Hidden time-nonlocal Floquet symmetries
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AbstractWe 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 summaryQuantum 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} }► References [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 [6] J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gossard. ``Landau-Zener-Stückelberg interferometry of a single electron charge qubit''. Phys. Rev. B 86, 121303(R) (2012). https://doi.org/10.1103/PhysRevB.86.121303 [7] F. Forster, G. Petersen, S. Manus, P. Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig. ``Characterization of qubit dephasing by Landau-Zener-Stückelberg-Majorana interferometry''. Phys. Rev. Lett. 112, 116803 (2014). https://doi.org/10.1103/PhysRevLett.112.116803 [8] Mika Sillanpää, Teijo Lehtinen, Antti Paila, Yuriy Makhlin, and Pertti Hakonen. ``Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box''. Phys. Rev. Lett. 96, 187002 (2006). https://doi.org/10.1103/PhysRevLett.96.187002 [9] David M. Berns, Mark S. Rudner, Sergio O. Valenzuela, Karl K. Berggren, William D. Oliver, Leonid S. Levitov, and Terry P. Orlando. ``Amplitude spectroscopy of a solid-state artificial atom''. Nature (London) 455, 51 (2008). https://doi.org/10.1038/nature07262 [10] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo. ``Dynamical control of matter-wave tunneling in periodic potentials''. Phys. Rev. Lett. 99, 220403 (2007). https://doi.org/10.1103/PhysRevLett.99.220403 [11] G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi. ``Visualization of coherent destruction of tunneling in an optical double well system''. Phys. Rev. Lett. 98, 263601 (2007). https://doi.org/10.1103/PhysRevLett.98.263601 [12] Georg Engelhardt, Gloria Platero, and Jianshu Cao. ``Discontinuities in driven spin-boson systems due to coherent destruction of tunneling: Breakdown of the Floquet-Gibbs distribution''. Phys. Rev. Lett. 123, 120602 (2019). https://doi.org/10.1103/PhysRevLett.123.120602 [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 [16] Michael Strass, Peter Hänggi, and Sigmund Kohler. ``Nonadiabatic electron pumping: Maximal current with minimal noise''. Phys. Rev. Lett. 95, 130601 (2005). https://doi.org/10.1103/PhysRevLett.95.130601 [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 [18] Oleh V. Ivakhnenko, Sergey N. Shevchenko, and Franco Nori. ``Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference''. Phys. Rep. 995, 1 (2023). https://doi.org/10.1016/j.physrep.2022.10.002 [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 [20] Vladimir V. Mangazeev, Murray T. Batchelor, and Vladimir V. Bazhanov. ``The hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 12LT01 (2021). https://doi.org/10.1088/1751-8121/abe426 [21] Cid Reyes-Bustos, Daniel Braak, and Masato Wakayama. ``Remarks on the hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 285202 (2021). https://doi.org/10.1088/1751-8121/ac0508 [22] Georg Engelhardt and Jianshu Cao. ``Dynamical symmetries and symmetry-protected selection rules in periodically driven quantum systems''. Phys. Rev. Lett. 126, 090601 (2021). https://doi.org/10.1103/PhysRevLett.126.090601 [23] Alexander Altland and Martin R. Zirnbauer. ``Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures''. Phys. Rev. B 55, 1142 (1997). https://doi.org/10.1103/PhysRevB.55.1142 [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 [6] J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gossard. ``Landau-Zener-Stückelberg interferometry of a single electron charge qubit''. Phys. Rev. B 86, 121303(R) (2012). https://doi.org/10.1103/PhysRevB.86.121303 [7] F. Forster, G. Petersen, S. Manus, P. Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig. ``Characterization of qubit dephasing by Landau-Zener-Stückelberg-Majorana interferometry''. Phys. Rev. Lett. 112, 116803 (2014). https://doi.org/10.1103/PhysRevLett.112.116803 [8] Mika Sillanpää, Teijo Lehtinen, Antti Paila, Yuriy Makhlin, and Pertti Hakonen. ``Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box''. Phys. Rev. Lett. 96, 187002 (2006). https://doi.org/10.1103/PhysRevLett.96.187002 [9] David M. Berns, Mark S. Rudner, Sergio O. Valenzuela, Karl K. Berggren, William D. Oliver, Leonid S. Levitov, and Terry P. Orlando. ``Amplitude spectroscopy of a solid-state artificial atom''. Nature (London) 455, 51 (2008). https://doi.org/10.1038/nature07262 [10] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo. ``Dynamical control of matter-wave tunneling in periodic potentials''. Phys. Rev. Lett. 99, 220403 (2007). https://doi.org/10.1103/PhysRevLett.99.220403 [11] G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi. ``Visualization