Abstract
Topological insulators—materials that are insulating in the bulk but allow electrons to flow on their surface—are striking examples of materials in which topological invariants are manifested in robustness against perturbations such as defects and disorder1. Their most prominent feature is the emergence of edge states at the boundary between areas with different topological properties. The observable physical effect is unidirectional robust transport of these edge states. Topological insulators were originally observed in the integer quantum Hall effect2 (in which conductance is quantized in a strong magnetic field) and subsequently suggested3,4,5 and observed6 to exist without a magnetic field, by virtue of other effects such as strong spin–orbit interaction. These were systems of correlated electrons. During the past decade, the concepts of topological physics have been introduced into other fields, including microwaves7,8, photonic systems9,10, cold atoms11,12, acoustics13,14 and even mechanics15. Recently, topological insulators were suggested to be possible in exciton-polariton systems16,17,18 organized as honeycomb (graphene-like) lattices, under the influence of a magnetic field. Exciton-polaritons are part-light, part-matter quasiparticles that emerge from strong coupling of quantum-well excitons and cavity photons19. Accordingly, the predicted topological effects differ from all those demonstrated thus far. Here we demonstrate experimentally an exciton-polariton topological insulator. Our lattice of coupled semiconductor microcavities is excited non-resonantly by a laser, and an applied magnetic field leads to the unidirectional flow of a polariton wavepacket around the edge of the array. This chiral edge mode is populated by a polariton condensation mechanism. We use scanning imaging techniques in real space and Fourier space to measure photoluminescence and thus visualize the mode as it propagates. We demonstrate that the topological edge mode goes around defects, and that its propagation direction can be reversed by inverting the applied magnetic field. Our exciton-polariton topological insulator paves the way for topological phenomena that involve light–matter interaction, amplification and the interaction of exciton-polaritons as a nonlinear many-body system.
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Data availability
The research data that support this publication can be accessed at https://doi.org/10.17630/4a62cbdd-bcae-45d7-a556-3cda53c0a656. Additional data related to this paper may be requested from the corresponding authors.
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Acknowledgements
We thank R. Thomale for discussions. S.K. acknowledges the European Commission for the H2020 Marie Skłodowska-Curie Actions (MSCA) fellowship (Topopolis). S.K., S.H. and M.S. are grateful for financial support by the JMU-Technion seed money programme. S.H. also acknowledges support by the EPSRC “Hybrid Polaritonics” grant (EP/M025330/1). The Würzburg group acknowledges support by the ImPACT Program, Japan Science and Technology Agency and the ENB programme (Tols 836315) of the State of Bavaria. T.C.H.L. and R.G. were supported by the Ministry of Education (Singapore) grant no. 2017-T2-1-001.
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S.K., M.S., C.S. and S.H. initiated the study and guided the work. S.K., T.H., K.W., M.E. and S.H. designed and fabricated the device. S.K. and T.H. performed optical measurements. S.K., T.H., O.A.E. and C.S. analysed and interpreted the experimental data. O.A.E., R.G., T.C.H.L., M.A.B. and M.S. developed the theory. S.K., T.H., O.A.E., T.C.H.L., C.S., M.S. and S.H. wrote the manuscript, with input from all co-authors.
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Extended data figures and tables
Extended Data Fig. 1 White-light reflectivity measurements as a function of the detuning.
Reflectivity measurements are shown as a function of the detuning. A distinct anticrossing behaviour with a Rabi splitting of 2ħΩR = 4.3 meV can be observed. The measurements were performed on a sample piece with approximately 15 mirror pairs removed from the top DBR to increase the signal quality. Inset, fitted peak positions versus detuning. UP and LP stand for upper polariton and lower polariton.
Extended Data Fig. 2 Zeeman splitting and TE–TM splitting for a III–V microcavity hosting In0.04Ga0.96As quantum wells.
Left, Zeeman splitting with regard to the magnetic field, including second-order polynomial fits as a guide to the eye. Sample A is the one used in this experiment. Inset, example of central emission energies for a λ/4-series with a sine fit. Right, experimentally determined TE–TM splitting at various detunings ΔE for sample A including fits with modified photonic Hopfield coefficients (red).
Extended Data Fig. 3 Microscopy image of a zigzag-edge polariton honeycomb lattice with an intentional defect.
Shown is an image of the honeycomb lattice with d = 2.0 µm and v = 0.85, with an intentional defect in the zigzag chain, selected for experiments in an external magnetic field.
