Next Article in Journal
Entropic Dynamics in Neural Networks, the Renormalization Group and the Hamilton-Jacobi-Bellman Equation
Next Article in Special Issue
What Is so Special about Quantum Clicks?
Previous Article in Journal
Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network
Previous Article in Special Issue
Specifying the Unitary Evolution of a Qudit for a General Nonstationary Hamiltonian via the Generalized Gell-Mann Representation
Open AccessArticle

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

1
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, Ciudad de México 04510, Mexico
2
Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, 141700 Moscow Region, Russia
3
Lebedev Physical Institute, Leninskii Prospect 53, 119991 Moscow, Russia
4
Department of Physics, Tomsk State University, Lenin Avenue 36, 634050 Tomsk, Russia
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(5), 586; https://doi.org/10.3390/e22050586
Received: 27 April 2020 / Revised: 19 May 2020 / Accepted: 20 May 2020 / Published: 23 May 2020
(This article belongs to the Special Issue Quantum Probability and Randomness II)
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 Schrödinger 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. View Full-Text
Keywords: Gaussian states; integrals of motion; parametric processes; nonunitary evolution; quantization; invariant states; covariance matrix Gaussian states; integrals of motion; parametric processes; nonunitary evolution; quantization; invariant states; covariance matrix
Show Figures

Figure 1

MDPI and ACS Style

López-Saldívar, J.A.; Man’ko, M.A.; Man’ko, V.I. Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States. Entropy 2020, 22, 586.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop