We study the relaxation of a two-dimensional (2D) ultracold Bose gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a nonzero coarse-grained vorticity field. The stream function of this emergent quasiclassical 2D flow is governed by a Poisson-Boltzmann equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large-scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Poisson-Boltzmann equation. The results allow us to characterize the spectral energy condensate in a 2D quantum fluid that bears striking similarity to similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.
We study the relaxation of a two-dimensional (2D) ultracold Bose gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a nonzero coarse-grained vorticity field. The stream function of this emergent quasiclassical 2D flow is governed by a Poisson-Boltzmann equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large-scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Poisson-Boltzmann equation. The results allow us to characterize the spectral energy condensate in a 2D quantum fluid that bears striking similarity to similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.
In this work the relaxation of a two-dimensional Bose-gas from a non-equilibrium \ninitial condition consisting of vortices is studied. To focus on the role of the vortex \nexcitations on the time evolution of the system, a point vortex model is used. The \nrelaxation of the vortex gas is seen to lead to clustering of like-signed vortices that \ncan be explained in terms of negative temperature states. The nature of the Coulomb \ninteractions between vortices, precludes a well-defined thermodynamic limit. The large \nscale \nflow structures, therefore strongly depend on the shape of the geometry. These \nstructures can be explained in terms of a maximum entropy principle for the vortex \ngas that leads to the Boltzmann-Poisson equation. For a square region the maximum \nentropy configuration corresponds to a monopole. This configuration results in the \nspontaneous acquisition of angular momentum by the \now. However, by stretching \nthe square domain into a rectangle, the configuration which maximises the entropy \nswitched to a dipole where like-signed vortices tend to equally occupy the two halves \nof the domain. In this case, the mean \nflow has zero angular momentum. A direct qualitative \nand quantitative comparison between the predictions of the mean-field theory \nand dynamical simulations of a point vortex model are presented. In particular, we \nshow that vortex-antivortex annihilation results in evaporative heating of the vortex \ngas and the subsequent migration of the system into the negative temperature regime. \nMoreover, the study is extended to the dynamics of quantised vortices in the same \nconfined geometries in a two-dimensional Bose-Einstein condensate described by the \nGross-Pitaevskii equation. Despite the coexistence of phonons with vortex excitations \nthat interact together, the above predictions continue to apply in this more realistic \nmodel of a two-dimensional superfl \nuid.
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.
The emergence of coherent rotating structures is a phenomenon characteristic of both classical and quantum two-dimensional (2D) turbulence. In this work we show theoretically that the coherent vortex structures that emerge in decaying 2D quantum turbulence can approach quasiclassical rigid-body rotation, obeying the Feynman rule of constant average areal vortex density while remaining spatially disordered. By developing a rigorous link between the velocity probability distribution and the quantum kinetic energy spectrum over wave number k, we show that the coherent vortex structures are associated with a k3 power law in the infrared region of the spectrum, and a well-defined spectral peak that is a physical manifestation of the largest structures. We discuss the possibility of realizing coherent structures in Bose-Einstein condensate experiments and present Gross-Pitaevskii simulations showing that this phenomenon, and its associated spectral signatures, can emerge dynamically from feasible initial vortex configurations.
Nonequilibrium interacting systems can evolve to exhibit large-scale structure and order. In two-dimensional turbulent flow, the seemingly random swirling motion of a fluid can evolve toward persistent large-scale vortices. To explain such behavior, Lars Onsager proposed a statistical hydrodynamic model based on quantized vortices. Here, we report on the experimental confirmation of Onsager's model. We dragged a grid barrier through an oblate superfluid Bose-Einstein condensate to generate nonequilibrium distributions of vortices. We observed signatures of an inverse energy cascade driven by the evaporative heating of vortices, leading to steady-state configurations characterized by negative absolute temperatures. Our results open a pathway for quantitative studies of emergent structures in interacting quantum systems driven far from equilibrium.
In a 2D turbulent fluid containing pointlike vortices, Lars Onsager predicted that adding energy to the fluid can lead to the formation of persistent clusters of like-signed vortices, i.e., Onsager vortex (OV) clusters. In the evolution of 2D superfluid turbulence in a uniform disk-shaped Bose-Einstein condensate (BEC), it was discovered that a pair of OV clusters with opposite signs can form without any energy input. This striking spontaneous order was explained as being due to a vortex evaporative-heating mechanism, i.e., annihilations of vortex-antivortex pairs which remove the lowest-energy vortices and thereby boost the mean energy per vortex. However, in our search for exotic OV states in a boundaryless 2D spherical BEC, we found that OV clusters never form despite the annihilations of vortex pairs. Our analysis reveals that contrary to the general belief, vortex-pair annihilation emits intense sound waves, which damp the motion of all vortices and hence suppress the formation of OV clusters. We also present unequivocal evidence showing that the true mechanism underlying the observed spontaneous OV state is the vortices exiting the BEC boundaries. Uncovering this mechanism paves the way for a comprehensive understanding of emergent vortex orders in 2D manifolds of superfluids driven far from equilibrium.
Fluids subjected to suitable forcing will exhibit turbulence, with characteristics strongly affected by the fluid's physical properties and dimensionality. In this work, we explore two-dimensional (2D) quantum turbulence in an oblate Bose-Einstein condensate confined to an annular trapping potential. Experimentally, we find conditions for which small-scale stirring of the condensate generates disordered 2D vortex distributions that dissipatively evolve toward persistent currents, indicating energy transport from small to large length scales. Simulations of the experiment reveal spontaneous clustering of same-circulation vortices and an incompressible energy spectrum with k(-5/3) dependence for low wave numbers k. This work links experimentally observed vortex dynamics with signatures of 2D turbulence in a compressible superfluid.