Observation of emergent ${\mathbb{Z}}_{2}$ gauge invariance in a superconducting circuit

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Observation of emergent $Z_{2}$ gauge invariance in a superconducting circuit

Zhan Wang, Zi-Yong Ge, Zhongcheng Xiang, Xiaohui Song, Rui-Zhen Huang, Pengtao Song, Xue-Yi Guo, Luhong Su, Kai Xu, Dongning Zheng, and Heng Fan

Phys. Rev. Research 4, L022060 – Published 17 June 2022

Abstract

Lattice gauge theories (LGTs) are one of the most fundamental subjects in many-body physics, and has recently attracted considerable research interests in quantum simulations. Here we experimentally investigate the emergent $Z_{2}$ gauge invariance in a 1D superconducting circuit with 10 transmon qubits. By precisely adjusting staggered longitudinal and transverse fields to each qubit, we construct an effective Hamiltonian containing an LGT and gauge-broken terms. The corresponding matter sector can exhibit a localization, and there also exists a 3-qubit operator, of which the expectation value can retain nonzero for a long time in low-energy regimes. The above localization can be regarded as the confinement of matter fields, and the 3-body operator is the $Z_{2}$ gauge generator. These experimental results demonstrate that, despite the absence of gauge structure in the effective Hamiltonian, $Z_{2}$ gauge invariance can still emerge in low-energy regimes. Our work provides a method for both theoretically and experimentally studying the rich physics in quantum many-body systems with emergent gauge invariance.

Received 17 November 2021
Revised 16 May 2022
Accepted 7 June 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.L022060

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum information with solid state qubits Quantum simulation

Superconducting qubits

Lattice gauge theory

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Zhan Wang^1,2,*, Zi-Yong Ge ^1,2,*, Zhongcheng Xiang^1,*, Xiaohui Song¹, Rui-Zhen Huang³, Pengtao Song^1,2, Xue-Yi Guo¹, Luhong Su^1,2, Kai Xu^1,4,5, Dongning Zheng^1,2,4,5,†, and Heng Fan^1,4,5,6,‡

¹Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
³Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
⁴Songshan Lake Materials Laboratory, Dongguan 523808, Guangdong, China
⁵CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100190, China
⁶Beijing Academy of Quantum Information Sciences, Beijing 100193, China

^*These authors contributed equally to this work.
^†dzheng@iphy.ac.cn
^‡hfan@iphy.ac.cn

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Vol. 4, Iss. 2 — June - August 2022

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Figure 1
Set-up and protocol. (a) Optical micrograph of the superconducting circuit. There are 10 transmon qubits arranged into a chain, where the odd and even qubits are used for realizing $s$ and $τ$ spins in the following experiment, respectively. Each qubit couples to a microwave line for the $X Y$ driving ( ${XY}_{j}$ ), a flux bias line for the $Z$ pulse ( $Z_{j}$ ), and a readout resonator for measurement ( $R_{j}$ ). The NN qubits are directly coupled via a capacitor, and the capacitance of each qubit itself contributes to the NNN coupling. The parameters of the device are presented in the Supplemental Material (SM) in detail [44]. (b) Lattice skeleton of the effective Hamiltonian (2). There are two types of spins, i.e., $s$ (red sites) and $τ$ (blue sites), corresponding to matter and gauge fields, respectively. The purple arrows represent the three-body couplings; the orange arrows are direct NNN two-body couplings, and the blue arrows are longitudinal/transverse fields. (c) Pulse sequences of the experiment. The solid and dashed lines represent XY drivings and Z pulses, respectively. First, all qubits are at their idle frequencies for preparing the initial state $| ψ_{0} 〉$ via XY driving. Then, each qubit is biased to the working point with $Z$ pulses, and meanwhile, the XY driving is applied to each even qubit. After the system evolves for time $t$ , all qubits are biased back to idle points for readout.
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Figure 2
Spreading of $s$ spins up to $1 μ s$ for (a) $h_{x} / 2 π = 2$ MHz, $V_{2 ℓ} / 2 π = 0$ MHz, $θ = - π / 3$ ; (b) $h_{x} / 2 π = 6$ MHz, $V_{2 ℓ} / 2 π = 15$ MHz, $θ = - π / 3$ ; (c) $h_{x} / 2 π = 6$ MHz, $V_{2 ℓ} / 2 π = 15$ MHz, $θ = - π / 2$ ; and (d) $h_{x} / 2 π = 6$ MHz, $V_{2 ℓ} / 2 π = 15$ MHz, $θ = π$ . Here the $s$ spin shows localization in the cases of (b) and (c) indicating the presence of confinement, while it transports freely in (a) and (d) indicating the absence of confinement. Each point is the average of 8000 single-shot readouts.
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Figure 3
Extended imbalance $I$ versus initial states. (a) Time evolution of $I$ with $h_{x} / 2 π = 6$ MHz, $V_{2 ℓ} / 2 π = 15$ MHz, and different $θ$ , i.e., cases in Figs. 2 and 2. The dots are experimental data, while the solid curves are the corresponding numerical results. (b) The relation between $I_{\infty}$ and $θ$ . Here $I_{\infty}$ is calculated as the average of $I$ from 0.2 to $1 μ s$ , and the standard deviation is the corresponding variance. Using Gaussian fitting, we obtain the peak of experimental result is at $θ = θ_{m} \approx - 0.35$ . The numerical results are obtained via Hamiltonian in Eq. (1), where decoherence and dephasing are both neglected.
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Figure 4
Gauge invariance. (a) The relation between ${〈 {\hat{G}}_{3} 〉}_{\infty}$ and $α$ with the parameters identical to the cases in Figs. 2. Here ${〈 {\hat{G}}_{3} 〉}_{\infty}$ , calculated as the average of $I$ from 0.2 to $1 μ s$ with the standard deviation being the error bar, represents the steady expectation value of 3-qubit operator ${\hat{G}}_{3} (α)$ , defined in Eq. (6). The curve is almost a sine-shape curve with the valley at $α \approx - 1.19$ . (b) Time evolution of 3-qubit operator $〈 {\hat{G}}_{3} (α = - 1.19) 〉$ up to $1 μ s$ . It shows that $〈 {\hat{G}}_{3} 〉$ can retain a nonzero value for a long time, where the oscillation is from the high-order term [44].
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Physical Review Research

Observation of emergent Z2 gauge invariance in a superconducting circuit

Zhan Wang, Zi-Yong Ge, Zhongcheng Xiang, Xiaohui Song, Rui-Zhen Huang, Pengtao Song, Xue-Yi Guo, Luhong Su, Kai Xu, Dongning Zheng, and Heng Fan

Phys. Rev. Research 4, L022060 – Published 17 June 2022

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Observation of emergent $Z_{2}$ gauge invariance in a superconducting circuit