Extended Data Fig. 4 Polariton chiral edge mode propagating around a corner.
a–c, Propagation dynamics of edge modes injected coherently into the topological gap and calculated within model 1. Shown here is the right-moving propagation for the positive-value splitting ΔB = +0.8 meV (|Δeff| = 0.2 meV) and β = 0.20 meV μm2 (βeff = 0.15 meV μm2). A linearly polarized narrow coherent seeding beam injects both polarization components into the region marked by the red circle. At t ≈ 100 ps, the mode propagates around the corner from the zigzag edge into the armchair one.
Extended Data Fig. 5 Polariton chiral edge mode propagating and avoiding a defect.
Propagation dynamics of edge modes injected coherently into the topological gap and calculated within model 1 are shown for the right-moving propagation for the positive-value splitting ΔB = +0.8 meV (|Δeff| = 0.2 meV) and β = 0.20 meV μm2 (βeff = 0.15 meV μm2). A linearly polarized narrow coherent seeding beam injects both polarization components into the mesa. At t ≈ 85 ps, the mode propagates around the defect in the zigzag chain, marked by the red circle.
Extended Data Fig. 6 Topological gap measurement.
A λ/4-plate measurement at B = 0 T (blue) and B = +5 T (red) at the K point yields a bandgap of Eg = 108 ± 32 μeV. PL, photoluminescence.
Extended Data Fig. 7 Input/output characteristics and linewidth behaviour as a function of pump power.
Below threshold, the gap and bulk mode cannot be distinguished. At a typical threshold Pth ≈ 1.8 mW, a distinct nonlinear increase in intensity as well as a sudden decrease in linewidth can be observed. Here, the populated gap modes show similar behaviour to the bulk mode.
Extended Data Fig. 8 Driven-dissipative Gross–Pitaevskii calculation of polariton condensation into topological edge mode.
a, Band structure of polaritons in a honeycomb lattice. The dotted curves represent the dispersion of the linear eigenmodes of a strip, colour-coded to represent localization on the bottom edge (red), upper edge (green) and in the bulk (blue). The shaded region represents the energy and momentum of the polariton steady state obtained from solving the driven-dissipative Gross–Pitaevskii equation. b, Imaginary components of the linear eigenmodes. The largest imaginary part corresponds to an edge state (the colour coding is the same as in a), suggesting that the edge state is most likely to be populated with increasing pumping. c, Edge state obtained from solution of the driven-dissipative Gross–Pitaevskii equation. Parameters: Δeff = 0.3 meV, βeff = 0.2 meV μm2. The effective mass m was taken as 1.3 × 10−4 of the free electron mass; the potential of depth 0.5 meV was constructed from a honeycomb lattice of cylinders of radius 1 μm and centre-to-centre separation 1.7 μm; the pump spot was taken as a Gaussian centred on the strip edge with extent 7.5 μm in the y direction. A spatially uniform decay rate of 0.2 meV was supplemented with a 1.7 meV decay in the region outside the cylinders.
Extended Data Fig. 9 Real-space mode tomographies of a polariton condensate at B = 0 T and B = −5 T.
a–d, Measurements at B = 0 T. a, Real-space spectrum in x direction perpendicular to the zigzag edge along the straight white line in d. The real-space x axis is consistent between a, b and d. The dashed white line marks the physical edge of the lattice. Only a trivial S-band condensate can be observed throughout the structure. b, Mode tomography displaying the topologically trivial S-band condensate at ES = 1.4673–1.4675 eV. A relatively homogeneous condensate within the pump spot diameter of 40 μm is observed. The inset shows a microscopy image of the structure. c, d, Mode tomography of the energy Eedge = 1.4678 eV for comparison at the corner position (c) and at the edge (d) of the sample. Without magnetic field, no localized edge mode can be observed. e–h, Measurements at B = −5 T (fully analogous to Fig. 3d–g). e, Real-space spectrum in the x direction perpendicular to the zigzag edge along the straight white line in h. The real-space x axis is consistent between e, f and h. The dashed white line marks the physical edge of the lattice. A trivial S-band condensate can be observed throughout the structure. At E = 1.4678 eV, again we observe the appearance of a localized mode, well separated from the bulk and located at the zigzag edge. f, Mode tomography displaying the topologically trivial S-band condensate at ES = 1.4673–1.4675 eV. A relatively homogeneous condensate within the pump spot diameter of 40 μm is observed. The inset shows a microscopy image of the structure. g, h, Mode tomography of the topological edge mode at Eedge = 1.4678 eV at the corner position (g) and at the edge (h) of the sample, showing clearly that the mode extends around the corner from the zigzag to the armchair configuration and avoids the intentional defect, both without bulk scattering.
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Klembt, S., Harder, T.H., Egorov, O.A. et al. Exciton-polariton topological insulator. Nature 562, 552–556 (2018). https://doi.org/10.1038/s41586-018-0601-5
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DOI: https://doi.org/10.1038/s41586-018-0601-5
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