Letter | Published:

Photonic topological boundary pumping as a probe of 4D quantum Hall physics

Nature volume 553, pages 5962 (04 January 2018) | Download Citation

Abstract

When a two-dimensional (2D) electron gas is placed in a perpendicular magnetic field, its in-plane transverse conductance becomes quantized; this is known as the quantum Hall effect1. It arises from the non-trivial topology of the electronic band structure of the system, where an integer topological invariant (the first Chern number) leads to quantized Hall conductance. It has been shown theoretically that the quantum Hall effect can be generalized to four spatial dimensions2,3,4, but so far this has not been realized experimentally because experimental systems are limited to three spatial dimensions. Here we use tunable 2D arrays of photonic waveguides to realize a dynamically generated four-dimensional (4D) quantum Hall system experimentally. The inter-waveguide separation in the array is constructed in such a way that the propagation of light through the device samples over momenta in two additional synthetic dimensions, thus realizing a 2D topological pump5,6,7,8. As a result, the band structure has 4D topological invariants (known as second Chern numbers) that support a quantized bulk Hall response with 4D symmetry7. In a finite-sized system, the 4D topological bulk response is carried by localized edge modes that cross the sample when the synthetic momenta are modulated. We observe this crossing directly through photon pumping of our system from edge to edge and corner to corner. These crossings are equivalent to charge pumping across a 4D system from one three-dimensional hypersurface to the spatially opposite one and from one 2D hyperedge to another. Our results provide a platform for the study of higher-dimensional topological physics.

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Acknowledgements

We thank H. M. Price, M. Lohse, C.-X. Liu, W. Benalcazar, E. Prodan and T. Ozawa for their comments and feedback on the manuscript. O.Z. thanks the Swiss National Science Foundation for financial support. M.C.R. acknowledges the National Science Foundation under award number ECCS-1509546, the Charles E. Kaufman Foundation, a supporting organization of the Pittsburgh Foundation, and the Alfred P. Sloan Foundation under fellowship number FG-2016-6418. K.P.C. acknowledges the National Science Foundation under award numbers ECCS-1509199 and DMS-1620218.

Author information

Author notes

    • Yaacov E. Kraus

    Deceased.

Affiliations

  1. Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland

    • Oded Zilberberg
  2. Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA

    • Sheng Huang
    • , Mohan Wang
    •  & Kevin P. Chen
  3. Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

    • Jonathan Guglielmon
    •  & Mikael C. Rechtsman
  4. Department of Physics, Holon Institute of Technology, Holon 5810201, Israel

    • Yaacov E. Kraus

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Contributions

O.Z., J.G., Y.E.K. and M.C.R. performed the theoretical analysis; S.H. developed the laser fabrication process and characterized the samples with the assistance of J.G. and M.W., under the supervision of K.P.C. and M.C.R.; O.Z. and M.C.R. designed the experiment, wrote the manuscript and supervised the project.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Oded Zilberberg or Mikael C. Rechtsman.

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https://doi.org/10.1038/nature25011

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