Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory

We provide a general theory for the structure of the quantum flow near 3-d nodal lines, i.e. one-dimensional loci where the 3-d wavefunction becomes equal to zero. In suitably defined co- ordinates (co-moving with the nodal line) the generic structure of the flow implies the formation of 3-d quantum vortices. We show that such vortices are accompanied by nearby invariant lines of the co-moving quantum flow, called X-lines, which are normally hyperbolic. Furthermore, the stable and unstable manifolds of the X-lines produce chaotic scatterings of nearby quantum (Bohmian) trajectories, thus inducing an intricate form of the quantum current in the neighborhood of each 3-d quantum vortex. Generic formulas describing the structure around 3-d quantum vortices are provided, applicable to an arbitrary choice of 3-d wavefunction. We also give specific numerical examples, as well as a discussion of the physical consequences of chaos near 3-d quantum vortices.