waveguides. Our results introduce photonic lattices as a versatile platform for the experimental study of topological defects and their braiding, and complement ongoing efforts in the study of solid-state systems and cold atomic gases.

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Fig. 1: Schematic of waveguide array.
Fig. 2: Experimentally measured vortex mode.
Fig. 3: Interferometric measurement of Berry phase associated with braiding.

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Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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Acknowledgements

M.C.R. and J.N. acknowledge support from the US National Science Foundation (NSF, grant no. ECCS-1509546) and the Penn State Materials Research Science and Engineering Center, Center for Nanoscale Science (grant no. NSF DMR-1420620). M.C.R. acknowledges support from the US Office of Naval Research (grant no. N00014-18-1-2595) and the Packard Foundation (fellowship no. 2017-66821). J.N. acknowledges support by the Corning Incorporated Office of STEM Graduate Research. T.S. acknowledges support from the US NSF Graduate Research Fellowship Program (grant no. DGE 1752814). T.I. acknowledges support from the Laboratory for Physical Sciences, Microsoft and a Joint Quantum Institute postdoctoral fellowship, as well as Iowa State University start-up funds. K.P.C., S.H. and M.W. acknowledge support from the US NSF (grant nos. ECCS-1509199 and DMS-1620218). C.C. is supported by the US Department of Energy (grant no. DE-FG02-06ER46316).

Author information

Author notes

  1. These authors contributed equally: Jiho Noh, Thomas Schuster.

Authors and Affiliations

  1. Department of Physics, The Pennsylvania State University, University Park, PA, USA

    Jiho Noh & Mikael C. Rechtsman

  2. Department of Physics, University of California, Berkeley, CA, USA

    Thomas Schuster

  3. Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD, USA

    Thomas Iadecola

  4. Department of Physics and Astronomy, Iowa State University, Ames, IA, USA

    Thomas Iadecola

  5. Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Sheng Huang, Mohan Wang & Kevin P. Chen

  6. Physics Department, Boston University, Boston, MA, USA

    Claudio Chamon

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Contributions

J.N. carried out the optical probing experiment and performed data analysis with guidance from M.C.R. J.N. and T.S. performed numerical simulations and designed the sample. T.S., T.I. and C.C. performed the theoretical analysis and helped to guide the experiment. S.H. and M.W. developed the laser fabrication process and fabricated and characterized the samples under the supervision of K.P.C. The manuscript was written by J.N., T.S., T.I, C.C. and M.C.R. M.C.R. supervised the project.

Corresponding author

Correspondence to Mikael C. Rechtsman.

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Supplementary information

Supplementary Video 1

Movies of the two waveguide lattices throughout the braiding stage, with each frame representing a constant-z slice of the waveguide array. The waveguides (filled circles) are displaced from their honeycomb positions (empty circles) at an angle equal to the phase of the Kekulé order parameter Δr(z) (arrows, drawn parallel to the displacements and colored according to their orientation). The order parameter in each lattice contains a vortex of charge –1 (central red square) near the lattice center. The overall offset α of the order parameter’s phase in each lattice is varied as a function of z and can be interpreted as the angle between the central anti-vortex and a fictitious vortex ‘at infinity’ (outer red square) that resides outside the waveguide array. In this first experiment, α → α + π in both lattices, corresponding to braiding a vortex at infinity counterclockwise about each lattice.

Supplementary Video 2

In this second experiment, α → α + π in the lattice on the left, and α → α − π in the lattice on the right, corresponding to braiding a vortex at infinity counterclockwise about the left lattice but clockwise about the right lattice. The final configurations of the two lattices differ by a phase 2π and are identical.

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Noh, J., Schuster, T., Iadecola, T. et al. Braiding photonic topological zero modes. Nat. Phys. 16, 989–993 (2020). https://doi.org/10.1038/s41567-020-1007-5

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