Ergodic-Localized Junctions in a Periodically Driven Spin Chain

Chen Zha, V. M. Bastidas, Ming Gong, Yulin Wu, Hao Rong, Rui Yang, Yangsen Ye, Shaowei Li, Qingling Zhu, Shiyu Wang, Youwei Zhao, Futian Liang, Jin Lin, Yu Xu, Cheng-Zhi Peng, J. Schmiedmayer, Kae Nemoto, Hui Deng, W. J. Munro, Xiaobo Zhu, and Jian-Wei Pan

Phys. Rev. Lett. 125, 170503 – Published 22 October 2020

Abstract

We report the analog simulation of an ergodic-localized junction by using an array of 12 coupled superconducting qubits. To perform the simulation, we fabricated a superconducting quantum processor that is divided into two domains: one is a driven domain representing an ergodic system, while the second is localized under the effect of disorder. Because of the overlap between localized and delocalized states, for a small disorder there is a proximity effect and localization is destroyed. To experimentally investigate this, we prepare a microwave excitation in the driven domain and explore how deep it can penetrate the disordered region by probing its dynamics. Furthermore, we perform an ensemble average over 50 realizations of disorder, which clearly shows the proximity effect. Our work opens a new avenue to build quantum simulators of driven-disordered systems with applications in condensed matter physics and material science.

Received 2 March 2020
Revised 16 July 2020
Accepted 11 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.170503

Physics Subject Headings (PhySH)

Quantum simulation

Disordered systems Floquet systems Superconducting qubits

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Chen Zha^1,2,3, V. M. Bastidas^4,5, Ming Gong ^1,2,3, Yulin Wu^1,2,3, Hao Rong^1,2,3, Rui Yang^1,2,3, Yangsen Ye^1,2,3, Shaowei Li^1,2,3, Qingling Zhu^1,2,3, Shiyu Wang^1,2,3, Youwei Zhao^1,2,3, Futian Liang^1,2,3, Jin Lin^1,2,3, Yu Xu^1,2,3, Cheng-Zhi Peng^1,2,3, J. Schmiedmayer ⁶, Kae Nemoto ⁵, Hui Deng^1,2,3, W. J. Munro ^4,5,*, Xiaobo Zhu^1,2,3,†, and Jian-Wei Pan^1,2,3

¹Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
²Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
³Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
⁴NTT Basic Research Laboratories and Research Center for Theoretical Quantum Physics, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan
⁵National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁶Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria

^*Corresponding author. bilmun@qis1.ex.nii.ac.jp
^†Corresponding author. xbzhu16@ustc.edu.cn

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Vol. 125, Iss. 17 — 23 October 2020

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Images

Figure 1
Ergodic-localized junction with superconducting qubits. (a) When the driven domain is coupled to a disordered one, there is an overlap between localized and delocalized states. (b) Depicts stadium and circular billiards, which exhibit ergodic and regular behavior, respectively. If one creates a junction between the two billiards, the destruction of conserved quantities creates a proximity effect, and the system explores all the available configurations. (c) Optical micrograph of the superconducting chip with 12 Xmon variants of transmon qubits arranged in a 1D lattice. Each qubit has a line for the $X Y$ control and a flux bias line for the $Z$ control. In addition, each qubit is dispersively coupled to a resonator for state readout. (d) Experimental waveform sequences to generate the ergodic-localized junctions. The ergodic domain is formed by qubits $Q_{1} - Q_{6}$ , which are periodically driven, and the localized domain is designed by adding disorder to $Q_{7} - Q_{12}$ .
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Figure 2
Dynamics of populations and correlations. Comparison between theory and experiment for cosine and flat potentials of qubits $Q_{7}$ to $Q_{12}$ . To explore localization due to the competition between kinetic and potential energy, in panels (a)–(e) we show the results for a cosine potential for $Q_{7}$ to $Q_{12}$ . Similarly, to investigate localization due to disorder, we consider a flat potential for $Q_{7}$ to $Q_{12}$ in panels (f)–(j) for $W = 0$ and (k)–(o) for $W = 5 J$ . (a),(f),(k) show the qubit frequency setups used in the experiment. The shaded areas represent the time-periodic modulation of the on-site energies. (b),(c),(g),(h),(l),(m) depict the population dynamics $⟨ {\hat{n}}_{l} ⟩$ . Correspondingly, (d),(e),(i),(j),(n),(o) show the dynamics of the correlation function $C_{Z Z} (l, 7, t)$ . Because of the shape of the cosine potential, the excitation can penetrate a small region of the domain $Q_{7}$ to $Q_{12}$ . In contrast, without disorder, the excitation propagates ballistically.
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Figure 3
Ensemble average of population dynamics. Motivated by the discussion on spectral signatures of the proximity effect, we explore the dynamics of the ensemble averages for a microwave excitations initially prepared at sites $l = 3$ and $l = 9$ in the ergodic and localized domains, respectively. (a) Depicts the frequency setup for several realizations of disorder with strength $W = 3 J$ . The shaded areas represent the time-periodic modulation of the on-site energies. On average, the potential of qubits $Q_{7}$ to $Q_{12}$ is flat. Panels (b),(d) show the theoretical prediction for the averaged populations, and panels (c),(e) the experimental result. (f)–(h) depict, respectively, the dynamics of the ensemble-averaged localization length $\bar{ξ} (t)$ , the participation ratio $\bar{P} (t)$ , and the population of the ergodic region ${\bar{N}}_{ergodic} (t)$ as a function of time for an excitation initially prepared at site $l = 9$ . The blue and red curves correspond to the theoretical and experimental results. (i),(j) illustrate the experimental result for the ensemble-averaged Fourier spectrum of $⟨ {\hat{n}}_{l} (t) ⟩$ at different sites $l$ corresponding to (c),(e), respectively. The insets in (i),(j) show the spatial average of the Fourier spectrum. The low-frequency dip is a signature of localization [44, 45].
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