Defect production by pure phase twist injection as Aharonov-Bohm effect

In this paper we demonstrate that new phase defects of the Gross-Pitaevskii equation (GPE) can be produced as a Aharonov-Bohm effect resulting from pure phase twist injection on existing defects. This is a phenomenon that has physical justification in the hydrodynamic interpretation of GPE. Here we give an analytical proof of its effects by using Fermi-Walker transport and Biot-Savart induction law. An analytical derivation of the dispersion relation is derived from the superposition of phase twist on the fundamental state. Since the extra twist corresponds to a topological change of the total linking number of the system, we show that the production of new defects is just another manifestation of the Aharonov-Bohm effect. We propose a laboratory experiment for Bose-Einstein condensates to test this phenomenon and to show that it can have useful applications in science and technology.

Figure 1

(a) Initial condition at $t=0$ given by a planar vortex ring visualized by the isodensity tubular surface $\rho =0.1$, the arrow indicating vorticity direction. (b) Twist $Tw=1$ is superposed by prescribing a full rotation of an isophase $\overline{\chi }$ as shown by the phase contour in the $(y,z)$ plane. (c) The presence of twist induces an instantaneous production of a new, central defect, here shown at time $t=20$ (adapted from Ref. [2]).

Figure 2

(a) Uniform twist visualized by the uniform rotation of the ribbon spanwise unit vector $\widehat{\mathbf{U}}$ around the ribbon straight axis. (b) Frenet frame $\{\widehat{\mathbf{T}},\widehat{\mathbf{N}},\widehat{\mathbf{B}}\}$ on a base space curve (thick, black line); (c) parallel transport frame $\{\widehat{\mathbf{T}},\widehat{\mathbf{P}},\widehat{\mathbf{Q}}\}$; (d) phase twist frame $\{\widehat{\mathbf{T}},\widehat{\mathbf{U}},\widehat{\mathbf{V}}\}$.

Figure 3

Hydrodynamic interpretation of the phase twist on a vortex ring. (a) Generation of a secondary, central flow field without singularity by the action of a pure azimuthal velocity ${\mathbf{u}}_{\theta }$ on the vortex ring; (b) generation of a new singularity along the ring axis by the combined action of an azimuthal and an axial velocity field ${\mathbf{u}}_{\theta }$ and ${\mathbf{u}}_{\xi }$ along the ring centreline.