Ground-state chiral currents in the synthetic Hall tube
Authors: Guan Xin , Chen Gang , Pan Jing , You Xiu-Fen , Gui Zhi-Guo ,   Department of Materials and Chemical Engineering, Taiyuan University, Taiyuan 030032, China,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China,   Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China,   Collaborative Innovation Center of Light Manipulations and Applications, School of Physics and Elecctronics Shandong Normal University, Jinan 250358, China
Published: May 2022

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 <inline-formula><tex-math id="M1">\begin{document}$xy$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M1.png"/></alternatives></inline-formula> plane (<inline-formula><tex-math id="M2">\begin{document}$xy$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M2.png"/></alternatives></inline-formula>-M), vortex current on <inline-formula><tex-math id="M3">\begin{document}$xy$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M3.png"/></alternatives></inline-formula> plane (<inline-formula><tex-math id="M4">\begin{document}$xy$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M4.png"/></alternatives></inline-formula>-V), vortex current on <inline-formula><tex-math id="M5">\begin{document}$xz$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M5.png"/></alternatives></inline-formula> plane (<inline-formula><tex-math id="M6">\begin{document}$xz$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M6.png"/></alternatives></inline-formula>-V), and vortex current on both <inline-formula><tex-math id="M7">\begin{document}$xy$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M7.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$xz$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M8.png"/></alternatives></inline-formula> planes (DV). For distinguishing these chiral currents, four order parameters <inline-formula><tex-math id="M9">\begin{document}$J_{C//}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M9.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$J_{AB}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M10.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M11">\begin{document}$J_{BC}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M11.png"/></alternatives></inline-formula>), and <inline-formula><tex-math id="M12">\begin{document}$J_{CA}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M12.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M13">\begin{document}$\tilde{t}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M13.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$t_{CA}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220293_M14.png"/></alternatives></inline-formula>. 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.