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Chaos An Interdisciplinary Journal of Nonlinear Science

Dynamical classic limit: Dissipative vs conservative systems

Gaspar Gonzalez, Angelo Plastino, A.M. Kowalski

Published Jan 1, 2023• 3 citations
Publisher

Abstract

We analyze the nonlinear dynamics of a quartic semiclassical system able to describe the interaction of matter with a field. We do it in both dissipative and conservative scenarios. In particular, we study the classical limit of both frameworks and compare the associated features. In the two environments, we heavily use a system's invariant, related to the Uncertainty Principle, that helps to determine how the dynamics tends to the pertinent classical limit. We exhibit the convergence to the classical limit and also verify that the Uncertainty Principle is complied with during the entire process, even in the presence of dissipation.

DOI
10.1063/5.0126040

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Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant Er. We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with Er we define the threshold above which chaos appears, and the interval during which both regimes co-exist.
We study the behavior of a nonlinear semiclassical system using Shannon entropy and two approaches to statistical complexity. These systems involve the interaction between classical variables (representing the environment) and quantum ones. Both conservative and dissipative regimes are explored. To calculate the information metrics, probability distributions are derived from the temporal evolution via the Bandt-Pompe permutation method. Additionally, we describe the classical limit in terms of a motion invariant linked to the uncertainty principle. Our analysis reveals three distinct regions, including a mesoscopic one, along with other notable findings.
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We argue for the universality of quantum theory using a dynamical consistency argument, within a specific Hamiltonian setting. We analyse two different types of coupling between simple quantum harmonic oscillators. Each illustrates an aspect of the free and interacting quantum fields and shows the inadequacy of semiclassical models. In particular, we establish that requiring the canonical algebra to be preserved under joint unitary dynamics rules out specific hybrid classical-quantum models. We apply our reasoning to the gravitational field in the linear regime, coupled to the quantised electromagnetic field and, separately, to quantised matter. We conclude with a comparison to DeWitt's analysis of quantum measurement, in which the apparatus, if classical, must be at least stochastic to preserve the Heisenberg Uncertainty Principle. We also note that stochastic models are inconsistent with the strict version of conservation principles, even if they comply with a probabilistic (on average) conservation.