{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T00:13:51Z","timestamp":1784160831497,"version":"3.55.0"},"reference-count":54,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T00:00:00Z","timestamp":1776643200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"The Poincar\u00e9 recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary&apos;s spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.<\/jats:p>","DOI":"10.22331\/q-2026-04-20-2074","type":"journal-article","created":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T09:19:38Z","timestamp":1776676778000},"page":"2074","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Quantum recurrences and the arithmetic of Floquet dynamics"],"prefix":"10.22331","volume":"10","author":[{"given":"Amit","family":"Anand","sequence":"first","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dinesh","family":"Valluri","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jack","family":"Davis","sequence":"additional","affiliation":[{"name":"QAT team, DIENS, \u00c9cole Normale Sup\u00e9rieure, PSL University, CNRS, INRIA, 45 rue d'Ulm, Paris 75005, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shohini","family":"Ghose","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5"},{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, Canada N2L 2Y5"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2026,4,20]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Henri Poincar\u00e9 ``Sur le probl\u00e8me des trois corps et les \u00e9quations de la dynamique'' Acta Mathematica 13, VII (1890).","DOI":"10.1007\/BF02392505"},{"key":"1","doi-asserted-by":"publisher","unstructured":"V Gimenoand J M Sotoca ``Upper bounds for the Poincar\u00e9 recurrence time in quantum mixed states'' Journal of Physics A Mathematical and Theoretical 50 (2017).","DOI":"10.1088\/1751-8121\/aa67fe"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Adam R. Brownand Leonard Susskind ``Second law of quantum complexity'' Physical Review D 97, 086015 (2018).","DOI":"10.1103\/PhysRevD.97.086015"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Jonathon Riddell, Nathan J. Pagliaroli, and \u00c1lvaro M. Alhambra, ``Concentration of quantum equilibration and an estimate of the recurrence time'' SciPost Phys. 15, 165 (2023).","DOI":"10.21468\/SciPostPhys.15.4.165"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Bernhard Rauer, Sebastian Erne, Thomas Schweigler, Federica Cataldini, Mohammadamin Tajik, and J\u00f6rg Schmiedmayer, ``Recurrences in an isolated quantum many-body system'' Science 360, 307\u2013310 (2018).","DOI":"10.1126\/science.aan7938"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Michael H. Freedman ``Quantum Detection of Recurrent Dynamics'' arXiv:2407.16055 (2024).","DOI":"10.48550\/arXiv.2407.16055"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Dominique Levesqueand Nicolas Sourlas ``Time Irreversibility in Statistical Mechanics'' Journal of Statistical Physics 192 (2025).","DOI":"10.1007\/s10955-025-03467-0"},{"key":"7","doi-asserted-by":"publisher","unstructured":"K Ropotenko ``The Poincar\u00e9 recurrence time for the de Sitter space with dynamical chaos'' arXiv:0712.0993 (2025).","DOI":"10.48550\/arXiv.0712.0993"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Marcin Kotowskiand Micha\u0142 Oszmaniec ``Tight bounds on recurrence time in closed quantum systems'' arXiv:2601.10409 (2026).","DOI":"10.48550\/arXiv.2601.10409"},{"key":"9","doi-asserted-by":"publisher","unstructured":"P Bocchieriand A. Loinger ``Quantum Recurrence Theorem'' Physical Review 107, 337\u2013338 (1957).","DOI":"10.1103\/PhysRev.107.337"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Lorenzo Campos Venuti ``The recurrence time in quantum mechanics'' arXiv:1509.04352 (2015).","DOI":"10.48550\/arXiv.1509.04352"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Adam Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M Preiss, and Markus Greiner, ``Quantum thermalization through entanglement in an isolated many-body system'' Science 353, 794\u2013800 (2016).","DOI":"10.1126\/science.aaf6725"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Shriya Paiand Michael Pretko ``Dynamical Scar States in Driven Fracton Systems'' Physical Review Letters 123, 136401 (2019).","DOI":"10.1103\/PhysRevLett.123.136401"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Bhaskar Mukherjee, Sourav Nandy, Arnab Sen, Diptiman Sen, and K. Sengupta, ``Collapse and revival of quantum many-body scars via Floquet engineering'' Physical Review B 101, 245107 (2020).","DOI":"10.1103\/PhysRevB.101.245107"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Kaoru Mizuta, Kazuaki Takasan, and Norio Kawakami, ``Exact Floquet quantum many-body