A geometric view of dynamics in quantum phase space
Classical phase space dynamics is governed by a continuity equation. The same is true for quantum dynamics in phase space: Wigner's quantum phase space current, J, governs the evolution of Wigner's phase space distribution W.
Unlike the classical case, W is typically negative in some regions. These negative regions represent quantum coherences.
Also, the current's velocity field is ill-defined. It is singular when W = 0. shows that this implies that there are no trajectories in quantum phase space.
With these velocity singularities Liouvillian phase space volumes feature singular changes too. It is demonstrated that this is necessary in order for quantum dynamics to create coherences. This is worth knowing particularly for numerical investigations.
Stationary points of J are important for both, classical and quantum dynamics, maybe even more so in the quantum case, since the dynamics can move and split or merge these stagnation points. But the existence of stagnation points is constrained by a topological conservation law which quantum mechanics has to obey (see Figure).
While velocity fields and trajectories are ill-defined in quantum phase space, it is demonstrated that the current J is always well behaved. It can be studied, providing new insights, and it can be numerically integrated to study the dynamics.
One may wonder how quantum dynamics suppresses the formation of very fine detail in phase space over long evolution times? It turns out that Wigner's current J is 'viscous'.
Steuernagel describes the mechanism that lies behind this observation.
This 'viscosity' induces a characteristic polarisation pattern in quantum phase space that quantifies quantum dynamics' detail suppression. Used as a measure, it singles out special states: Wigner current can be used as a sensitive probe.
The video illustrates topological charge conservation in the system that were studied.