Wigner Flow Reveals Topological Order in Quantum Phase Space Dynamics

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.

Figure 1

Flow field around various types of stagnation points of Wigner flow with associated winding numbers. This list is nonexhaustive.

Figure 2

Wigner function, Wigner flow, and momentum distribution of state $\Psi$ (parameters $\hslash =1$, $m=1/2$, $\alpha =0.5$, $\Delta E=0.5$). (a) Wigner function $W(x,p;T/4)$, projection to bottom shows its contours. Projections onto background walls show momentum and position probability distribution (blue and rose filled curves, respectively, in arbitrary units). Inset: Plot of Caticha-potential with wave functions for lowest two energy eigenstates shifted to their respective energy levels (dashed lines). (b) $x$ component of Wigner flow $J_x(X_S,p;t)$ at barrier top at position $X_S$; bottom projection shows its contours (note its phase shift at $t=T/2$). Projections onto background walls show time and momentum projections (blue and rose filled curves, $⟨J_x(X_S,p){⟩}_T={\int }_0^{T}d\tau J_x(X_S,p;\tau )$ and ${\widehat{ȷ}}_x(X_S;t)$ of Eq. (4), respectively, in arbitrary units). The projection ${\widehat{ȷ}}_x$ features the sinusoidal variation $\propto \sin (2\pi t/T)$ expected of the tunneling current of a two-state system.

Figure 3

Features of Wigner flow of an asymmetric double well potential (same parameters as in Fig. 1, at $t=T/4$). Contour plot of Wigner function $W(x,p;T/4)$ (black contour lines) with an overlay of colored arrows showing normalized Wigner flow $\mathit{J}/|\mathit{J}|$ (red arrows for $W>0$ and green for $W<0$). The classical phase portrait is shown as a collection of thick white lines. All locations of Wigner flow stagnation points are highlighted by symbols (cyan and red circles centre on clockwise and anti-clockwise vortices, respectively, yellow diamonds on separatrix intersections, and the blue square on a $p$-directed saddle flow. The quantum-displacement of the vortices near the potential minima, towards the center, is clearly visible.

Figure 4

Wigner flow’s stagnation points’ positions across phase space as a function of time. Same parameters as in Fig. 2. The tube surfaces show where in phase space the magnitude of Wigner flow is small $\mathit{J}(x,p;t{)}^2=3\times {10}^{-5}$. They are displayed over 120% of one oscillation period ($t=-0.1T,\dots ,1.1T$); the rainbow spectrum is matched to $T$, red-orange for $t=0$, via green, cyan at $t=0.5T$, through blue and purple back to red-orange. Because of the periodicity of the two-state scenario, the red-orange-yellow torus is seen twice at the beginning and end of the time window. At the core of all tubes lie time lines of stagnation points. Their movement through phase space leads to their mergers and splits. The gray flow integration lines are guiding the eye past vortices and separatrices, they do not represent physical flow since they integrate $\mathit{J}(x,p;t=0.05)$ at fixed time. The $x$ coordinate is shown from the left well minimum at $X_L=-2.095$ to the right well minimum at $X_R=1.514$, the position of the associated vortices ((L) in left well and $\circledR$ in right), at $p=0$ and just inside the plot region, confirms the inward quantum-displacement of these stagnation points. The remnant of the classical separatrix stagnation point at position $X_S+\delta x_S(t)\approx -0.3$ is labeled (S)$$. It follows a bent path and becomes displaced, forming part of the torus, when it coalesces with or splits from other stagnation points. All mergers or splits are constrained by topological charge [20] conservation; see Fig. 5.

Figure 5

Same system parameters as for Fig. 2: Contour plots of Wigner function with superimposed normalized flow field $\mathit{J}/|\mathit{J}|$ at times $t=0$, $0.075T$, $0.0875T$, $0.15T$, $0.2T$, and $0.325T$ in panels $\mathbf{a}$ to $\mathbf{f}$ demonstrates mergers and splitting of stagnation points and their pinning to zeros of the Wigner function. Each vertical black line shows the position $X_S$ of the tunnelling barrier top. Small white loops delineate areas where ${\mathit{J}}^2=2\times {10}^{-6}$ and encircle stagnation points. We chose the region where the torus closes and several stagnation points coalesce (compare Fig. 4). The cross sections of the torus are visible in the right half of panels (a), (b), and (c), a convex loop $\mathcal{L}$ encircling both torus points only, such as the dotted yellow ellipse, yields $\omega =0$. This value is conserved over time throughout the merger which leads to the disappearance of the torus [panel (d)]. The mergers of the four stagnation points outside the yellow loop (two vortices and two separatrix crossings) in panels (d), (e), and (f) into one vortex and one separatrix crossing evidently carry a conserved total topological charge of zero throughout.