Helicity conservation under quantum reconnection of vortex rings

Abstract

Here we show that under quantum reconnection, simulated by using the three-dimensional GrossPitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion and intrinsic twist of the reconnecting vortex rings.

Helicity within the vortex filament model

“…Therefore, whenever we refer to φ in what follows, we will assume this represents only this component of the phase. The direction of the constant phase (or velocity potential) of the complex wavefunction has also been proposed by others as a possible direction that can be used to define N for superfluid vortices161718. This choice for the vector N turns out to be equivalent to using the so-called Seifert framing.…”

“…In a microscopic model, such as that given by the Gross-Pitaevskii equation, they are made possible by the “quantum-pressure” term which acts to couple the incompressible and compressible modes of the system. Using the Gross-Pitaevskii model, the effect of reconnections on helicity has recently been analyzed in several papers1617181925.…”

Helicity, topology, and Kelvin waves in reconnecting quantum knots

“…However, numerical studies of Burgers-type vortices indicate that helicity is not conserved [20]. While experiments studying helicity in quantum flows have not been done yet, the recent experimental creation of quantum knots in a Bose-Einstein condensate in the laboratory [21] is a significant step in that direction.Recently, quantum flows have been used as a testbed for many of these ideas [17,19], as vorticity in a quantum flow is concentrated along vortex lines with quantized circulation, and as these vortex lines can reconnect without dissipation. However, the lack of a fluid-like definition of helicity for a quantum flow requires complex topological measurements of the linking and knottedness of vortex lines [17], or artificial filtering of the fields [19] to prevent spurious values of helicity resulting from the singularity near quantum vortices.…”

“…Recently, quantum flows have been used as a testbed for many of these ideas [17,19], as vorticity in a quantum flow is concentrated along vortex lines with quantized circulation, and as these vortex lines can reconnect without dissipation. However, the lack of a fluid-like definition of helicity for a quantum flow requires complex topological measurements of the linking and knottedness of vortex lines [17], or artificial filtering of the fields [19] to prevent spurious values of helicity resulting from the singularity near quantum vortices.…”

“…However, the lack of a fluid-like definition of helicity for a quantum flow requires complex topological measurements of the linking and knottedness of vortex lines [17], or artificial filtering of the fields [19] to prevent spurious values of helicity resulting from the singularity near quantum vortices. Moreover, helicity in quantum flows has an interest per se, as reconnection events in superfluid turbulence can excite Kelvin waves [22].…”