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Direct observation of Landau levels in silicon photonic crystals

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Abstract

When electrons are confined to a two-dimensional plane and are subjected to an out-of-plane magnetic field, they move in circular cyclotron orbits as a result of the Lorentz force. In the quantum domain, this cyclotron motion is quantized, and as a consequence, the energy spectrum of the electrons splits into discrete, highly degenerate states called Landau levels. These flat bands are the origin of the integer and fractional quantum Hall effects1,2. Although photons do not experience the Lorentz force because they do not carry charge, they can be made to experience ‘pseudomagnetic fields’3,4 as a result of periodicity-breaking strain. In this work, we experimentally observe photonic Landau levels that arise due to a strain-induced pseudomagnetic field in a silicon photonic crystal slab. The Landau levels are dispersive (that is, they are not flat bands) due to the distortion of the unit cell by the strain. We employ an additional strain of a different form that induces a pseudoelectric potential to flatten them. By acting akin to cavities that are delocalized across space, flat bands such as these have the potential to strongly enhance light–matter interaction as a result of the photonic structure. The analytical framework that we develop here for understanding the effects of inhomogeneous strain in photonic crystals via gauge fields can help to guide the design of multiscale non-periodic photonic structures.

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Fig. 1: Schematic illustrating the effect of strain on the Dirac cone.
Fig. 2: Unstrained and strained photonic crystals and their calculated band structures.
Fig. 3: Observation of Landau levels in the spectrum of a strained photonic crystal.
Fig. 4: Landau levels are excited at increasing ky in the band structure as the input beam moves from left to right through the sample.
Fig. 5: Introduction of a pseudoelectric potential acts to flatten the Landau-level bands.

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Data availability

Data are available from the corresponding author upon reasonable request.

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Acknowledgements

We gratefully acknowledge funding support from the Office of Naval Research MURI program under agreement no. N00014-20-1-2325; the Air Force Office of Scientific Research MURI program under agreement no. FA9550-22-1-0339; and the Kaufman and Packard foundations under grant nos. KA2020-114794 and 2017-66821, respectively. This research was also supported in part by National Science Foundation grants DMS-1620422 (M.C.R.), DMS-1620418 (M.I.W.), DMS-1908657 (M.I.W.) and DMS-1937254 (M.I.W.), as well as Simons Foundation Math + X Investigator Award no. 376319 (M.I.W.). We acknowledge the Nanofabrication Lab within the Materials Research Institute at Penn State and the help of M. Labella, as well as seed funding from the Center for Nanofabricated Optics at Penn State University. F.G. thanks GenISys, particularly R. McCay for his help in optimizing the fracturing of the electron-beam patterns. M.B. thanks S. Mukherjee and A. Cerjan for fruitful discussions in the early stages of the project and help with numerical optimization.

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Author notes

  1. These authors contributed equally: Maria Barsukova, Fabien Grisé, Zeyu Zhang.

Authors and Affiliations

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

    Maria Barsukova, Zeyu Zhang, Sachin Vaidya, Jonathan Guglielmon & Mikael C. Rechtsman

  2. Department of Astronomy & Astrophysics, The Pennsylvania State University, State College, PA, USA

    Fabien Grisé & Randall McEntaffer

  3. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Sachin Vaidya

  4. Department of Applied Physics and Applied Mathematics and Department of Mathematics, Columbia University, New York, NY, USA

    Michael I. Weinstein

  5. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA

    Li He & Bo Zhen

Authors
  1. Maria Barsukova
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  2. Fabien Grisé
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  10. Mikael C. Rechtsman
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Contributions

M.B. built the experimental setup and carried out all the experimental observations, with advice from L.H., B.Z. and M.C.R. M.B. and F.G. developed the device designs with input from Z.Z., S.V., J.G., R.M. and M.C.R. F.G. carried out the device fabrication. Z.Z. carried out the numerical and analytical calculations with input from J.G., M.B., S.V., M.I.W. and M.C.R. All authors contributed to writing the manuscript. M.C.R. supervised the project.

Corresponding author

Correspondence to Mikael C. Rechtsman.

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Nature Photonics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Sections 1–5 and Figs. 1–13.

Supplementary Video 1

Evolution of the relative excitation of the Landau-level states as the beam moves from left to right (Fig. 4d).

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Barsukova, M., Grisé, F., Zhang, Z. et al. Direct observation of Landau levels in silicon photonic crystals. Nat. Photon. (2024). https://doi.org/10.1038/s41566-024-01425-y

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