{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:50:16Z","timestamp":1760244616876,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T00:00:00Z","timestamp":1671494400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the evolution of the energy of a harmonic oscillator when its frequency slowly varies with time and passes through a zero value. We consider both the classical and quantum descriptions of the system. We show that after a single frequency passage through a zero value, the famous adiabatic invariant ratio of energy to frequency (which does not hold for a zero frequency) is reestablished again, but with the proportionality coefficient dependent on the initial state. The dependence on the initial state disappears after averaging over the phases of initial states with the same energy (in particular, for the initial vacuum, the Fock and thermal quantum states). In this case, the mean proportionality coefficient is always greater than unity. The concrete value of the mean proportionality coefficient depends on the power index of the frequency dependence on a time near the zero point. In particular, the mean energy triplicates if the frequency tends to zero linearly. If the frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of time intervals between zero points, so that the mean energy behavior becomes quasi-stochastic after many passages through a zero value. The original Born\u2013Fock theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many Fock states, whose weights do not depend on time in the new adiabatic regime. When the mean energy triplicates, the initial Nth Fock state becomes a superposition of, roughly speaking, 6N states, distributed nonuniformly. The initial vacuum and low-order Fock states become squeezed, as well as the initial thermal states with low values of the mean energy.<\/jats:p>","DOI":"10.3390\/e25010002","type":"journal-article","created":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T03:56:08Z","timestamp":1671508568000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adiabatic Amplification of the Harmonic Oscillator Energy When the Frequency Passes through Zero"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7599-209X","authenticated-orcid":false,"given":"Viktor V.","family":"Dodonov","sequence":"first","affiliation":[{"name":"Institute of Physics, University of Brasilia, P.O. Box 04455, Brasilia 70919-970, DF, Brazil"},{"name":"International Center of Physics, University of Brasilia, Brasilia 70910-900, DF, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7142-7453","authenticated-orcid":false,"given":"Alexandre V.","family":"Dodonov","sequence":"additional","affiliation":[{"name":"Institute of Physics, University of Brasilia, P.O. Box 04455, Brasilia 70919-970, DF, Brazil"},{"name":"International Center of Physics, University of Brasilia, Brasilia 70910-900, DF, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,20]]},"reference":[{"key":"ref_1","unstructured":"Landau, L.D., and Lifschitz, E.M. (1966). Mechanics, Mir."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1103\/PhysRev.106.205","article-title":"Adiabatic invariant of the harmonic oscillator","volume":"106","author":"Kulsrud","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1515\/zna-1966-0603","article-title":"The variation of the adiabatic invariant of the harmonic oscillator","volume":"21","author":"Knorr","year":"1966","journal-title":"Z. Naturforschung A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1922","DOI":"10.1063\/1.1664783","article-title":"Quantum harmonic oscillator with time-dependent frequency","volume":"10","author":"Solimeno","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1063\/1.1666360","article-title":"Linear adiabatic invariants and coherent states","volume":"14","author":"Malkin","year":"1973","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0003-4916(91)90243-2","article-title":"Changes in adiabatic invariants","volume":"205","author":"Keller","year":"1991","journal-title":"Ann. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L35","DOI":"10.1088\/0305-4470\/39\/1\/L05","article-title":"Exact analysis of adiabatic invariants in the time-dependent harmonic oscillator","volume":"39","author":"Robnik","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s11080-006-8222-0","article-title":"Energy evolution in time-dependent harmonic oscillator","volume":"13","author":"Robnik","year":"2006","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3011","DOI":"10.1103\/PhysRevA.49.3011","article-title":"Theory of adiabatic cooling in cavities","volume":"49","author":"Zaugg","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1088\/0031-8949\/55\/5\/005","article-title":"On wave-packet dynamics in a decaying quadratic potential","volume":"55","author":"Henriksen","year":"1997","journal-title":"Phys. Scripta"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1143\/ptp\/9.4.381","article-title":"Miscellanea in elementary quantum mechanics. II","volume":"9","author":"Husimi","year":"1953","journal-title":"Prog. Theor. Phys."},{"key":"ref_12","first-page":"1375","article-title":"Parametric excitation of a quantum oscillator","volume":"56","author":"Popov","year":"1969","journal-title":"Zhurn. Eksp. Teor. Fiz."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1458","DOI":"10.1063\/1.1664991","article-title":"An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field","volume":"10","author":"Lewis","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1103\/PhysRevD.2.1371","article-title":"Coherent states and transition probabilities in a time-dependent electromagnetic field","volume":"2","author":"Malkin","year":"1970","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1016\/j.aop.2010.02.020","article-title":"Quantum integrals of motion for variable quadratic Hamiltonians","volume":"325","author":"Suazo","year":"2010","journal-title":"Ann. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8401","DOI":"10.1038\/s41598-018-26759-w","article-title":"KvN mechanics approach to the time-dependent frequency harmonic oscillator","volume":"8","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1976","DOI":"10.1063\/1.1664532","article-title":"Class of exact invariants for classical and quantum time-dependent harmonic oscillators","volume":"9","author":"Lewis","year":"1968","journal-title":"J. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3927","DOI":"10.1088\/0305-4470\/27\/11\/039","article-title":"A class of exactly solved time-dependent quantum harmonic oscillators","volume":"27","author":"Kim","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_19","unstructured":"Gradshtein, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Academic. [7th ed.]."},{"key":"ref_20","unstructured":"Erd\u00e9lyi, A. (1953). Bateman Manuscript Project: Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1103\/PhysRev.35.1303","article-title":"The penetration of a potential barrier by electrons","volume":"35","author":"Eckart","year":"1930","journal-title":"Phys. Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1073\/pnas.16.10.627","article-title":"Reflection of waves in an inhomogeneous absorbing medium","volume":"16","author":"Epstein","year":"1930","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/BF01343193","article-title":"Beweis des Adiabatensatzes","volume":"51","author":"Born","year":"1928","journal-title":"Z. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1364\/OL.4.000205","article-title":"Sub-Poissonian photon statistics in resonance fluorescence","volume":"4","author":"Mandel","year":"1979","journal-title":"Opt. Lett."},{"key":"ref_25","first-page":"103","article-title":"Invariants and correlated states of nonstationary quantum systems","volume":"Volume 183","author":"Markov","year":"1989","journal-title":"Invariants and the Evolution of Nonstationary Quantum Systems"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/0030-4018(88)90322-7","article-title":"Principal squeezing of vacuum fluctuations","volume":"67","year":"1988","journal-title":"Opt. Commun."},{"key":"ref_27","first-page":"713","article-title":"Principal squeezing","volume":"74","author":"Hradil","year":"1988","journal-title":"Acta Phys. Polon. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/0375-9601(94)90444-8","article-title":"Geometrical squeezed states of a charged particle in a time-dependent magnetic field","volume":"188","author":"Dodonov","year":"1994","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"7721","DOI":"10.1088\/0305-4470\/33\/43\/305","article-title":"Universal integrals of motion and universal invariants of quantum systems","volume":"33","author":"Dodonov","year":"2000","journal-title":"J. Phys. A Math. Gen."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/2\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:44:28Z","timestamp":1760147068000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,20]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010002"],"URL":"https:\/\/doi.org\/10.3390\/e25010002","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,12,20]]}}}