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This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called \u201cefficient basis\u201d over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.<\/jats:p>","DOI":"10.3390\/e25010008","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T02:59:06Z","timestamp":1671677946000},"page":"8","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Chaos and Thermalization in the Spin-Boson Dicke Model"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4012-6826","authenticated-orcid":false,"given":"David","family":"Villase\u00f1or","sequence":"first","affiliation":[{"name":"Instituto de Ciencias Nucleares, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Apdo. Postal 70-543, Mexico City 04510, Mexico"},{"name":"Instituto de Investigaciones en Matem\u00e1ticas Aplicadas y en Sistemas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Mexico City 04510, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9843-6491","authenticated-orcid":false,"given":"Sa\u00fal","family":"Pilatowsky-Cameo","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1552-4101","authenticated-orcid":false,"given":"Miguel A.","family":"Bastarrachea-Magnani","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad Aut\u00f3noma Metropolitana-Iztapalapa, Av. Ferrocarril San Rafael Atlixco 186, Mexico City 09340, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5289-7698","authenticated-orcid":false,"given":"Sergio","family":"Lerma-Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Facultad de F\u00edsica, Universidad Veracruzana, Circuito Aguirre Beltr\u00e1n s\/n, Xalapa 91000, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9400-2709","authenticated-orcid":false,"given":"Lea F.","family":"Santos","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Connecticut, Storrs, CT 06269, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2170-9903","authenticated-orcid":false,"given":"Jorge G.","family":"Hirsch","sequence":"additional","affiliation":[{"name":"Instituto de Ciencias Nucleares, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Apdo. Postal 70-543, Mexico City 04510, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/BF01339852","article-title":"Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik","volume":"57","year":"1929","journal-title":"Zeitschrift f\u00fcr Physik"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1140\/epjh\/e2010-00008-5","article-title":"Proof of the ergodic theorem and the H-theorem in quantum mechanics","volume":"35","year":"2010","journal-title":"Eur. Phys. J. H"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1140\/epjh\/e2010-00007-7","article-title":"Long-time behavior of macroscopic quantum systems","volume":"35","author":"Goldstein","year":"2010","journal-title":"Eur. Phys. J. 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