Berry Curvature and Bulk-Boundary Correspondence from Transport Measurement for Photonic Chern Bands

Chao Chen, Run-Ze Liu, Jizhou Wu, Zu-En Su, Xing Ding, Jian Qin, Lin Wang, Wei-Wei Zhang, Yu He, Xi-Lin Wang, Chao-Yang Lu, Li Li, Barry C. Sanders, Xiong-Jun Liu, and Jian-Wei Pan

Phys. Rev. Lett. 131, 133601 – Published 25 September 2023

Abstract

Berry curvature is a fundamental element to characterize topological quantum physics, while a full measurement of Berry curvature in momentum space was not reported for topological states. Here we achieve two-dimensional Berry curvature reconstruction in a photonic quantum anomalous Hall system via Hall transport measurement of a momentum-resolved wave packet. Integrating measured Berry curvature over the two-dimensional Brillouin zone, we obtain Chern numbers corresponding to $- 1$ and 0. Further, we identify bulk-boundary correspondence by measuring topology-linked chiral edge states at the boundary. The full topological characterization of photonic Chern bands from Berry curvature, Chern number, and edge transport measurements enables our photonic system to serve as a versatile platform for further in-depth study of novel topological physics.

Received 14 November 2022
Accepted 17 August 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.133601

Physics Subject Headings (PhySH)

Quantum simulation Quantum walks Topological effects in photonic systems Topological phase transition Topological phases of matter

Interdisciplinary PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Chao Chen ^1,2,3,*, Run-Ze Liu^1,2,*, Jizhou Wu ^4,*, Zu-En Su⁵, Xing Ding^1,2, Jian Qin^1,2, Lin Wang ⁶, Wei-Wei Zhang ⁷, Yu He⁸, Xi-Lin Wang³, Chao-Yang Lu^1,2, Li Li^1,2,†, Barry C. Sanders ^1,2,9,‡, Xiong-Jun Liu ^10,11,12,§, and Jian-Wei Pan^1,2

¹Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
²CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
³National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
⁴Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China
⁵The Physics Department and the Solid State Institute, Technion–Israel Institute of Technology, Haifa 3200003, Israel
⁶Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
⁷School of Computer Science, Northwestern Polytechnical University, Xi’an 710129, China
⁸Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁹Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada
¹⁰International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
¹¹CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
¹²International Quantum Academy, Shenzhen 518048, China

^*These authors contributed equally to this work.
^†eidos@ustc.edu.cn
^‡sandersb@ucalgary.ca
^§xiongjunliu@pku.edu.cn

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Vol. 131, Iss. 13 — 29 September 2023

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Images

Figure 1
Translation operations in 2DQW, energy band spectrum, phase diagram, and transversal Hall shift. (a) Spin-dependent translations $T_{x}$ , $T_{y}$ , and $T_{x y}$ . The walker of spin up and down shifts in the opposite directions. (b) Typical open-gap two-band quasienergy spectrum. An external force along the $y$ direction drives the eigenstate wave packets (balls) moving in the $k_{y}$ direction. (c) Phase diagram of 2DQW in the parameter space $(θ_{1}, θ_{2})$ . Different topological phases are denoted by the Chern number of $\pm 1$ and 0 for the lower band. (d) Transversal wave-packet drift when the external force is along the $y$ direction. The wave packets of the lower and upper bands drift in the opposite directions.
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Figure 2
Experimental setup, temporal-spatial coordinate mapping, and electro-optical modulator signals. (a) Experimental setup of time-bin encoded 2DQW. A picosecond laser pulse of 905 nm generated by Mira-900 is injected into the circuit by an AOM. Spin-dependent translation in the $x$ ( $y$ ) direction is implemented by a two-polarizing-beam-splitter fiber (free-space) optical delay. Between two translations, the light is switched by pairs of AOMs either for ISP (dashed red line) or for 2DQW evolution (dot-dashed red line). (b) Sketch of laser pulse distribution in temporal modes after the ISP. (c) Mapping the temporal distribution in (b) to 2D position space. (d) Schematic phase EOM voltage signals $V$ versus loop number $n$ . The evolution time $t$ is labeled as 0 when 2DQW starts after 15 loops’ ISP and increases by 1 for every two loops. During the 2DQW, an upward and downward ladder-type modulation is applied on the $y$ -phase EOM to implement an effective force in the $\pm y$ direction, whose strength $F$ is tuned by the two-step difference. The initial $k_{c}$ is tuned by voltages during ISP. $V_{π}$ is the half wave voltage.
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Figure 3
Measured transversal Hall drifts, Chern number, and the wave-packet recurrence under the Bloch oscillation in $k_{y}$ with $F δ t = π$ . (a) Measured probability distribution $P (x, y)$ after ten steps 2DQW under $\pm F$ forces. We set $θ = [- (π / 2), (π / 2)]$ and $k_{x_{c}} \approx 0.46 π$ . The inset shows the marginal distributions $P (x)$ for the measured $P (x, y)$ , where the distance between the wave-packet c.m. under $\pm F$ is labeled. (b) Same as (a) except for $θ = [0, (5 π / 6)]$ . (c) Measured (simulated) $Λ (k_{x_{c}})$ versus $k_{x_{c}}$ with $θ$ of (a). The $Λ (k_{x_{c}})$ marked by the triangle is obtained from the displacement shown in (a). By summing up $Λ (k_{x_{c}})$ , we get the Chern number $- 1.01 \pm 0.02$ . (d) Same as (c) except for the $θ$ of (b). In this case, $C = 0.00 \pm 0.02$ is obtained. The $Λ (k_{x_{c}})$ marked by the red circle corresponds to the situation shown in (b). (e) Recurrence of the wave packet in the $y$ direction during the Bloch oscillation when $θ$ of (a) is chosen.
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Figure 4
Reconstruction of Berry curvature in the 2D Brillouin zone. (a) Theoretically simulated (left) and measured (right) Berry curvature when $θ = [- (π / 2), (π / 2)]$ marked by the red star in the phase diagram. By scanning $k_{c}$ in steps of $(π / 11)$ in the BZ, we obtain the local Berry curvature $Ω (k_{c})$ from the measured $Λ (k_{c})$ . The contour lines of Berry curvature being $- 0.68$ are a guide for the eye. Measured Berry curvature configurations agree with the theoretical ones in general. The momentum resolution can be promoted by using long-time initial-state evolution to prepare the Gaussian wave packet with smaller momentum distribution width. (b) Theoretically simulated and measured Berry curvature when $θ = [(π / 2), (π / 2)]$ .
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Figure 5
Chiral edge states. (a) Measured probability distribution $P (x, y)$ after 12 steps’ inhomogeneous 2DQW with an edge along the $y$ direction between $x < 0$ ( $C = 0$ ) and $x \geq 0$ ( $C = - 1$ ). Parameters $θ^{left} = [- (π / 4), π]$ (orange diamond) and $θ^{right} = [- (π / 4), (π / 5)]$ (green diamond) are chosen. A spin-up polarized walker starts at site (0,0) marked by a white circle. The red arrow denotes the propagation direction of the chiral edge states. (b) Measured $P (x, y)$ when $θ^{left} = [(π / 4), (π / 5)]$ and $θ^{right} = [(π / 4), π]$ . Parameters for bulk topology measurement in Figs. 3 and 4 are represented by red stars.
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