Here we present supplementary online video material concerning the following publication:
Niklas Rasch,1, ∗ Lauriane Chomaz,2 and Thomas Gasenzer1, 3
1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
2Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
3Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
The emergence of distinctly sub-diffusive scaling in the vicinity of an anomalous non-thermal fixed point is discussed in a quasi-two-dimensional dipolar Bose gas in the superfluid phase, carrying ensembles of vortices and antivortices with zero net angular momentum. The observed scaling behavior reflects coarsening dynam- ics driven by the mutual annihilation of vortices and antivortices, with the mean inter-defect distance growing algebraically over time as lv(t) ∼ tβ. A sub-diffusive (β < 1/2) exponent β ≈ 0.2 is extracted for various pa- rameter regimes, initial conditions, and dipolar configurations from both scaling occupation-number spectra and the evolution of inter-defect distances as well as the corresponding total vortex densities. As vortex-antivortex annihilation progresses, excitations of the background condensate increase. This gives rise to a transition in the scaling behavior at late times, toward a non-thermal fixed point governed by diffusion-type scaling with β ≈ 1/2 as expected for the mutual annihilation of well-separated vortex-antivortex dipoles. While the temporal scaling with β does not depend significantly on the strength and anisotropy of the dipolar interactions and thus under- lines the universality of the anomalous as well as diffusion-type non-thermal fixed points, we find distinctly different vortex patterns resulting in the dipolar case. While in the superfluid with contact interactions only, same-sign vortices tend to cluster and form large-scale eddies, in the dipolar and tilted cases, roton excitations appear to prevent such motion, giving rather rise to a maximisation of distances between vortices of either sign.
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All videos by Niklas Rasch.
The video shows the approach of an ultradilute 2D dipolar Bose gas with relative dipolar to contact interaction strength εdd = 0.5 towards the anomalous non-thermal fixed point, starting from an 8 x 8 lattice of non-elementary vortices with winding number |w|=6. The anomalous fixed point implies subdiffusive coarsening with a temporal power-law growth of the characteristic length scale as lv ~ t1/5, which is a measure for the mean interdefect distance. The mean vortex number decreases, accordingly, as Nv ~ t –2/5. Snapshots of the run are shown in Fig. 1 in the article. The run belongs to the data set, for which spectra are shown in Fig. 2a, the time evolution of the exponents in Fig. 9a.
The video shows the evolution of an ultradilute 2D dipolar Bose gas with relative dipolar to contact interaction strength εdd = 1.5 and a tilting of the dipole moment by θ = π/4 towards the anomalous non-thermal fixed point, starting from 1000 randomly distributed elementary (anti)vortices with winding number |w| = 1 and net zero winding number. The anomalous fixed point implies subdiffusive coarsening with a temporal power-law growth of the interdefect distance scale as lv ~ t1/5 and decay of the mean vortex number as Nv ~ t –2/5. The run belongs to the data set, for which the time evolution of the exponents is shown in Fig. 3b,e (red lines), the scaling evolution of lv and Nv in Fig. 4b,e, and the clustering evolution in Fig. 7b,e.
The video shows the evolution of an 2D dipolar Bose gas in the `quantum' regime, i.e., with experimentally realistic diluteness, with relative dipolar to contact interaction strength εdd = 1.47 and no tilting of the dipole moment towards the anomalous non-thermal fixed point, starting from 1000 randomly distributed elementary (anti)vortices with winding number |w| = 1 and net zero winding number. The anomalous fixed point implies subdiffusive coarsening with a temporal power-law growth of the interdefect distance scale as lv ~ t1/5 and decay of the mean vortex number as Nv ~ t –2/5. The run belongs to the data set, for which snapshots of the single-particle spectrum are shown in Fig. 2b, the time evolution of the exponents is shown in Fig. 3c,f (blue lines), the scaling evolution of lv and Nv in Fig. 4c,f, the clustering evolution in Fig. 7c,f, and the anisotropy in Fig. 10a,b. A snapshot of clusters is shown in Fig. 11c.
The video shows the evolution of an 2D dipolar Bose gas in the `quantum' regime, i.e., with experimentally realistic diluteness, with relative dipolar to contact interaction strength εdd = 1.47 and a tilting of the dipole moment by θ = π/4 towards the anomalous non-thermal fixed point, starting from 1000 randomly distributed elementary (anti)vortices with winding number |w| = 1 and net zero winding number. The tilting leads to asymmetric vortex excitations, which give rise to a striped pattern of strong roton modulations on the top of the superfluid density, at later times centred around the vortices remaining. The run belongs to the data set, for which the time evolution of the exponents is shown in Fig. 3c,f (red lines), the scaling evolution of lv and Nv in Fig. 4c,f, the clustering evolution in Fig. 7c,f, and the anisotropy in Fig. 10a,b. A snapshot of clusters is shown in Fig. 11d. Snapshots of the density are shown in Fig. 11e-h.
