Emergent Parallel Transport And Curvature In Hermitian And Non-Hermitian Quantum Mechanics

Chia-Yi Ju^1,2, Adam Miranowicz^3,4, Yueh-Nan Chen^5,6,7, Guang-Yin Chen⁸, and Franco Nori^4,9,10

¹Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
²Center for Theoretical and Computational Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
³Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
⁴Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama, 351-0198, Japan
⁵Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
⁶Center for Quantum Frontiers of Research & Technology, NCKU, Tainan 70101, Taiwan
⁷Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
⁸Department of Physics, National Chung Hsing University, Taichung 40227, Taiwan
⁹Quantum Computing Center, RIKEN, Wakoshi, Saitama, 351-0198, Japan
¹⁰Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA

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Abstract

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

Popular summary

In this study, we show that if a system depends on a continuous parameter, the quantum states vary with the parameter described by a Schroedinger-like equation, which formally resembles a parallel transport or evolution equation along the dimension described by the parameter. Moreover, we derive the governing equation for the geometry/metric of the underlying Hilbert space along the parameter-formed dimension. Rather than solely engaging in a formal study of the properties of these emergent dimensions, we also explore their applications across various fields in quantum physics.

► BibTeX data

@article{Ju2024emergentparallel, doi = {10.22331/q-2024-03-13-1277}, url = {https://doi.org/10.22331/q-2024-03-13-1277}, title = {Emergent parallel transport and curvature in {H}ermitian and non-{H}ermitian quantum mechanics}, author = {Ju, Chia-Yi and Miranowicz, Adam and Chen, Yueh-Nan and Chen, Guang-Yin and Nori, Franco}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1277}, month = mar, year = {2024} }

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Cited by

[1] Ievgen I. Arkhipov, Adam Miranowicz, Fabrizio Minganti, Şahin K. Özdemir, and Franco Nori, “Dynamically crossing diabolic points while encircling exceptional curves: A programmable symmetric-asymmetric multimode switch”, Nature Communications 14, 2076 (2023).

[2] Miloslav Znojil, “Hybrid form of quantum theory with non-Hermitian Hamiltonians”, Physics Letters A 457, 128556 (2023).

[3] Miloslav Znojil, “Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation”, Physics Letters A 462, 128655 (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-03-23 23:58:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2024-03-23 23:58:37).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

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