Geometrical structure of the Wigner flow information quantifiers and hyperbolic stability in the phase-space framework

TL;DR

This work develops a geometric, phase-space approach to quantum dynamics by embedding the Weyl-Wigner formulation within a differential-geometric framework of Wigner currents. By defining and linking stationarity, classicality, purity, and vorticity to hyperbolic stability properties, it provides a concrete method to quantify how quantum fluctuations modify nonlinear phase-space dynamics, including explicit results for Gaussian ensembles and Harper-like systems. The approach yields tractable expressions for the quantum corrections to stationarity and stability and demonstrates how quantum vortices modulate fixed-point structure, offering a pathway to differentiate quantum- from classical-driven behavior in complex systems. The findings have implications for quantum control, chaos, and topological aspects of quantum phase-space flows, and point toward extensions to Husimi and other representations and to more general Hamiltonians and hybrid discrete-continuous systems.

Abstract

Quantifiers of stationarity, classicality, purity and vorticity are derived from phase-space differential geometrical structures within the Weyl-Wigner framework, after which they are related to the hyperbolic stability of classical and quantum-modified Hamiltonian (non-linear) equations of motion. By examining the equilibrium regime produced by such an autonomous system of ordinary differential equations, a correspondence between Wigner flow properties and hyperbolic stability boundaries in the phase-space is identified. Explicit analytical expressions for equilibrium-stability parameters are obtained for quantum Gaussian ensembles, wherein information quantifiers driven by Wigner currents are identified. Illustrated by an application to a Harper-like system, the results provide a self-contained analysis for identifying the influence of quantum fluctuations associated to the emergence of phase-space vorticity in order to quantify equilibrium and stability properties of Hamiltonian non-linear dynamics.

Figures (4)

• Figure 1: (Color online) Classical portrait of Harper-like lattice. Phase-space trajectories and corresponding lattice designs are for $\hbox{Max}\{\nu^2-1,0\} < \vert\epsilon\vert < \nu^2+1$ corresponding to closed trajectories for $\epsilon > 0$ (black dashed lines), and for $\epsilon < 0$ (red thin lines), and with $0 < \vert\epsilon\vert < \nu^2-1$ corresponding to opened trajectories (blue thick lines), when they exist. The limiar (opened-closed) value is given by $\vert\epsilon\vert = \nu^2 -1$. The plots are for $\nu^2=2$ (first plot), with $\vert\epsilon\vert = 5/2,\, 2,\,3/2,\,\dots,\,0$, $\nu^2=1$ (second plot), with $\vert\epsilon\vert = 5/4,\, 1,\,3/4,\,\dots,\,0$ and $\nu^2=1/2$ (third plot), with $\vert\epsilon\vert = 5/4,\, 1,\,3/4,\,\dots,\,0$. Plots are similar to those from Ref. 2021A, with the classical energy identified by $\mathcal{H}_H \to \epsilon$, and with arbitrary values for phenomenological parameter, $\nu$.
• Figure 2: (Color online) Wigner flow quantifier pattern for Gaussian ensembles in the $x - k$ plane. First column: Stationarity quantifier, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \hbox{\boldmath $\mathcal{J}$}$, described according to the background color scheme. The results are for the increasing spreading characteristic of the Gaussian function, from $\gamma =1/4$ (first row), $1/2$ (second row) and $1$ (third row). Peaked Gaussian distributions ($\gamma =1$ in ) localizes the quantum distortions which result into non-stationarity. Second column: Liouvillian quantifier, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w}$, depicted through the background color scheme, from darker regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} \sim 0$, to lighter regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} > 0$. Third column: Circulation quantifier, $\Sigma$, for Gaussian ensembles which do not exhibit neither vortices nor stagnation points, in a kind of camouflage of the quantum distortions. The classical pattern is shown as a collection of black lines.
• Figure 3: (Color online) Velocity vector field pattern, $\mathbf{w}^{\Theta}$, evolving terms of $\Theta$-rotations, as parameterized by the canonical transformations from Eqs. \ref{['ads1']}-\ref{['ads2']}, from a divergenceless regime at equilibrium points ($\Theta = 0$) to an almost completely irrotational regime at $\Theta = 7\pi/16$ in the $x - k$ plane (with $\Theta = n \pi/16, with n=0,\,1,\,\dots,\,7$ from left to right, and from top to bottom). The classical pattern is shown as a collection of black lines. Results are for $\gamma = 1/4$ -- a choice which qualitatively does not affect the above interpretation
• Figure 4: (Color online) $\Theta$-rotational invariant quantifier, $Inv(x,\,k)$, for quantum ($\alpha = 1/4$, first column) and classical ($\alpha \to 0$, second column) for isotropic ($\nu=1$, first row) and anisotropic ($\nu=0.8$, second row) regimes. The classical pattern is shown as a collection of black lines in both plots.