Abstract
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 . 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.
Highlights
The study of Gaussian states has been of essential interest in the last few decades
The problems of the new developments of the foundations of quantum mechanics and applications of new results in quantum information and quantum probabilities, as well as in areas like mathematical finance and economics have attracted the attention of the researchers; they are intensely discussed in the literature [3,4,5]
There exists increasing interest in quantum foundations since a deeper understanding of the essence and formalism of quantum theory is needed for the development of quantum technologies and the possibilities to extend the applications of quantum formalism in physics to all other areas of science like the economy, finance, and social disciplines
Summary
The study of Gaussian states has been of essential interest in the last few decades. These types of states, associated with classical random fields, were considered as a possibility to connect covariance matrices of the states as quantum density matrices and, with this definition, to study the quantum–classical relation of randomness with the quantization procedure [1,2]. To obtain this evolution, we make use of the derivatives of the covariance matrix, the mean values, and the parameters of the density operator; we define and obtain invariant states for this system. We make use of the derivatives of the covariance matrix, the mean values, and the parameters of the density operator; we define and obtain invariant states for this system It is known that there exist invariants linearly dt depending on the quadrature operators pand q, i.e., R(t) = λ1 (t) p + λ2 (t)q + λ3 (t) By substituting this expression into the von Neumann equation, which determines the dynamics of R, d R(t) i.
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