{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T01:23:01Z","timestamp":1772155381790,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,23]],"date-time":"2020-05-23T00:00:00Z","timestamp":1590192000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006087","name":"Direcci\u00f3n General de Asuntos del Personal Acad\u00e9mico, Universidad Nacional Aut\u00f3noma de M\u00e9xico","doi-asserted-by":"publisher","award":["IN101619"],"award-info":[{"award-number":["IN101619"]}],"id":[{"id":"10.13039\/501100006087","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian     H ^    . We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schr\u00f6dinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.<\/jats:p>","DOI":"10.3390\/e22050586","type":"journal-article","created":{"date-parts":[[2020,5,25]],"date-time":"2020-05-25T06:43:40Z","timestamp":1590389020000},"page":"586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5489-698X","authenticated-orcid":false,"given":"Julio","family":"L\u00f3pez-Sald\u00edvar","sequence":"first","affiliation":[{"name":"Instituto de Ciencias Nucleares, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Apdo. Postal 70-543, Ciudad de M\u00e9xico 04510, Mexico"},{"name":"Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, 141700 Moscow Region, Russia"}]},{"given":"Margarita","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, 119991 Moscow, Russia"}]},{"given":"Vladimir","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, 141700 Moscow Region, Russia"},{"name":"Lebedev Physical Institute, Leninskii Prospect 53, 119991 Moscow, Russia"},{"name":"Department of Physics, Tomsk State University, Lenin Avenue 36, 634050 Tomsk, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2009). 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