Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Damped GPE simulations reveal that such OKC states can emerge dynamically, via aggregation of small-scale clusters into giant OKC clusters, as the end states of decaying 2D quantum turbulence in a compressible, finite-temperature superfluid. These statistical equilibrium states should be accessible in atomic Bose-Einstein condensate experiments.
Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Damped GPE simulations reveal that such OKC states can emerge dynamically, via aggregation of small-scale clusters into giant OKC clusters, as the end states of decaying 2D quantum turbulence in a compressible, finite-temperature superfluid. These statistical equilibrium states should be accessible in atomic Bose-Einstein condensate experiments.
Turbulence in classical fluids is a ubiquitous non-equilibrium phenomenon, yet a complete theoretical description for turbulent flow remains a challenging problem. A useful simplification for ideal two-dimensional (2D) fluids is to describe the turbulent flow with long-range-interacting point vortices, each possessing quantised circulation. In 1949, Onsager applied statistical mechanics to determine the equilibria of this model. He showed that at sufficiently high energies, like-circulation vortices preferentially aggregate into large-scale clusters, and are characterised by a negative absolute temperature. Onsager's theory has been highly influential, providing understanding of diverse quasi-2D systems such as turbulent soap films, guiding-centre plasmas, and self-gravitating systems. It also predicts the striking tendency of 2D turbulence to spontaneously form large-scale, long-lived vortices -- Jupiter's Great Red Spot is a well-known example. However, Onsager's theory doesn't quantitatively apply to classical fluids where vorticity is continuous, and experimental systems demonstrating Onsager's point-vortex statistical mechanics have remained elusive. Here we realise high energy, negative-temperature vortex clusters in a uniform superfluid Bose-Einstein condensate. Our results confirm Onsager's prediction of negative temperature clustered phases of quantum vortices, and demonstrate the utility of point-vortex statistical mechanics in 2D quantum fluids. This work opens future directions for the study of turbulent dynamics and we anticipate exploring the entire phase diagram of 2D quantum vortices, including the formation of clusters from 2D quantum turbulence.
We numerically model decaying quantum turbulence in two-dimensional disk-shaped Bose--Einstein condensates, and investigate the effects of finite temperature on the turbulent dynamics. We prepare initial states with a range of condensate temperatures, and imprint equal numbers of vortices and antivortices at randomly chosen positions throughout the fluid. The initial states are then subjected to unitary time-evolution within the c-field methodology. For the lowest condensate temperatures, the results of the zero temperature Gross--Pitaevskii theory are reproduced, whereby vortex evaporative heating leads to the formation of Onsager vortex clusters characterised by a negative absolute vortex temperature. At higher condensate temperatures the dissipative effects due to vortex--phonon interactions tend to drive the vortex gas towards positive vortex temperatures dominated by the presence of vortex dipoles. We associate these two behaviours with the system evolving toward an anomalous non-thermal fixed point, or a Gaussian thermal fixed point, respectively.
Copyright A. J. Groszek et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation.We numerically model decaying quantum turbulence in two-dimensional disk-shaped Bose–Einstein condensates, and investigate the effects of finite temperature on the turbulent dynamics. We prepare initial states with a range of condensate temperatures, and imprint equal numbers of vortices and antivortices at randomly chosen positions throughout the fluid. The initial states are then subjected to unitary time-evolution within the c-field methodology. For the lowest condensate temperatures, the results of the zero temperature Gross–Pitaevskii theory are reproduced, whereby vortex evaporative heating leads to the formation of Onsager vortex clusters characterised by a negative absolute vortex temperature. At higher condensate temperatures the dissipative effects due to vortex–phonon interactions tend to drive the vortex gas towards positive vortex temperatures dominated by the presence of vortex dipoles. We associate these two behaviours with the system evolving toward an anomalous non-thermal fixed point, or a Gaussian thermal fixed point, respectively.