scars under Rydberg blockade'' Physical Review Research 2, 033284 (2020).","DOI":"10.1103\/PhysRevResearch.2.033284"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Rahul Royand Fenner Harper ``Floquet topological phases with symmetry in all dimensions'' Physical Review B 95 (2017).","DOI":"10.1103\/PhysRevB.95.195128"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Krzysztof Sachaand Jakub Zakrzewski ``Time crystals: a review'' Reports on Progress in Physics 81, 016401 (2017).","DOI":"10.1088\/1361-6633\/aa8b38"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Vedika Khemani, Roderich Moessner, and S. L. Sondhi, ``A Brief History of Time Crystals'' arXiv:1910.10745 (2019).","DOI":"10.48550\/arXiv.1910.10745"},{"key":"18","doi-asserted-by":"publisher","unstructured":"F. M. Izrailevand D. L. Shepelyanskii ``Quantum resonance for a rotator in a nonlinear periodic field'' Theoretical and Mathematical Physics 43, 553\u00e2\u0080\u0093561 (1980).","DOI":"10.1007\/BF01029131"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Shmuel Fishman, D. R. Grempel, and R. E. Prange, ``Chaos, Quantum Recurrences, and Anderson Localization'' Physical Review Letters 49, 509\u2013512 (1982).","DOI":"10.1103\/PhysRevLett.49.509"},{"key":"20","doi-asserted-by":"publisher","unstructured":"G. Floquet ``Sur les \u00e9quations diff\u00e9rentielles lin\u00e9aires \u00e0 coefficients p\u00e9riodiques'' Annales scientifiques de l'\u00c9cole Normale Sup\u00e9rieure 12, 47\u201388 (1883).","DOI":"10.24033\/asens.220"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Milena Grifoniand Peter H\u00e4nggi ``Driven quantum tunneling'' Physics Reports 304, 229\u2013354 (1998).","DOI":"10.1016\/S0370-1573(98)00022-2"},{"key":"22","doi-asserted-by":"publisher","unstructured":"T. Hoggand B. A. Huberman ``Recurrence Phenomena in Quantum Dynamics'' Physical Review Letters 48, 711\u2013714 (1982).","DOI":"10.1103\/PhysRevLett.48.711"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Tanmoy Pandit, Alaina M Green, C Huerta Alderete, Norbert M Linke, and Raam Uzdin, ``Bounds on the recurrence probability in periodically-driven quantum systems'' Quantum 6, 682\u2013682 (2022).","DOI":"10.22331\/q-2022-04-06-682"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Micha\u0142 Oszmaniec, Marcin Kotowski, Micha\u0142 Horodecki, and Nicholas Hunter-Jones, ``Saturation and Recurrence of Quantum Complexity in Random Local Quantum Dynamics'' Physical Review X 14 (2024).","DOI":"10.1103\/PhysRevX.14.041068"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Amit Anand, Jack Davis, and Shohini Ghose, ``Quantum recurrences in the kicked top'' Physical Review Research 6, 023120 (2024).","DOI":"10.1103\/PhysRevResearch.6.023120"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Changyuan Lyu, Sayan Choudhury, Chenwei Lv, Yangqian Yan, and Qi Zhou, ``Eternal discrete time crystal beating the Heisenberg limit'' Phys. Rev. Res. 2, 033070 (2020).","DOI":"10.1103\/PhysRevResearch.2.033070"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Zhixing Zouand Jiao Wang ``Pseudoclassical Dynamics of the Kicked Top'' Entropy 24, 1092 (2022).","DOI":"10.3390\/e24081092"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Zhixing Zou, Jiangbin Gong, and Weitao Chen, ``Enhancing quantum metrology by quantum resonance dynamics'' Physical Review Letters 134 (2025).","DOI":"10.1103\/lkrt-lvng"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Hillol Biswasand Sayan Choudhury ``The Floquet central spin model: A platform to realize eternal time crystals, entanglement steering, and multiparameter metrology'' arXiv:2501.18472 (2025).","DOI":"10.48550\/arXiv.2501.18472"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Jens Bolte ``Some studies on arithmetical chaos in classical and qauntum mechanics'' International Journal of Modern Physics B 07, 4451\u20134553 (1993).","DOI":"10.1142\/S0217979293003759"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Eugene B Bogomolny, Bertrand Georgeot, M-J Giannoni, and Charles Schmit, ``Arithmetical chaos'' Physics Reports 291, 219\u2013324 (1997).","DOI":"10.1016\/S0370-1573(97)00016-1"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Jens Marklof ``Arithmetic quantum chaos'' Encyclopedia of Mathematical Physics 1, 212\u2013220 (2006).","DOI":"10.1016\/B0-12-512666-2\/00449-1"},{"key":"33","unstructured":"David S. Dummitand Richard M. Foote ``Abstract Algebra'' John Wiley & Sons (2003)."