of coherent destruction of tunneling in an optical double well system''. Phys. Rev. Lett. 98, 263601 (2007). https://doi.org/10.1103/PhysRevLett.98.263601 [12] Georg Engelhardt, Gloria Platero, and Jianshu Cao. ``Discontinuities in driven spin-boson systems due to coherent destruction of tunneling: Breakdown of the Floquet-Gibbs distribution''. Phys. Rev. Lett. 123, 120602 (2019). https://doi.org/10.1103/PhysRevLett.123.120602 [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 [16] Michael Strass, Peter Hänggi, and Sigmund Kohler. ``Nonadiabatic electron pumping: Maximal current with minimal noise''. Phys. Rev. Lett. 95, 130601 (2005). https://doi.org/10.1103/PhysRevLett.95.130601 [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 [18] Oleh V. Ivakhnenko, Sergey N. Shevchenko, and Franco Nori. ``Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference''. Phys. Rep. 995, 1 (2023). https://doi.org/10.1016/j.physrep.2022.10.002 [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 [20] Vladimir V. Mangazeev, Murray T. Batchelor, and Vladimir V. Bazhanov. ``The hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 12LT01 (2021). https://doi.org/10.1088/1751-8121/abe426 [21] Cid Reyes-Bustos, Daniel Braak, and Masato Wakayama. ``Remarks on the hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 285202 (2021). https://doi.org/10.1088/1751-8121/ac0508 [22] Georg Engelhardt and Jianshu Cao. ``Dynamical symmetries and symmetry-protected selection rules in periodically driven quantum systems''. Phys. Rev. Lett. 126, 090601 (2021). https://doi.org/10.1103/PhysRevLett.126.090601 [23] Alexander Altland and Martin R. Zirnbauer. ``Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures''. Phys. Rev. B 55, 1142 (1997). https://doi.org/10.1103/PhysRevB.55.1142Cited by[1] O. A. Ilinskaya and S. N. Shevchenko, "Resonant excitation of single and coupled qubits for coherent quantum control and microwave detection", arXiv:2603.27610, (2026). [2] 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 SAO/NASA ADS (last updated successfully 2026-05-21 11:07:50). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-21 11:07:49: Could not fetch cited-by data for 10.22331/q-2026-05-21-2112 from Crossref. This is normal if the DOI was registered recently.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. AbstractWe 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 summaryQuantum 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} }► References [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 [6] J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gossard. ``Landau-Zener-Stückelberg interferometry of a single electron charge qubit''. Phys. Rev. B 86, 121303(R) (2012). https://doi.org/10.1103/PhysRevB.86.121303 [7] F. Forster, G. Petersen, S. Manus, P. Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig. ``Characterization of qubit dephasing by Landau-Zener-Stückelberg-Majorana interferometry''. Phys. Rev. Lett. 112, 116803 (2014). https://doi.org/10.1103/PhysRevLett.112.116803 [8] Mika Sillanpää, Teijo Lehtinen, Antti Paila, Yuriy Makhlin, and Pertti Hakonen. ``Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box''. Phys. Rev. Lett. 96, 187002 (2006). https://doi.org/10.1103/PhysRevLett.96.187002 [9] David M. Berns, Mark S. Rudner, Sergio O. Valenzuela, Karl K. Berggren, William D. Oliver, Leonid S. Levitov, and Terry P. Orlando. ``Amplitude spectroscopy of a solid-state artificial atom''. Nature (London) 455, 51 (2008). https://doi.org/10.1038/nature07262 [10] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo. ``Dynamical control of matter-wave tunneling in periodic potentials''. Phys. Rev. Lett. 99, 220403 (2007). https://doi.org/10.1103/PhysRevLett.99.220403 [11] G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi. ``Visualization of coherent destruction of tunneling in an optical double well system''. Phys. Rev. Lett. 98, 263601 (2007). https://doi.org/10.1103/PhysRevLett.98.263601 [12] Georg Engelhardt, Gloria Platero, and Jianshu Cao. ``Discontinuities in driven spin-boson systems due to coherent destruction of tunneling: Breakdown of the Floquet-Gibbs distribution''. Phys. Rev. Lett. 123, 120602 (2019). https://doi.org/10.1103/PhysRevLett.123.120602 [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 [16] Michael Strass, Peter Hänggi, and Sigmund Kohler. ``Nonadiabatic electron pumping: Maximal current with minimal noise''. Phys. Rev. Lett. 95, 130601 (2005). https://doi.org/10.1103/PhysRevLett.95.130601 [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 [18] Oleh V. Ivakhnenko, Sergey N. Shevchenko, and Franco