The two-dimensional point vortex model was first introduced in the nineteenth century by Helmholtz and Kirchhoff. The methods of statistical physics were used to apply it to the problem of two-dimensional fluid flow and turbulence by Onsager in the 1940s, and it has provided many qualitative insights into the emergence of large scale vortices from turbulence. However, until recently there has been no quantitative comparison of its predictions with experimental measurements. Here we present the results of two sets of superfluid dynamics experiments for which the point vortex model provides quantitative insights. The first is in a planar quasi-two-dimensional atomic gas Bose-Einstein condensate cooled to nanokelvin temperatures [1]. The superfluid is confined and manipulated in a potential formed by laser light reflected from a Digital Micromirror Device (DMD) identical to those found in digital video projectors. Two different stirring procedures inject vortices into the gas. For the first stirring method the vortex number decays relatively quickly. However, the second stirring method leads to the formation of two macroscopic vortex clusters of opposite signs. The system quickly relaxes into a deeply negative temperature Onsager vortex state that persists in quasi-equilibrium for more than fifty cluster turnover times. The second experiment is in a superfluid helium thin film [2]. Optical stirring generates a macroscopic superfluid flow in the system that lifts the degeneracy in frequency of a number of third sound modes. These frequency splitting are subsequently observed to decay. Modelling the system using the point vortex model with a small amount of dissipation leads to a consistent interpretation of the measurements. We infer that a pinned persistent current slowly decays as a cluster of opposite sign vortices slowly annihilate the central charge. Improvements in the experimental apparatus are expected to lead to the capability of tracking the dynamics of a single vortex in real time. [1] Negative-Temperature Onsager Vortex Clusters in a Quantum Fluid, G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, T. W. Neely, to appear in Science (2019). arXiv:1801.06951 [2] Coherent vortex dynamics in a strongly-interacting superfluid on a silicon chip, Yauhen P. Sachkou, Christopher G. Baker, Glen I. Harris, Oliver R. Stockdale, Stefan Forstner, Matthew T. Reeves, Xin He, David L. McAuslan, Ashton S. Bradley, Matthew J. Davis, Warwick P. Bowen. arXiv:1902.04409
We have studied computationally turbulent non-equilibrium dynamics of quasi-two-dimensional superfluid Bose-Einstein condensates. Beginning with a random out-of-equilibrium distribution of vortices and antivortices in the condensate, we have observed emergence of ordered vortex structures corresponding to absolute negative temperature states in this conservative Hamiltonian system. We explain the spontaneous self-organization of the singly quantized vortices into two``Onsager vortex''clusters of like-signed vortices in terms of an``evaporative heating''mechanism of the vortex gas. These results provide a clear pathway to observing Onsager vortex structures and their unusual thermodynamics in a superfluid Bose gas
We have theoretically [1-5] and experimentally [6] studied statistical mechanics of vortices in two-dimensional superfluids in confined geometries. We are particularly interested in the part of the phase diagram where negative absolute temperature states of the vortex degrees of freedom are realized. The vortex dynamics is driven by two opposing mechanisms: a vortex evaporative heating due to vortex-antivortex annihilations drives the system toward low entropy negative temperature states while the coupling to the phonon bath results in cooling of the vortices driving the system toward vortex dipole dominated positiver temperature states. Furthermore, we have observed, both theoretically and experimentally, signatures of anomalous dynamical scaling and evolution toward non-thermal fixed points. For sufficiently high energies the Onsager vortex clusters are predicted to undergo Einstein-Bose condensation - a negative temperature counterpart to the Berezinskii-Kosterlitz-Thouless transition. [1] T Simula, MJ Davis, K Helmerson, Emergence of Order from Turbulence in an Isolated Planar Superfluid , Physical Review Letters 113, 165302, (2014). [2] AJ Groszek, TP Simula, DM Paganin, K Helmerson, Onsager vortex formation in Bose-Einstein condensates in two-dimensional power-law traps, Physical Review A 93, 043614, (2016). [3] RN Valani, AJ Groszek, TP Simula, Einstein–Bose condensation of Onsager vortices , New Journal of Physics 20, 053038, (2018). [4] AJ Groszek, MJ Davis, DM Paganin, K Helmerson, TP Simula, Vortex thermometry for turbulent two-dimensional fluids, Physical Review Letters 120, 034504, (2018). [5] AJ Groszek, MJ Davis, TP Simula, Decaying quantum turbulence in a two-dimensional Bose-Einstein condensate at finite temperature , arXiv:1903.05528. [6] Shaun P Johnstone, Andrew J Groszek, Philip T Starkey, Christopher J Billington, Tapio P Simula, Kristian Helmerson, Order from chaos: Observation of large-scale flow from turbulence in a two-dimensional superfluid, arXiv:1801.06952.