Topology has recently become a focus in condensed matter physics, arising in the context of the quantum Hall effect and topological insulators. In both of these cases, the topology of the system is defined through bulk properties (‘topological invariants’) but detected through surface properties. Here we measure three topological invariants of a quantum Hall material—photonic Landau levels in curved space—through local electromagnetic and gravitational responses of the bulk material. Viewing the material as a many-port circulator, the Chern number (a topological invariant) manifests as spatial winding of the phase of the circulator. The accumulation of particles near points of high spatial curvature and the moment of inertia of the resultant particle density distribution quantify two additional topological invariants—the mean orbital spin and the chiral central charge. We find that these invariants converge to their global values when probed over increasing length scales (several magnetic lengths), consistent with the intuition that the bulk and edges of a system are distinguishable only for sufficiently large samples (larger than roughly one magnetic length). Our experiments are enabled by applying quantum optics tools to synthetic topological matter (here twisted optical resonators). Combined with advances in Rydberg-mediated photon collisions, our work will enable precision characterization of topological matter in photon fluids.
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The raw data the support the findings of this study are available from the corresponding author on reasonable request.
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We thank C. Kane and M. Levin for conversations. This work was supported by DOE grant DE-SC0010267 for apparatus construction and data collection and MURI grant FA9550-16-1-0323 for analysis.
The authors declare no competing interests.
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Extended data figures and tables
The LDOS in the second excited Landau level with an effective magnetic flux of ΦB/(2π) = −2/3 threading the cone tip highlights the improvements in the resonator design and the imaging system. The previous resonator28 (top left) exhibits substantial diagonal astigmatism, which has been removed in the resonator used here (top right). Images of modes in the lowest Landau level provide estimates of the expectation value of r2 (bottom), errors in which directly cause systematic errors in measurements of the shifted second moment. The substantial reduction in deviations from the ideal system enables measurements of the central charge and extensions to higher Landau levels. Error bars are calculated from the uncertainty in the centre location and waist size of the modes and are all smaller than the symbol size.
This 12-page document contains 8 sections and 9 figures. These provide additional details about the experimental methods, theoretical background for numerical calculations and the theoretical results connecting LDOS measurements to topological invariants.
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Schine, N., Chalupnik, M., Can, T. et al. Electromagnetic and gravitational responses of photonic Landau levels. Nature 565, 173–179 (2019). https://doi.org/10.1038/s41586-018-0817-4
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