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Special Issue:
SPECIAL TOPIC — Non-Hermitian physics
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| SPECIAL TOPIC—Non-Hermitian physics |
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Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics |
| Da-Jian Zhang(张大剑)1,†, Qing-hai Wang(王清海)2,‡, and Jiangbin Gong(龚江滨)2,§ |
1 Department of Physics, Shandong University, Jinan 250100, China;
2 Department of Physics, National University of Singapore, 117551, Singapore |
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Abstract A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.
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Received: 04 May 2021
Revised: 02 June 2021
Accepted manuscript online: 16 June 2021
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PACS:
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03.65.-w
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(Quantum mechanics)
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02.40.Ky
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(Riemannian geometries)
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03.65.Aa
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(Quantum systems with finite Hilbert space)
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| Fund: J.G. is supported by Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112) and by the Singapore NRF Grant No. NRFNRFI2017-04 (WBS No. R-144-000-378-281). Q.W. is supported by Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-352-112). |
Corresponding Authors:
Da-Jian Zhang, Qing-hai Wang, Jiangbin Gong
E-mail: zdj@sdu.edu.cn;qhwang@nus.edu.sg;phygj@nus.edu.sg
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Cite this article:
Da-Jian Zhang(张大剑), Qing-hai Wang(王清海), and Jiangbin Gong(龚江滨) Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics 2021 Chin. Phys. B 30 100307
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