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This is particularly true for quantum mechanical systems. We demonstrate here how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum system, one that can be viewed as a natural quantum analog of several previously studied classical models. In particular, we consider an optical mode passing through a reservoir composed of a large number of sequentially-encountered modes of the same frequency, each of which it interacts with through a beam splitter. We then analyze the dependence of the asymptotic state of this mode on the assumed stationary common initial state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03C3;<\/mml:mi><\/mml:math> of the reservoir modes and on the transmittance <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03C4;<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>cos<\/mml:mi><mml:mo>&amp;#x2061;<\/mml:mo><mml:mi>&amp;#x03BB;<\/mml:mi><\/mml:math> of the beam splitters. These results allow us to establish that at small <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03BB;<\/mml:mi><\/mml:math> such a mode will, starting from an arbitrary initial system state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03C1;<\/mml:mi><\/mml:math>, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized. We show in addition that, when the initial states are pure, the asymptotic state of the optical mode is maximally entangled with the reservoir and exhibits less nonclassicality than the state of the reservoir modes.<\/jats:p>","DOI":"10.22331\/q-2024-05-23-1360","type":"journal-article","created":{"date-parts":[[2024,5,23]],"date-time":"2024-05-23T15:50:32Z","timestamp":1716479432000},"page":"1360","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Rigorous results on approach to thermal equilibrium, entanglement, and nonclassicality of an optical quantum field mode scattering from the elements of a non-equilibrium quantum reservoir"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2445-2701","authenticated-orcid":false,"given":"Stephan","family":"De Bievre","sequence":"first","affiliation":[{"name":"Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire P. Painlev\u00e9, F-59000 Lille, France"}]},{"given":"Marco","family":"Merkli","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John&apos;s, NL, A1C 5S7, Canada"}]},{"given":"Paul E.","family":"Parris","sequence":"additional","affiliation":[{"name":"Missouri University of Science and Technology, Rolla, Missouri, 65409, USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,5,23]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Joshua M. Deutsch. Eigenstate thermalization hypothesis. Rep. Progr. Phys., 81(8):082001, 16, 2018. URL: https:\/\/doi.org\/10.1088\/1361-6633\/aac9f1, doi:10.1088\/1361-6633\/aac9f1.","DOI":"10.1088\/1361-6633\/aac9f1"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Joel L. Lebowitz. Boltzmann&apos;s Entropy and Time&apos;s Arrow. 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