Nori. ``Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference''. Phys. Rep. 995, 1 (2023). https://doi.org/10.1016/j.physrep.2022.10.002 [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 [20] Vladimir V. Mangazeev, Murray T. Batchelor, and Vladimir V. Bazhanov. ``The hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 12LT01 (2021). https://doi.org/10.1088/1751-8121/abe426 [21] Cid Reyes-Bustos, Daniel Braak, and Masato Wakayama. ``Remarks on the hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 285202 (2021). https://doi.org/10.1088/1751-8121/ac0508 [22] Georg Engelhardt and Jianshu Cao. ``Dynamical symmetries and symmetry-protected selection rules in periodically driven quantum systems''. Phys. Rev. Lett. 126, 090601 (2021). https://doi.org/10.1103/PhysRevLett.126.090601 [23] Alexander Altland and Martin R. Zirnbauer. ``Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures''. Phys. Rev. B 55, 1142 (1997). https://doi.org/10.1103/PhysRevB.55.1142 [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 [6] J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gossard. ``Landau-Zener-Stückelberg interferometry of a single electron charge qubit''. Phys. Rev. B 86, 121303(R) (2012). https://doi.org/10.1103/PhysRevB.86.121303 [7] F. Forster, G. Petersen, S. Manus, P. Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig. ``Characterization of qubit dephasing by Landau-Zener-Stückelberg-Majorana interferometry''. Phys. Rev. Lett. 112, 116803 (2014). https://doi.org/10.1103/PhysRevLett.112.116803 [8] Mika Sillanpää, Teijo Lehtinen, Antti Paila, Yuriy Makhlin, and Pertti Hakonen. ``Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box''. Phys. Rev. Lett. 96, 187002 (2006). https://doi.org/10.1103/PhysRevLett.96.187002 [9] David M. Berns, Mark S. Rudner, Sergio O. Valenzuela, Karl K. Berggren, William D. Oliver, Leonid S. Levitov, and Terry P. Orlando. ``Amplitude spectroscopy of a solid-state artificial atom''. Nature (London) 455, 51 (2008). https://doi.org/10.1038/nature07262 [10] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo. ``Dynamical control of matter-wave tunneling in periodic potentials''. Phys. Rev. Lett. 99, 220403 (2007). https://doi.org/10.1103/PhysRevLett.99.220403 [11] G. Della Valle, M. Ornigotti, E. Cianci, V. Foglietti, P. Laporta, and S. Longhi. ``Visualization of coherent destruction of tunneling in an optical double well system''. Phys. Rev. Lett. 98, 263601 (2007). https://doi.org/10.1103/PhysRevLett.98.263601 [12] Georg Engelhardt, Gloria Platero, and Jianshu Cao. ``Discontinuities in driven spin-boson systems due to coherent destruction of tunneling: Breakdown of the Floquet-Gibbs distribution''. Phys. Rev. Lett. 123, 120602 (2019). https://doi.org/10.1103/PhysRevLett.123.120602 [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 [16] Michael Strass, Peter Hänggi, and Sigmund Kohler. ``Nonadiabatic electron pumping: Maximal current with minimal noise''. Phys. Rev. Lett. 95, 130601 (2005). https://doi.org/10.1103/PhysRevLett.95.130601 [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 [18] Oleh V. Ivakhnenko, Sergey N. Shevchenko, and Franco Nori. ``Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference''. Phys. Rep. 995, 1 (2023). https://doi.org/10.1016/j.physrep.2022.10.002 [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 [20] Vladimir V. Mangazeev, Murray T. Batchelor, and Vladimir V. Bazhanov. ``The hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 12LT01 (2021). https://doi.org/10.1088/1751-8121/abe426 [21] Cid Reyes-Bustos, Daniel Braak, and Masato Wakayama. ``Remarks on the hidden symmetry of the asymmetric quantum Rabi model''. J. Phys. A: Math. Theor. 54, 285202 (2021). https://doi.org/10.1088/1751-8121/ac0508 [22] Georg Engelhardt and Jianshu Cao. ``Dynamical symmetries and symmetry-protected selection rules in periodically driven quantum systems''. Phys. Rev. Lett. 126, 090601 (2021). https://doi.org/10.1103/PhysRevLett.126.090601 [23] Alexander Altland and Martin R. Zirnbauer. ``Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures''. Phys. Rev. B 55, 1142 (1997). https://doi.org/10.1103/PhysRevB.55.1142Cited by[1] O. A. Ilinskaya and S. N. Shevchenko, "Resonant excitation of single and coupled qubits for coherent quantum control and microwave detection", arXiv:2603.27610, (2026). [2] 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 SAO/NASA ADS (last updated successfully 2026-05-21 11:07:50). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-21 11:07:49: Could not fetch cited-by data for 10.22331/q-2026-05-21-2112 from Crossref. This is normal if the DOI was registered recently.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.