Here we present supplementary online video material concerning the following publication:
Noel Cuadra,1, ∗ Alberto Villois,2,3 Thomas Gasenzer,1,4,5 and Davide Proment2,3,5
1Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
2School of Engineering, Mathematics and Physics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom
3Centre for Photonics and Quantum Science, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom
4Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
5ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung, Planckstrasse 1, 64291 Darmstadt, Germany
The nonlinear excitations underlying the onset of rotation in a dilute Bose–Einstein condensate confined to a thin spherical shell are studied. These excitations correspond to solitary waves rotating about the sphere at constant angular speed: at low speeds they appear as dipoles of singly quantized vortices with opposite circulation, while at higher speeds they evolve into vortex-free Jones–Roberts solitons. With further increase of the angular speed, these excitations hybridize with equatorially confined modes whose azimuthal wave number is set by the sphere radius measured in units of the healing length. The propagation speed of these modes is shown to play the role of a Landau critical velocity, thereby setting the upper limiting angular speed of the entire Jones–Roberts family.
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Movie showing the real time evolution of a Jones-Roberts vortex dipole; density field is at the top, argument (phase) field at the bottom. Since numerical time evolution inevitably introduces truncation and round-off errors, the persistence of the solitary wave throughout the simulation provides numerical test of its dynamical robustness.
The video shows the temporal evolution according to Eq. (2) using, as initial condition, the solitary wave depicted in Fig.1(b-c), for Ω = 0.03 μ/ℏ.
Movie showing the evolution of a Jones-Roberts (a) vortex dipole (b) density dip superimposed with a thermal distribution of randomly-excited Bogoliubov modes, see Supplemental Material/Appendices for more details; energy-momentum plot is at the top (blue dot is the real-time measure of the momentum and energy value during the simulation, the orange line identify the Jones-Roberts solitary wave branch), density field is at the middle, argument (phase) field at the bottom. The thermal noise is relatively strong, contributing around 12% to the solitary wave’s excitation energy (while leaving its momentum unchanged, as the sum of the thermal perturbations has zero net momentum). The vortex dipole appears to be remarkably long-lived, propagating with no appreciable changes in their structure for at least three full rotations around the z-axis, at which point the simulations are terminated.
The videos show the temporal evolution according to Eq. (2) using, as initial condition, solitary waves as depicted in Fig.2, for Ω = 0.08 μ/ℏ and Ω = 0.13 μ/ℏ respectively.
Movie showing the evolution and decay of an initial Jones-Roberts vortex dipole under the dissipative Gross-Pitaevskii equation with gamma=0.01, see Supplemental Material/Appendices for more details; energy-momentum plot is at the top (blue dot is the real-time measure of the momentum and energy value during the simulation, the orange line identify the Jones-Roberts solitary wave branch), density field is at the middle, argument (phase) field at the bottom. Due to dissipative effects, the initial vortex dipole looses energy and momentum, gradually transforming into a Jones-Roberts density dip. No appreciable sound emission is observed during the topological vortex to vortex-free transition.
The video shows the decay of a Jones-Roberts soliton according to Eq. (2) using, as initial condition, a solitary wave depicted in Fig. 2, with decay rate γ = 0.05.
Movie showing the real time evolution of a Jones-Roberts vortex dipole; density field is at the top, argument (phase) field at the bottom. Since numerical time evolution inevitably introduces truncation and round-off errors, the persistence of the solitary wave throughout the simulation provides numerical test of its dynamical robustness.
Movie showing the scattering of two Jones-Roberts vortex dipoles initially moving along the sphere's equator; density field is at the top, argument (phase) field at the bottom. After the first scattering the dipoles emit some sound and propagate towards the poles. They meet again and scatter at the equator: the sound emission prior to this scattering allows for the dipoles to transient into vortex-free density dips, which then scatter by passing one onto the other, continuing to move along the equator. Subsequent scatterings follow the same dynamics.
Movie showing the scattering of two Jones-Roberts density dips initially moving along the sphere's equator; density field is at the top, argument (phase) field at the bottom. After the first scattering the dips emit some sound scatter by passing one onto the other, continuing to move along the equator. Subsequent scatterings follow the same dynamics.
Movie showing the scattering of two small Jones-Robert vortex dipoles initially moving along the sphere's equator; energy-momentum plot is at the top (blue dot is the real-time measure of the momentum and energy value during the simulation, the orange line identify the Jones-Roberts solitary wave branch), density field is at the middle, argument (phase) field at the bottom. The sound emission prior to the first scattering allows for the dipoles to transient into vortex-free density dips, which then scatter by passing one onto the other, continuing to move along the equator. Subsequent scatterings follow the same dynamics.
Download Wolfram Mathematica Notebooks:
• RankinePoleVortex.nb: Wolfram Mathematica notebook showing the details of the calculations of angular momentum and kinetic energy of a fluid possessing the Rankine polar vortex solution, see Supplemental Material/Appendices for more details.
• LandauAngularSpeed.nb: Wolfram Mathematica notebook showing the details of the calculations of the Landau critical value lLandau and lim R → ∞, see Supplemental Material/Appendices for more details.