},{"key":"34","doi-asserted-by":"publisher","unstructured":"Fritz Haake, Marek Ku\u015b, and Rainer Scharf, ``Classical and quantum chaos for a kicked top'' Zeitschrift f\u00fcr Physik B Condensed Matter 65, 381\u2013395 (1987).","DOI":"10.1007\/BF01303727"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Joshua B. Ruebeck, Jie Lin, and Arjendu K. Pattanayak, ``Entanglement and its relationship to classical dynamics'' Physical Review E 95, 062222 (2017).","DOI":"10.1103\/PhysRevE.95.062222"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Udaysinh T. Bhosaleand M. S. Santhanam ``Periodicity of quantum correlations in the quantum kicked top'' Physical Review E 98, 052228 (2018).","DOI":"10.1103\/PhysRevE.98.052228"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Shruti Dogra, Vaibhav Madhok, and Arul Lakshminarayan, ``Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top'' Physical Review E 99, 062217 (2019).","DOI":"10.1103\/PhysRevE.99.062217"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Harshit Sharmaand Udaysinh T. Bhosale ``Exactly solvable dynamics and signatures of integrability in an infinite-range many-body Floquet spin system'' Physical Review B 109, 014412 (2024).","DOI":"10.1103\/PhysRevB.109.014412"},{"key":"39","doi-asserted-by":"publisher","unstructured":"Harshit Sharmaand Udaysinh T. Bhosale ``Exact Solvability Of Entanglement For Arbitrary Initial State in an Infinite-Range Floquet System'' Annals of Physics 486, 170327 (2026).","DOI":"10.1016\/j.aop.2025.170327"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Harshit Sharmaand Udaysinh T. Bhosale ``Signatures of quantum integrability and exactly solvable dynamics in an infinite-range many-body Floquet spin system'' Physical Review B 110, 064313 (2024).","DOI":"10.1103\/PhysRevB.110.064313"},{"key":"41","unstructured":"Meenu Kumari ``Quantum-Classical Correspondence and Entanglement in Periodically Driven Spin Systems'' University of Waterloo (2019)."},{"key":"42","doi-asserted-by":"publisher","unstructured":"L. C. Biedenharn, James D. Louck, and Peter A. Carruthers, ``Angular Momentum in Quantum Physics: Theory and Application'' Cambridge University Press (1984).","DOI":"10.1017\/CBO9780511759888"},{"key":"43","doi-asserted-by":"publisher","unstructured":"W. D\u00fcr, G. Vidal, and J. I. Cirac, ``Three qubits can be entangled in two inequivalent ways'' Physical Review A 62, 062314 (2000).","DOI":"10.1103\/PhysRevA.62.062314"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Aram W Harrow ``The church of the symmetric subspace'' arXiv:1308.6595 (2013).","DOI":"10.48550\/arXiv.1308.6595"},{"key":"45","doi-asserted-by":"publisher","unstructured":"O. Giraud, D. Braun, D. Baguette, T. Bastin, and J. Martin, ``Tensor Representation of Spin States'' Physical Review Letters 114, 080401 (2015).","DOI":"10.1103\/PhysRevLett.114.080401"},{"key":"46","doi-asserted-by":"publisher","unstructured":"F. T. Arecchi, Eric Courtens, Robert Gilmore, and Harry Thomas, ``Atomic Coherent States in Quantum Optics'' Physical Review A 6, 2211\u20132237 (1972).","DOI":"10.1103\/PhysRevA.6.2211"},{"key":"47","doi-asserted-by":"publisher","unstructured":"D. Baguette, T. Bastin, and J. Martin, ``Multiqubit symmetric states with maximally mixed one-qubit reductions'' Physical Review A 90, 032314 (2014).","DOI":"10.1103\/PhysRevA.90.032314"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Christoph Fleckensteinand Marin Bukov ``Prethermalization and thermalization in periodically driven many-body systems away from the high-frequency limit'' Physical Review B 103, L140302 (2021).","DOI":"10.1103\/PhysRevB.103.L140302"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Zhihang Liuand Chao Zheng ``Recurrence Theorem for Open Quantum Systems'' arXiv:2402.19143 (2024).","DOI":"10.48550\/arXiv.2402.19143"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens, ``Reference frames, superselection rules, and quantum information'' Rev. Mod. Phys. 79, 555\u2013609 (2007).","DOI":"10.1103\/RevModPhys.79.555"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Ronnie Kosloff ``Quantum Thermodynamics: A Dynamical Viewpoint'' Entropy 15, 2100\u20132128 (2013).","DOI":"10.3390\/e15062100"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Stefano Scopa, Gabriel T. Landi, and Dragi Karevski, ``Lindblad-Floquet description of finite-time quantum heat engines'' Physical Review A 97, 062121 (2018).","DOI":"10.1103\/PhysRevA.97.062121"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Serge Lang ``Algebra'' Springer (2002).","DOI":"10.1007\/978-1-4613-0041-0"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2026-04-20-2074\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T09:19:45Z","timestamp":1776676785000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2026-04-20-2074\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,20]]},"references-count":54,"URL":"https:\/\/doi.org\/10.22331\/q-2026-04-20-2074","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,20]]},"article-number":"2074"}}