Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties of these driven settings. Most prominently, there are two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers being equal to zero. Here we realize such a system with bosonic atoms in a periodically driven honeycomb lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
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The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
The code that supports the plots within this paper are available from the corresponding author upon reasonable request.
Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).
Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).
Eckardt, A. Colloquium: atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).
Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996–999 (2011).
Aidelsburger, M. et al. Experimental realization of strong effective magnetic fields in an optical lattice. Phys. Rev. Lett. 107, 255301 (2011).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon 7, 1001–1005 (2013).
Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys. 13, 146–151 (2017).
McIver, J. W. et al. Light-induced anomalous hall effect in graphene. Nat. Phys. 16, 38–41 (2020).
Aidelsburger, M., Nascimbène, S. & Goldman, N. Artificial gauge fields in materials and engineered systems. C. R. Phys. 19, 394–432 (2018).
Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).
Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).
Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
Wu, Z. et al. Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates. Science 354, 83–88 (2016).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).
Mittal, S., Ganeshan, S., Fan, J., Vaezi, A. & Hafezi, M. Measurement of topological invariants in a 2D photonic system. Nat. Photon. 10, 180–183 (2016).
Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).
Tarnowski, M. et al. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat. Commun. 10, 1728 (2019).
Asteria, L. et al. Measuring quantized circular dichroism in ultracold topological matter. Nat. Phys. 15, 449–454 (2019).
Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993).
Qi, X.-L., Wu, Y.-S. & Zhang, S.-C. General theorem relating the bulk topological number to edge states in two-dimensional insulators. Phys. Rev. B 74, 045125 (2006).
Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).
Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).
Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet–Bloch systems. New J. Phys. 17, 125014 (2015).
Rudner, M. S. & Lindner, N. H. Band structure engineering and non-equilibrium dynamics in Floquet topological insulators. Nat. Rev. Phys. 2, 229–244 (2020).
Nathan, F., Abanin, D., Berg, E., Lindner, N. H. & Rudner, M. S. Anomalous Floquet insulators. Phys. Rev. B 99, 195133 (2019).
Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012).
Hu, W. et al. Measurement of a topological edge invariant in a microwave network. Phys. Rev. X 5, 011012 (2015).
Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).
Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).
D’Errico, A. et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica 7, 108–114 (2020).
Peng, Y.-G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Commun. 7, 13368 (2016).
Zenesini, A., Ciampini, D., Morsch, O. & Arimondo, E. Observation of Stückelberg oscillations in accelerated optical lattices. Phys. Rev. A 82, 065601 (2010).
Kling, S., Salger, T., Grossert, C. & Weitz, M. Atomic Bloch–Zener oscillations and Stückelberg interferometry in optical lattices. Phys. Rev. Lett. 105, 215301 (2010).
Quelle, A., Weitenberg, C., Sengstock, K. & Morais Smith, C. Driving protocol for a Floquet topological phase without static counterpart. New J. Phys. 19, 113010 (2017).
Ünal, F. N., Seradjeh, B. & Eckardt, A. How to directly measure floquet topological invariants in optical lattices. Phys. Rev. Lett. 122, 253601 (2019).
Greiner, M., Bloch, I., Mandel, O., Hänsch, T. W. & Esslinger, T. Exploring phase coherence in a 2D lattice of Bose–Einstein condensates. Phys. Rev. Lett. 87, 160405 (2001).
Bouhon, A., Black-Schaffer, A. M. & Slager, R.-J. Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry. Phys. Rev. B 100, 195135 (2019).
Simon, B. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167–2170 (1983).
Bellissard, J. Change of the Chern number at band crossings. Preprint at https://arxiv.org/abs/cond-mat/9504030(1995).
Leboeuf, P., Kurchan, J., Feingold, M. & Arovas, D. P. Topological aspects of quantum chaos. Chaos 2, 125–130 (1992).
Barelli, A. & Fleckinger, R. Semiclassical analysis of Harper-like models. Phys. Rev. B 46, 11559–11569 (1992).
Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406 (2009).
Price, H. M. & Cooper, N. R. Mapping the Berry curvature from semiclassical dynamics in optical lattices. Phys. Rev. A 85, 033620 (2012).
Dauphin, A. & Goldman, N. Extracting the Chern number from the dynamics of a Fermi gas: implementing a quantum Hall bar for cold atoms. Phys. Rev. Lett. 111, 135302 (2013).
Buchhold, M., Cocks, D. & Hofstetter, W. Effects of smooth boundaries on topological edge modes in optical lattices. Phys. Rev. A 85, 063614 (2012).
Goldman, N. et al. Direct imaging of topological edge states in cold-atom systems. Proc. Natl Acad. Sci. USA 11, 6736–6741 (2013).
Reichl, M. D. & Mueller, E. J. Floquet edge states with ultracold atoms. Phys. Rev. A 89, 063628 (2014).
Titum, P., Lindner, N. H., Rechtsman, M. C. & Refael, G. Disorder-induced floquet topological insulators. Phys. Rev. Lett. 114, 056801 (2015).
Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. Anomalous Floquet–Anderson insulator as a nonadiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).
Gopalakrishnan, S., Lamacraft, A. & Goldbart, P. M. Universal phase structure of dilute Bose gases with Rashba spin–orbit coupling. Phys. Rev. A 84, 061604 (2011).
Sedrakyan, T. A., Kamenev, A. & Glazman, L. I. Composite fermion state of spin–orbit-coupled bosons. Phys. Rev. A 86, 063639 (2012).
Sedrakyan, T. A., Galitski, V. M. & Kamenev, A. Statistical transmutation in Floquet driven optical lattices. Phys. Rev. Lett. 115, 195301 (2015).
We thank J. Bellissard, E. Berg, J. Dalibard, E. Demler and N. Lindner for inspiring discussions. The research in Munich and Dresden was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Research Unit FOR 2414 under project number 277974659. The work in Munich was further supported under Germany’s Excellence Strategy – EXC-2111 – 39081486. F.N.Ü. further acknowledges support from EPSRC grant number EP/P009565/1. The work in Belgium was supported by the ERC Starting Grant TopoCold, and the Fonds De La Recherche Scientifique (FRS-FNRS, Belgium).
The authors declare no competing interests.
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This Supplementary Information includes a presentation of the relevant calibration measurements and additional datasets, a detailed description of the theoretical model, a calculation of the edge states using a tight-binding model and a discussion of the theoretical connection between the topological charge and the Berry curvature.
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Wintersperger, K., Braun, C., Ünal, F.N. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0949-y
Nature Physics (2020)