Hydrodynamic View of Wave-Packet Interference: Quantum Caves

Wave-packet interference is investigated within the complex quantum Hamilton-Jacobi formalism using a hydrodynamic description. Quantum interference leads to the formation of the topological structure of quantum caves in space-time Argand plots. These caves consist of the vortical and stagnation tubes originating from the isosurfaces of the amplitude of the wave function and its first derivative. Complex quantum trajectories display counterclockwise helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. The string of alternating stagnation and vortical tubes is sufficient to generate divergent trajectories. Moreover, the average wrapping time for trajectories and the rotational rate of the nodal line in the complex plane can be used to define the lifetime for interference features.

Figure 1

Quantum caves for head-on collision of two Gaussian wave packets. These caves are formed with the isosurfaces $|\Psi (z,t)|=0.053$ (pink/lighter gray sheets) and $|\partial \Psi (z,t)/\partial z|=0.106$ (violet/darker gray sheets). The complex quantum trajectories launched from two branches of the isochrone reach the real axis at $t=5$.

Figure 2

(a) Trajectories 1, 2, 3, and 4 launched from the isochrone which arrive on the real axis at $t=5$ display hyperbolic deflection around the vortical curves (pink/light gray) and helical wrapping around the stagnation curves (violet/dark gray). Trajectories 4 and 5 diverge at position “$V$” near the vortical curve. (b) Divergence and vorticity of the QMF along trajectory 1.

Figure 3

Evolution of the nodal line at $t=0$, 2.5, 5, 7.5, 10 (black solid lines) and $t=\infty$ (black dashed line); the arrows indicate the rotation direction. Nodes and stagnation points are denoted by dots; nodal trajectories are shown as dotted lines passing through the nodal points.