中国物理B ›› 2021, Vol. 30 ›› Issue (10): 100307-100307.doi: 10.1088/1674-1056/ac0ba8

所属专题: 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. 1 Department of Physics, Shandong University, Jinan 250100, China;
    2 Department of Physics, National University of Singapore, 117551, Singapore
  • 收稿日期:2021-05-04 修回日期:2021-06-02 接受日期:2021-06-16 出版日期:2021-09-17 发布日期:2021-08-19
  • 通讯作者: Da-Jian Zhang, Qing-hai Wang, Jiangbin Gong E-mail:zdj@sdu.edu.cn;qhwang@nus.edu.sg;phygj@nus.edu.sg
  • 基金资助:
    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).

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. 1 Department of Physics, Shandong University, Jinan 250100, China;
    2 Department of Physics, National University of Singapore, 117551, Singapore
  • Received:2021-05-04 Revised:2021-06-02 Accepted:2021-06-16 Online:2021-09-17 Published:2021-08-19
  • Contact: Da-Jian Zhang, Qing-hai Wang, Jiangbin Gong E-mail:zdj@sdu.edu.cn;qhwang@nus.edu.sg;phygj@nus.edu.sg
  • Supported by:
    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).

摘要: 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.

关键词: time-dependent $\mathcal{PT}$-symmetric quantum mechanics, geometry, time-varying inner product, unconventional geometric phase

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.

Key words: time-dependent $\mathcal{PT}$-symmetric quantum mechanics, geometry, time-varying inner product, unconventional geometric phase

中图分类号:  (Quantum mechanics)

  • 03.65.-w
02.40.Ky (Riemannian geometries) 03.65.Aa (Quantum systems with finite Hilbert space)