Generalized phase-space description of nonlinear Hamiltonian systems and Harper-like dynamics

Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian HW (q, p) constrained by the partial differential 2HW / partial differential q partial differential 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 HW (q, p) = K(p) +V (q).

Automated summary

Phase-space features of one-dimensional systems with a constrained Hamiltonian are obtained analytically using Wigner functions and currents. Profiles for thermodynamic and Gaussian ensembles are identified, and the results are specialized to the Harper Hamiltonian system. This generalized Wigner approach serves as a probe for quantumness and classicality of Harper-like systems, and it can be extended to any quantum system described by specific Hamiltonians.