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Gain-induced topological response via tailored long-range interactions

Nature Physics volume 17, pages 704–709 (2021)Cite this article

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Abstract

The ability to tailor the hopping interactions between the constituent elements of a physical system could enable the observation of unusual phenomena that are otherwise inaccessible in standard settings1,2. In this regard, a number of recent theoretical studies have indicated that an asymmetry in either the short- or long-range complex exchange constants can lead to counterintuitive effects, for example, the possibility of a Kramer’s degeneracy, even in the absence of spin 1/2 or the breakdown of the bulk–boundary correspondence3,4,5,6,7,8. Here we show how such tailored asymmetric interactions can be realized in photonic integrated platforms by exploiting non-Hermitian concepts, enabling a class of topological behaviours induced by optical gain. As a demonstration, we implement the Haldane model, a canonical lattice that relies on asymmetric long-range hopping to exhibit quantum Hall behaviour without a net external magnetic flux. The topological response observed in this lattice is a result of gain and vanishes in a passive but otherwise identical structure. Our findings not only enable the realization of a wide class of non-trivial phenomena associated with tailored interactions, but also open up avenues to study the role of gain and nonlinearity in topological systems in the presence of quantum noise.

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Fig. 1: Haldane crystal model.
Fig. 2: The photonic topological Haldane lattice.
Fig. 3: Experimental results for two-element and three-element subunits.
Fig. 4: Topological Haldane lattice experimental results.

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Data availability

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

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Acknowledgements

This work was supported by DARPA (D18AP00058), the Office of Naval Research (N00014-20-1-2522, N00014-20-1-2789, N00014-16-1-2640, N00014-18-1-2347 and N00014-19-1-2052), the Army Research Office (W911NF-17-1-0481), the Air Force Office of Scientific Research (FA9550-14-1-0037 and FA9550-20-1-0322), the National Science Foundation (CBET 1805200, ECCS 2000538, ECCS 2011171), the US–Israel Binational Science Foundation (BSF; 2016381) and the Polish Ministry of Science and Higher Education (Mobility Plus, 1654/MOB/V/2017). Y.G.N.L. thanks the following individuals for their support: A. U. Hassan in design and analysis, F. O. Wu in analysis, W. E. Hayenga in characterization and fabrication, O. Hemmatyar in analysis and graphics, and M. P. Hokmabadi in design and fabrication. The fan-shaped couplers are named after F. O. Wu. D.N.C. and M.K. acknowledge the technical discussions with M. Segev.

Author information

Authors and Affiliations

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

    Yuzhou G. N. Liu & Mercedeh Khajavikhan

  2. CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, FL, USA

    Yuzhou G. N. Liu, Pawel S. Jung, Midya Parto, Demetrios N. Christodoulides & Mercedeh Khajavikhan

  3. Faculty of Physics, Warsaw University of Technology, Warsaw, Poland

    Pawel S. Jung

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Contributions

Y.G.N.L., D.N.C. and M.K. conceived the idea. Y.G.N.L. designed the structures and experiments. Y.G.N.L. fabricated and characterized the lattices and sublattices. Y.G.N.L., P.S.J. and M.P. performed the simulations. Y.G.N.L., P.S.J., M.P., D.N.C. and M.K. developed the theoretical analysis. All authors contributed to preparing the manuscript.

Corresponding authors

Correspondence to Demetrios N. Christodoulides or Mercedeh Khajavikhan.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Shuang Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Schematic of a two-element system with unidirectional microring resonators and a link structure.

The directional couplers provide a π phase difference between the coupling terms κ1→2 and κ2→1. The coupling phase kL can be changed by varying the length L = 2Lc + Lm.

Extended Data Fig. 2 Schematics and dimensions of the waveguides, resonators, fan-shape constructs and the link.

a, The triangular microring resonator. b, The fan-shaped structure to be incorporated in the triangular resonators. c, The perimeters of two adjacent resonators are denoted by LA and LB. d, Transverse electric field distribution in a single waveguide. The black arrows indicate the electric field vector.

Extended Data Fig. 3 Schematic of the fabrication procedure of microring lasers.

a, HSQ e-beam resist is spun onto the wafer. b, The wafer is patterned by e-beam lithography. c, A dry etching process to define the rings. d, The sample is immersed in BOE to remove the masking HSQ. e, A 2 μm layer of SiO2 is deposited via PECVD. f, The wafer is flipped upside-down and bonded to a glass substrate by SU-8 photoresist to provide mechanical support. g, Lastly, the InP substrate is wet etched by HCl.

Extended Data Fig. 4 Schematic of the μ-PL characterization set-up.

The microrings are pumped by a pulsed laser (15 ns pulse width, 290 kHz repetition rate). The pump beam is focused onto the sample with a 10× objective, this objective in turns also collects the emission from the samples. Light is then either directed to a linear array detector for spectral measurements or to an IR camera for intensity profile observation.

Supplementary information

Supplementary Information

Supplementary materials.

Source data

Source Data Fig. 3

Normalized data for spectra in Fig. 3b–d.

Source Data Fig. 4

Normalized data for spectra in Fig. 4h,l.

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Liu, Y.G.N., Jung, P.S., Parto, M. et al. Gain-induced topological response via tailored long-range interactions. Nat. Phys. 17, 704–709 (2021). https://doi.org/10.1038/s41567-021-01185-4

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