Generalized phase-space description of non-linear Hamiltonian systems and the Harper-like dynamics

TL;DR

This paper develops an analytical phase-space framework using Wigner functions to study non-linear Hamiltonian systems, focusing on Harper-like dynamics, quantum-classical transition, and quantum fluctuations.

Contribution

It introduces a generalized Wigner flow approach for non-linear Hamiltonian systems, specifically applied to Harper-like models, to analyze quantum modifications and classicality.

Findings

Analytical expressions for Wigner flow in constrained Hamiltonian systems.

Quantification of quantum fluctuations over classical regimes.

Framework applicable to a broad class of Hamiltonian systems.

Abstract

Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system which, besides working as a feasible test platform for the framework here introduced, admits a statistical description in terms of TD and Gaussian ensembles, where the Wigner flow properties are all obtained through analytical tools. Quantum fluctuations over the classical regime are therefore quantified through probability and information fluxes whenever the classical Hamiltonian background is provided. Besides allowing for a broad range of theoretical applications, our results suggest that such a generalized Wigner approach works as a probe for quantumness and classicality of Harper-like systems, in a framework which can be extended to any quantum system described by Hamiltonians in the form of $H^{W}(q,\,p) = K(p) + V(q)$.