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Hall tube is an important model to simulate the quantum Hall effect. However it hasn't been realized in superconducting circuits which have emerged as a promising platform for macro-controlling quantum effect. Taking advantage of the fine tunability of superconducting circuits, the three-chain superconducting transmon qubits with periodic boundary condition are designed in this paper. For constructing a synthetic Hall tube, ac magnetic fluxes are introduced to drive each transmon qubit. The gauge field emerged in this synthetic Hall tube can be tuned independently by properly choosing the driving phases. Then the ground-state chiral currents are discovered in this synthetic Hall tube, which are Meissner current on
$xy$ plane ($xy$ -M), vortex current on$xy$ plane ($xy$ -V), vortex current on$xz$ plane ($xz$ -V), and vortex current on both$xy$ and$xz$ planes (DV). For distinguishing these chiral currents, four order parameters$J_{C//}$ ,$J_{AB}$ ($J_{BC}$ ), and$J_{CA}$ are defined. Then the ground-state quantum phase diagrams are mapped out. The emergence of the different quantum phases is due to the competition between the coupling strengths$\tilde{t}$ and$t_{CA}$ . The Meissner and vortex currents emerging in this synthetic Hall tube also emerge in type II superconductor, which can generate an opposite field to weaken the influence of the applied field. Thus this synthetic Hall tube can be used as a diamagnet. At last we consider the influence of the imperfections in device fabrication. We proof when the strength of the imperfection is not large enough, the quantum phase diagrams shown in this paper remain valid. Moreover, the possible experimental observations of the ground-state chiral currents are addressed. The ground state of this synthetic Hall tube can be generated by applying microwave pulses. Then the corresponding density matrix can be constructed by the quantum state tomography. After constructing the density matrix, the order parameters can be obtained by calculating the trace. These results enrich the quantum currents in Hall tube and provide a new route to explore novel quantum phases.-
Keywords:
- synthetic Hall tube /
- superconducting circuits /
- chiral currents
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图 1 人造霍尔管模型 (a) 三条transmon比特链分别标记为A, B, C. 沿着每条链, 最近邻比特之间耦合. 三条链之间, 同一个元胞中的比特两两耦合. 这里所有的耦合器均为电容.
$ Q_{\nu j} $ 表示的是第ν条链上的第j个比特.$ C_{\nu j} $ 和$ E_{\nu j}^{{\rm{J}}} $ 分别是第ν条链上、第j个比特的有效电容和约瑟夫森能.$ C_{\nu ij} $ 是耦合第ν条链上、第i和第j个元比特的电容.$ \{C_{ABj}, C_{BCj}, C_{CAj}\} $ 是耦合第j个元胞中不同链间比特的电容.$ \phi _{\nu j} $ 是第ν条链上、第j个比特的约瑟夫森结的相位. transmon比特的约瑟夫森结由超导量子干涉仪(SQUID)形成,$ E_{\nu j}^{{\rm{J}}0} $ 是SQUID中每个约瑟夫森结的能量. 每个比特都受到外加磁通$ \varPhi_{\nu j}(t) $ 的调制. (b) 人造霍尔管示意图, 在面$ Q_{A(j-1)}Q_{Aj}Q_{Bj}Q_{B(j-1)} $ 和$ Q_{C(j-1)}Q_{Cj}Q_{Bj}Q_{B(j-1)} $ 上积累$ \varphi_0 $ 的相位, 在面$ Q_{A(j-1)}Q_{Aj}Q_{Cj}Q_{C(j-1)} $ 上积累$ 2\varphi_0 $ 的相位. 图中, 红色、黑色和蓝色的实心球分别表示A, B和C链上的比特Figure 1. Model of the synthetic Hall tube: (a) Three-leg (labeled respectively by A, B, C) superconducting circuits with transmon qubits. Along each leg, the qubits are coupled with their nearestneighbor sites, while they interact with each other between legs at the same unit cell. All couplers are capacitors.
$ C_{\nu j} $ and$ E_{\nu j}^{{\rm{J}}} $ are the effective capacitance and the Josephson energy of the qubit at the j th site on the ν th leg.$ C_{\nu ij} $ and$ \{C_{ABj}, C_{BCj}, C_{CAj}\} $ are the capacitors to couple the qubits at the jth site on the νth leg with its nearest-neighbor sites along each leg and between the legs, respectively.$ \phi _{\nu j} $ is the phase of the Josephson junction of the qubit at the jth site on the νth leg. The Josephson junction of the transmon qubit is a superconducting quantum interference device (SQUID).$ E_{\nu j}^{{\rm{J}}0} $ is the Josephson energy of SQUID. Each qubit is modulate by an external magnetic flux$ \varPhi_{\nu j}(t) $ . (b) Schematics of the synthetic Hall tube with an effective phase$ \varphi_0 $ in the plaquette$ Q_{A(j-1)}Q_{Aj}Q_{Bj}Q_{B(j-1)} $ and$Q_{C(j-1)}Q_{Cj}Q_{Bj} $ $ Q_{B(j-1)}$ and$ 2\varphi_0 $ in the plaquette$ Q_{A(j-1)}Q_{Aj}Q_{Cj}Q_{C(j-1)} $ . The red, black, and blue filled circles indicate the qubits in the A, B and C legs, respectively.
