Tuning Anomalous Floquet Topological Bands with Ultracold Atoms

Jin-Yi Zhang, Chang-Rui Yi, Long Zhang, Rui-Heng Jiao, Kai-Ye Shi, Huan Yuan, Wei Zhang, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan

Phys. Rev. Lett. 130, 043201 – Published 24 January 2023

Abstract

The Floquet engineering opens the way to create new topological states without counterparts in static systems. Here, we report the experimental realization and characterization of new anomalous topological states with high-precision Floquet engineering for ultracold atoms trapped in a shaking optical Raman lattice. The Floquet band topology is manipulated by tuning the driving-induced band crossings referred to as band inversion surfaces (BISs), whose configurations fully characterize the topology of the underlying states. We uncover various exotic anomalous topological states by measuring the configurations of BISs that correspond to the bulk Floquet topology. In particular, we identify an unprecedented anomalous Floquet valley-Hall state that possesses anomalous helical-like edge modes protected by valleys and a chiral state with high Chern number.

Received 16 May 2022
Revised 8 November 2022
Accepted 4 January 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.043201

Physics Subject Headings (PhySH)

Cold gases in optical lattices Exotic phases of matter Spin-orbit coupling Synthetic gauge fields Topological phases of matter

Bose-Einstein condensates Topological materials Ultracold gases

Optical lattices & traps

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Jin-Yi Zhang ^1,2,3, Chang-Rui Yi^1,2,3, Long Zhang ^8,4,5, Rui-Heng Jiao^1,2,3, Kai-Ye Shi ⁶, Huan Yuan ^1,2,3, Wei Zhang⁶, Xiong-Jun Liu^4,5,7, Shuai Chen ^1,2,3, and Jian-Wei Pan^1,2,3

¹Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
²Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
³Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
⁴International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
⁵Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
⁶Department of Physics, Renmin University of China, Beijing 100872, China
⁷Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁸School of Physics and Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

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Issue

Vol. 130, Iss. 4 — 27 January 2023

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Images

Figure 1
Scheme of shaking Raman lattices. (a) Cartoon of the optical Raman lattices with frequency modulation of a laser beam. (b) Sketch of the Floquet bands along the diagonal direction of the quasimomentum $q_{diag}$ . $k_{0}$ is the wave number of the lattice beams, and $(| ↑ ⟩, n ω)$ [ $(| ↓ ⟩, n ω)$ ] is the shifted copy of the static band $(| ↑ ⟩, 0)$ [ $(| ↓ ⟩, 0)$ ]. Solid curves are the Floquet bands which present 0 gap at the center of FBZ (black diamonds) and $π$ gap at the boundary of FBZ (green circles). (c) Experimental protocol and evolution of the spin textures ${⟨ σ_{z} (q, t) ⟩}_{z}$ . The upper (lower) row is spin textures without (with) periodic driving where the parameters $(V_{0}, Ω_{0}, δ_{0}) = (4.0, 1.0, 0.4) E_{r}$ [ $(V_{0}, Ω_{0}, δ_{0}, A_{F}) = (4.0, 1.0, 0.4, 0.8) E_{r}$ , $T = 400 μ s$ ]. Here, $E_{r} \approx h \times 3.7 kHz$ is the recoil energy. $Γ$ and $M$ is high symmetric momenta. The red (blue) color in the spin textures denotes spin up (down).
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Figure 2
Tunable topological states. The emergence of ring patterns from one to four in the measured (first row) spin textures $⟨ σ_{z} (q, t) ⟩$ and the Floquet bands along $q_{diag}$ as $T$ varies with parameters $(V_{0}, Ω_{0}, δ_{0}) = (4.0, 1.0, 0.1) E_{r}$ . And $(T, A_{F}, t) = (280 μ s, 0.8 E_{r}, 2 T)$ , $(400 μ s, 0.8 E_{r}, 3 T)$ , $(520 μ s, 0.8 E_{r}, 2 T)$ , $(740 μ s, 1.2 E_{r}, 2 T)$ from left to right. The dashed black (green) curves mark the 0 BISs ( $π$ BISs).
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Figure 3
Characterization of topological states. (a) Time-averaged spin textures ${\bar{⟨ σ_{z} (q, t) ⟩}}_{i}$ obtained in the experimental measurements (upper) and numerics with parameters in experiment (lower). The black and green curves represent the location of BISs. The dashed curves in the upper row are guides to the eye. The arrows along the BISs display the dynamical field. The BISs divide the Brillouin zone into three regions: two of them with $h_{F, z} < 0$ as labeled by $-$ , and one of them with $h_{F, z} > 0$ as denoted by $+$ . $ν_{0}^{j}$ ( $ν_{π}^{j}$ ) is the topological number of the $j$ th BIS in 0 gap ( $π$ gap). (b) The numerical band structures. The orange (green) curves denote edge states in the left (right) boundary. The parameters are $(V_{0}, Ω_{0}, δ_{0}, A_{F}, T) = (4.0 E_{r}, 1.0 E_{r}, 0.4 E_{r}, 0.8 E_{r}, 400 μ s)$ .
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Figure 4
Topological phase diagram. (a) Topological phase diagram with $(V_{0}, Ω_{0}) = (4.0, 1.0) E_{r}$ in the $T − δ_{0}$ plane. The blue circles with error bars are experimental measurements. The solid and dashed curves are calculated phase boundaries induced by 0 gap and $π$ gap, respectively. Roman numerals i to vi label 6 topological phases within the range of experimental parameters, whose topological invariants are listed in the table. (b) Spin textures ${⟨ σ_{z} (q, t) ⟩}_{z}$ marked by $☆$ with typical phases in (a). To clearly observe vi, the raw data are shown in Fig. S8 of [29]. The BIS configurations labeled by dashed black and green curves fully distinguish the topology of all phases. The following gray (orange) frame contains the bulk band structures, the aerial view of the lowest band and the band structures with open boundaries corresponding to the spin texture iv (vi). The orange and green curves denote “helical” edge modes.
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