Topological Maxwell Metal Bands in a Superconducting Qutrit

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Topological Maxwell Metal Bands in a Superconducting Qutrit

Xinsheng Tan, Dan-Wei Zhang, Qiang Liu, Guangming Xue, Hai-Feng Yu, Yan-Qing Zhu, Hui Yan, Shi-Liang Zhu, and Yang Yu

Phys. Rev. Lett. 120, 130503 – Published 28 March 2018

Abstract

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Threefold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one.

Received 15 September 2017
Revised 1 January 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.130503

Physics Subject Headings (PhySH)

Quantum simulation

Topological materials

Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Xinsheng Tan¹, Dan-Wei Zhang², Qiang Liu¹, Guangming Xue¹, Hai-Feng Yu^1,*, Yan-Qing Zhu¹, Hui Yan², Shi-Liang Zhu^1,2,3,†, and Yang Yu^1,3,‡

¹National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
²Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China
³Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

^*hfyu@nju.edu.cn
^†slzhu@nju.edu.cn
^‡yuyang@nju.edu.cn

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Vol. 120, Iss. 13 — 30 March 2018

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Images

Figure 1
Phase diagram and geometric illustration of the spin-1 Maxwell system. (a) Phase diagram of the Maxwell system with respect to the parameter $Λ$ . From left to right: the energy spectra for the Maxwell metal phase with a pair of Maxwell points denoted by $M_{\pm}$ ( $Λ = 0$ ), the topological transition point with the merging of the two points ( $Λ = 1$ ), and the trivial insulator phase with band gaps ( $Λ = 2$ ). (b) Geometric illustrations of the topological difference between the two distinct phases when the spherical manifold $S$ moves from the degeneracy in the $z$ direction by distance $Λ$ . The Berry flux vectors are schematically presented by arrows, showing the different signature textures for the topological and trivial phases: the vectors fully (do not) wind around in the topological (trivial) case with $Λ = 0$ ( $Λ > 1$ ), giving the Chern number $C_{+} = 2$ ( $C_{+} = 0$ ).
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Figure 2
(a) Schematic of experimental setup for controlling and measuring a 3D superconducting transmon qutrit. The microwaves for manipulating and measuring the qutrit are applied to the sample. We use digital heterodyne for data acquisition. Amplifiers and filters are used to increase signal-to-noise ratio and isolate qutrit from external noise. (b) Schematic of the relevant energy levels of the transmon for spectroscopy measurement. Levels $| 1 ⟩, | 2 ⟩$ , and $| 3 ⟩$ form the spin-1 basis ${(1, 0, 0)^{T}, (0, 1, 0)^{T}, (0, 0, 1)^{T}}$ . $| 0 ⟩$ serves as the reference level to measure the spectroscopy by sweeping the frequency of the probe microwave (schematically illustrated by the dashed arrow). (c) Time profile for spectroscopy measurement. The control microwave pulses $R_{x}$ and $R_{z}$ drive the system to form the eigenstates, which are empty. Then the probe microwave pulse pumps the system from $| 0 ⟩$ to the eigenstates when its frequency matches the level spacing. By sending a detect pulse to the cavity, we can readout the population in the eigenstates as resonant peaks. By collecting the resonant peaks we obtain the energy spectrum of the Maxwell metal.
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Figure 3
Measured Maxwell bands. (a), (b), and (c) are band structures in the first Brillouin zone for $Λ = 0$ , 1, 2, respectively. (d), (e), and (f) The corresponding cross sections of band structures containing Maxwell points in the $E - k_{z}$ ( $k_{x} \approx 0$ ) plane of (a) to (c). A linear dispersion is observed in the Maxwell metal phase. The theoretical calculations are plotted with the red dashed lines. The spectra are plotted by dropping the overall energy constant $ω_{01}$ with the energy unit $Ω = 10 MHz$ .
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Figure 4
(a) Time profile for the measurement of Chern number. The qutrit state is initialized at $(| 2 ⟩ + i | 3 ⟩) / \sqrt{2}$ and then evolves quasiadiabatically during a nonadiabatic ramp, which is followed by state tomography. (b) Three lowest energy levels ${| 1 ⟩, | 2 ⟩, | 3 ⟩}$ coupled by pluses $R_{x, y, z}$ are used to construct the spin-1 Hamiltonian, and the pulse sequence results in a parameter-space motion along the $ϕ = 0$ meridian ( $R_{y} = 0$ ) on the spherical manifold. (c) and (d) The measured and simulated (with the measured decoherence time of the transmon) Berry curvature $F_{θ ϕ}$ as functions of $θ$ and $Λ$ . The oscillation pattern suggests a nonadiabatic response. (e) The measured (circles and diamonds) and simulated (solid line) Chern numbers as a function of $Λ$ for the Maxwell points. For $| Λ | < 1$ , $| C_{\pm} | = 2$ indicates the Maxwell points in the topological Maxwell metal phase; for $| Λ | > 1$ , $| C_{\pm} | = 0$ indicates the system in the trivial insulator phase.
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