Hidden time-nonlocal Floquet symmetries

Sigmund Kohler; Jesús Casado-Pascual

doi:10.22331/q-2026-05-21-2112

Hidden time-nonlocal Floquet symmetries

Sigmund Kohler^1,2 and Jesús Casado-Pascual^3,4

¹Quantum Advanced Research Center (QuARC), CSIC, Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain
²Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain
³Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
⁴Multidisciplinary Unit for Energy Science, Universidad de Sevilla, E-41080 Sevilla, Spain

Published:	2026-05-21, volume 10, page 2112
Editor:	Géza Giedke
Eprint:	arXiv:2601.05783v2
Doi:	https://doi.org/10.22331/q-2026-05-21-2112
Citation:	Quantum 10, 2112 (2026).

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Abstract

We investigate the Floquet spectrum of a detuned, driven two-level system and show that it exhibits exact quasienergy crossings when the detuning is an integer multiple of the energy quantum of the driving field. This behavior can be explained by a hidden time-nonlocal parity, which allows the Floquet modes to be classified as even or odd. Then a generic feature is the emergence of exact crossings between quasienergies of different parity. A constructive proof of the existence of the symmetry is based on a scalar recurrence relation. Moreover, we present a general scheme for its numerical computation, which can be applied to models beyond the two-level system. Analytical results are illustrated with numerical data.

Featured image: Floquet spectrum of the driven two-level system with detuning equal to the energy quantum of the driving. Quasienergies with different hidden parity may form exact crossings, while equal parity leads to avoided crossings.

Popular summary

Quantum systems under periodic driving display behaviors that are impossible in static situations. We study one of the simplest examples: a two-level quantum system exposed to an oscillating field. Under particular resonance conditions, we find that different Floquet modes can have exactly the same quasienergy instead of the generic level repulsion. We show that this effect is protected by a hidden symmetry of the dynamics that separates the states into two distinct classes. Our results clarify how symmetries govern quantum dynamics under periodic driving and may be useful for the control of quantum devices.

► BibTeX data

@article{Kohler2026hiddentimenonlocal,
  doi = {10.22331/q-2026-05-21-2112},
  url = {https://doi.org/10.22331/q-2026-05-21-2112},
  title = {Hidden time-nonlocal {F}loquet symmetries},
  author = {Kohler, Sigmund and Casado-Pascual, Jes{\'{u}}s},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2112},
  month = may,
  year = {2026}
}

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[1] Jon H. Shirley. ``Solution of the Schrödinger equation with a Hamiltonian periodic in time''. Phys. Rev. 138, B979 (1965).
https://doi.org/10.1103/PhysRev.138.B979

[2] Hideo Sambe. ``Steady states and quasienergies of a quantum-mechanical system in an oscillating field''. Phys. Rev. A 7, 2203 (1973).
https://doi.org/10.1103/PhysRevA.7.2203

[3] Peter Hänggi. ``Driven quantum systems''. In Quantum Transport and Dissipation. Chapter 5, pages 249–286. Wiley-VCH, Weinheim (1998).

[4] Frank Grossmann, Thomas Dittrich, Peter Jung, and Peter Hänggi. ``Coherent destruction of tunneling''. Phys. Rev. Lett. 67, 516 (1991).
https://doi.org/10.1103/PhysRevLett.67.516

[5] Frank Großmann and Peter Hänggi. ``Localization in a driven two-level dynamics''. Europhys. Lett. 18, 571 (1992).
https://doi.org/10.1209/0295-5075/18/7/001

[13] Sigmund Kohler. ``Quantum dissipation at conical intersections of quasienergies''. Phys. Rev. A 110, 052218 (2024).
https://doi.org/10.1103/PhysRevA.110.052218

[14] Fritz Haake, Sven Gnutzmann, and Marek Kuś. ``Quantum signatures of chaos''. Springer Series in Synergetics. Springer. Cham (2018). 4th edition.
https://doi.org/10.1007/978-3-319-97580-1

[15] Asher Peres. ``Dynamical quasidegeneracies and quantum tunneling''. Phys. Rev. Lett. 67, 158 (1991).
https://doi.org/10.1103/PhysRevLett.67.158

[17] S. Ashhab, J. R. Johansson, A. M. Zagoskin, and Franco Nori. ``Two-level systems driven by large-amplitude fields''. Phys. Rev. A 75, 063414 (2007).
https://doi.org/10.1103/PhysRevA.75.063414

[19] S. Ashhab. ``Attempt to find the hidden symmetry in the asymmetric quantum Rabi model''. Phys. Rev. A 101, 023808 (2020).
https://doi.org/10.1103/PhysRevA.101.023808

Cited by

[1] Chen Wei and Frank Großmann, "Higher order Magnus expansion for driven two-level quantum dynamics", The Journal of Chemical Physics 164 22, 224121 (2026).

[2] O. A. Ilinskaya and S. N. Shevchenko, "Resonant excitation of single and coupled qubits for coherent quantum control and microwave detection", Low Temperature Physics 52 6, 685 (2026).

[3] Xiufeng Cao, Chen Wang, Georg Engelhardt, Hang Zheng, and Dahai He, "Steady-state quantum coherence in driven open quantum system: An optimal transformation analysis", Journal of Chemical Physics 164 16, 164307 (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 16:19:10) and SAO/NASA ADS (last updated successfully 2026-07-15 16:19:11). The list may be incomplete as not all publishers provide suitable and complete citation data.

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