Wigner Flow Reveals Topological Order in Quantum Phase Space Dynamics

Abstract

The behavior of classical mechanical systems is characterized by their phase portraits, the collections of their trajectories. Heisenberg's uncertainty principle precludes the existence of sharply defined trajectories, which is why traditionally only the time evolution of wave functions is studied in quantum dynamics. These studies are quite insensitive to the underlying structure of quantum phase space dynamics. We identify the flow that is the quantum analog of classical particle flow along phase portrait lines. It reveals hidden features of quantum dynamics and extra complexity. Being constrained by conserved flow winding numbers, it also reveals fundamental topological order in quantum dynamics that has so far gone unnoticed.

Topological origin of quantum mechanical vacuum transitions and tunneling

“…Our results reveal that the tunneling dynamics can be brought to stationary configurations that mimics the complete standstill known as coherent destruction of tunneling [20,21]. The evolution of both non-stationary and non-unitary quantum perturbations supported by non-equivalent topological profiles are described through the Wigner's quasi-probability distribution function [22].…”

Non-classicality from the phase-space flow analysis of the Weyl-Wigner quantum mechanics

“…Conditions that circumstantially imply that ∇ ξ · w = 0 are very helpful in identifying approximately Liouvillian-like trajectories in the phase-space [16].…”

“…Turning back to the fluid-analog framework, our analysis is particularly concerned to the quantum aspects of physical systems [16,17] , 19], that drives the flow of W (x, p; τ ) in the phase-space. For the flow field identified by the phase-space component directions, J = J xx + J pp , wherep =p x , the equivalent of the Schrödinger equation in phase-space can be written in terms of a continuity equation [5,7,16]:…”

“…For stationary states, one has ∂W/∂τ = 0, and Eq. (23) describes the way that classical paths are deformed by quantum effects that are introduced by local fluid perturbations associated to flow stagnation points [16].…”

“…For the flow field identified by the phase-space component directions, J = J xx + J pp , wherep =p x , the equivalent of the Schrödinger equation in phase-space can be written in terms of a continuity equation [5,7,16]:…”