The topology of two-dimensional materials traditionally manifests itself through the quantization of the Hall conductance, which is revealed in transport measurements1,2,3. Recently, it was predicted that topology can also give rise to a characteristic spectroscopic response on subjecting a Chern insulator to a circular drive: comparing the frequency-integrated depletion rates associated with drives of opposite orientation leads to a quantized response dictated by the topological Chern number of the populated Bloch band4,5. Here we experimentally demonstrate this intriguing topological effect using ultracold fermionic atoms in topological Floquet bands. In addition, our depletion-rate measurements also provide an experimental estimation of the Wannier-spread functional, a fundamental geometric property of Bloch bands related to the quantum metric6,7. Our results establish topological spectroscopic responses as a versatile probe, which could be applied to access the geometry and topology of many-body quantum systems, such as fractional Chern insulators8.
Access optionsAccess options
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & Nijs, M. D. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Tran, D. T., Dauphin, A., Grushin, A. G., Zoller, P. & Goldman, N. Probing topology by ‘heating’: quantized circular dichroism in ultracold atoms. Sci. Adv. 3, e1701207 (2017).
Tran, D. T., Cooper, N. R. & Goldman, N. Quantized Rabi oscillations and circular dichroism in quantum Hall systems. Phys. Rev. A 97, 061602(R) (2018).
Marzari, N. & Vanderbilt, D. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847 (1997).
Ozawa, T. & Goldman, N. Extracting the quantum metric tensor through periodic driving. Phys. Rev. B 97, 201117(R) (2018).
Neupert, T., Chamon, C., Iadecola, T., Santos, L. H. & Mudry, C. Fractional (Chern and topological) insulators. Phys. Scr. T164, 014005 (2015).
Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
Fläschner, N. et al. High-precision multiband spectroscopy of ultracold fermions in a nonseparable optical lattice. Phys. Rev. A 97, 051601(R) (2018).
Souza, I. & Vanderbilt, D. Dichroic f-sum rule and the orbital magnetization of crystals. Phys. Rev. B 77, 054438 (2008).
Bennett, H. S. & Stern, E. A. Faraday effect in solids. Phys. Rev. 137, A448–A461 (1965).
Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).
De Juan, F., Grushin, A. G., Morimoto, T. & Moore, J. E. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. 8, 15995 (2017).
Wang, Y. & Gedik, N. Circular dichroism in angle-resolved photoemission spectroscopy of topological insulators. Phys. Status Solidi Rapid Res. Lett. 7, 64–71 (2013).
Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nat. Mater. 14, 290–294 (2015).
Gullans, M. J., Taylor, J. M., Imamoğlu, A., Ghaemi, P. & Hafezi, M. High-order multipole radiation from quantum Hall states in Dirac materials. Phys. Rev. B 95, 235439 (2017).
Liu, Y., Yang, S. A. & Zhang, F. Circular dichroism and radial Hall effects in topological materials. Phys. Rev. B 97, 035153 (2018).
Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61, 2015 (1988).
Jotzu, G. et al. Experimental realisation of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).
Tarnowski, M. et al. Characterizing topology by dynamics: Chern number from linking number. Preprint at https://arxiv.org/abs/1709.01046 (2017).
Eckardt, A. Colloquium: Atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).
Becker, C. et al. Ultracold quantum gases in triangular optical lattices. New J. Phys. 12, 065025 (2010).
Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996–999 (2011).
Soltan-Panahi, P. et al. Multi-component quantum gases in spin-dependent hexagonal lattices. Nat. Phys. 7, 434–440 (2011).
Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).
Weinberg, M. et al. Multiphoton excitations of quantum gases in driven optical lattices. Phys. Rev. A 92, 043621 (2015).
Wu, Z., Taylor, E. & Zaremba, E. Probing the optical conductivity of trapped charge-neutral quantum gases. Eur. Phys. Lett. 110, 26002 (2015).
Anderson, R. et al. Optical conductivity of a quantum gas. Preprint at https://arxiv.org/abs/1712.09965 (2017).
Sugawa, S., Salces-Carcoba, F., Perry, A. R., Yue, Y. & Spielman, I. B. Second Chern number of a quantum-simulated non-Abelian Yang monopole. Science 360, 1429–1434 (2018).
Lohse, M., Schweitzer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).
Schüler, M. & Werner, P. Tracing the nonequilibrium topological state of Chern insulators. Phys. Rev. B 96, 155122 (2017).
Repellin, C. & Goldman, N. Detecting fractional Chern insulators through circular dichroism. Preprint at https://arxiv.org/abs/1811.08523 (2018).
Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406(R) (2009).
Zheng, W. & Zhai, H. Floquet topological states in shaking optical lattices. Phys. Rev. A 89, 061603(R) (2014).
Thonhauser, T. & Vanderbilt, D. Insulator/Chern-insulator transition in the Haldane model. Phys. Rev. B 74, 235111 (2006).
The authors acknowledge discussions with N. R. Cooper, M. Dalmonte, A. Dauphin, A. G. Grushin, C. Repellin and P. Zoller, and they also thank P. Zoller for his insightful comments on the manuscript. This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Research Unit FOR 2414 under project number 277974659 and via the Excellence Cluster ‘The Hamburg Centre for Ultrafast Imaging - Structure, Dynamics and Control of Matter at the Atomic Scale’ under project number 194651731. B.S.R. acknowledges financial support from the European Commission (Marie Curie Fellowship). Work in Brussels is supported by the FRS-FNRS (Belgium) and the ERC Starting Grant TopoCold. T.O. is supported by the Interdisciplinary Theoretical and Mathematical Sciences Program at RIKEN.
About this article
Nature Communications (2019)