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Teoreticheskaya i Matematicheskaya Fizika, 2020, Volume 204, Number 2, Pages 211–225
DOI: https://doi.org/10.4213/tmf9860
(Mi tmf9860)
 

This article is cited in 2 scientific papers (total in 2 papers)

Optimal Evolution Time Generated by Pseudo-Hermitian Hamiltonian

W H. Wanga, Z. L. Chenb, Y. Songc, Y. J. Fand

a School of Ethnic Nationalities Education, Shaanxi Normal University, Xi'an, China
b School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, China
c School of Computer Science, Shaanxi Normal University, Xi'an, China
d School of Mathematics and Information Science, North Minzu University, Yinchuan, China
Full-text PDF (419 kB) Citations (2)
References:
Abstract: If an initial state $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ and a final state $|\psi_{\scriptscriptstyle\mathrm{F}}\rangle$ are given, then there exist many Hamiltonians under whose action $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ evolves into $|\psi_{\scriptscriptstyle\mathrm F}\rangle$. In this case, the problem of the transition of $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ to $|\psi_{\scriptscriptstyle \mathrm F}\rangle$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ and $|\psi_{\scriptscriptstyle\mathrm F}\rangle$ are orthogonal. But for a $PT$-symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.
Keywords: optimum time, Hermitian Hamiltonian, pseudo-Hermitian Hamiltonian, inner product, unitary evolution.
Funding agency Grant number
National Natural Science Foundation of China 11871318
11771009
11601300
11571213
61602291
Fundamental Research Funds for the Central Universities of China GK202003093
China Scholarship Council
This research is supported by the National Natural Science Foundation of China (Grant Nos. 11871318, 11771009, 11701011, 11601300, 11571213, and 61602291), the FRF for the Central Universities (Grant No. GK202003093), and the State Scholarship Fund of China Scholarship Council.
Received: 14.12.2019
Revised: 20.02.2020
English version:
Theoretical and Mathematical Physics, 2020, Volume 204, Issue 2, Pages 1020–1032
DOI: https://doi.org/10.1134/S0040577920080048
Bibliographic databases:
Document Type: Article
PACS: 03.65.-w, 03.65.Er, 03.65.Vf
Language: Russian
Citation: W H. Wang, Z. L. Chen, Y. Song, Y. J. Fan, “Optimal Evolution Time Generated by Pseudo-Hermitian Hamiltonian”, TMF, 204:2 (2020), 211–225; Theoret. and Math. Phys., 204:2 (2020), 1020–1032
Citation in format AMSBIB
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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