图 2 霍尔管上的流以及比特的态密度分布. 箭头的粗细表示流的强弱, 红色、蓝色和黑色实心小球的大小分别表示A链、B链和C链上比特态密度分布的多少. 参数选择分别为 (a)
$ \tilde{t} = 1.1 $ 和$ t_{CA} = 4.5 $ ; (b)$ \tilde{t} = 0.4 $ 和$ t_{CA} = 1.8 $ ; (c)$ \tilde{t} = 4 $ 和$ t_{CA} = 4.4 $ ; (d)$ \tilde{t} = 2 $ 和$ t_{CA} = 2 $ . 这四个图共同的参数是$ \varphi_0 = 2\pi/3 $ , 并且所有参数均以$ t_0 $ 为单位Figure 2. Currents and densities distributions on the Hall tube with (a)
$ \tilde{t} = 1.1 $ and$ t_{CA} = 4.5 $ ; (b)$ \tilde{t} = 0.4 $ and$ t_{CA} = 1.8 $ ; (c)$ \tilde{t} = 4 $ and$ t_{CA} = 4.4 $ ; (d)$ \tilde{t} = 2 $ and$ t_{CA} = 2 $ . The thicknesses of the arrow denote the currents strength. The size of the red, black, and blue filled circles indicate the densities on the A, B, and C legs respectively. Here$ t_0 $ is used as the unit. The common parameter is$ \varphi_0 = 2\pi/3 $ .
图 3 参数空间中的相图 (a)
$J_{C//}$ ; (b)$ J_{CA} $ ; (c)$ J_{AB} $ 和$ J_{BC} $ ; (d)参数空间中$ \tilde{t} $ -$ t_{CA} $ 不同的相. 这四个图共同的参数是$ \varphi_0 = 2\pi/3 $ , 并且所有参数均以$ t_0 $ 为单位Figure 3. Phase diagram of (a)
$J_{C//}$ , (b)$ J_{CA} $ , (c)$ J_{AB} $ and$ J_{BC} $ as functions of$ \tilde{t} $ and$ t_{CA} $ . Different phases are indicated in (d). Here$ t_0 $ is used as the unit. The common parameter is$ \varphi_0 = 2\pi/3 $ .
图 4 不同参数取值下序参量
$J_{C//}$ ,$ J_{AB} $ ,$ J_{BC} $ 和$ J_{CA} $ 随$\lg W$ 的变化 (a)$ \tilde{t} = 1 $ 和$ t_{CA} = 4 $ ; (b)$ \tilde{t} = 0.5 $ 和$ t_{CA} = 2.5 $ ; (c)$ \tilde{t} = 2.5 $ 和$ t_{CA} = 3 $ ; (d)$ \tilde{t} = 2 $ 和$ t_{CA} = 2 $ . 这四个图共同的参数是$ \varphi_0 = 2\pi/3 $ , 并且所有参数均以$ t_0 $ 为单位Figure 4.
$J_{C//}$ ,$ J_{AB} $ ,$ J_{BC} $ , and$ J_{CA} $ as functions of$\lg W$ with (a)$ \tilde{t} = 1 $ and$ t_{CA} = 4 $ , (b)$ \tilde{t} = 0.5 $ and$ t_{CA} = 2.5 $ ), (c)$ \tilde{t} = 2.5 $ and$ t_{CA} = 3 $ , (d)$ \tilde{t} = 2 $ and$ t_{CA} = 2 $ . The common parameter is$ \varphi_0 = 2\pi/3 $ . Here$ t_0 $ is used as the unit.
图 5 第ν条链的第j个比特的态
$ \vert \psi \rangle _{\nu j} $ 在布洛赫球上的表示.$ \theta _{\nu j} $ 和$ \chi_\nu j $ 分别表示$ \vert \psi \rangle _{\nu j} $ 在布洛赫球上与z轴和x轴的夹角Figure 5. The jth qubit located on the ν leg described in Bloch sphere.
$ \theta _{\nu j} $ ($ \chi_\nu j $ ) is the angle between the state$ \vert \psi \rangle _{\nu j} $ and z (x) axis in the Bloch sphere.表 1 不同手性流的判别
Table 1. Discriminant of the chiral currents.
手性流的类别 $xy$-M $xy$-V $xz$-V DV $J_{C\Vert }$ 非0 非0 0 非0 $J_{AB }$($J_{BC}$) 0 0 非0 非0 $J_{CA}$ 0 非0 1非0 非0 
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[1] Tai M E, Lukin A, Rispoli M, Menke T, Borgnia D, Preiss P M, Grusdt F, Kaufman A M, Greiner M 2017 Nature 546 519
Google Scholar
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Google Scholar
Liu B, Zhou X F, Chen G, Jia S T 2020 Acta Phys. Sin. 69 080501
Google Scholar
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Google Scholar
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Google Scholar
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