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A statistical mechanical approach of self-organization of a quantised vortex gas in a two-dimensional superfluid

Davide Maestrini

Published Sep 1, 2016• 1 citation
OpenAlex

Abstract

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.

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•Mar 12, 2020•0 citations
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.
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 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 dynamics of an isolated zero temperature quasi-two-dimensional superfluid Bose-Einstein condensate that is imprinted with a spatially random distribution of quantum vortices. Following a period of vortex annihilation the remaining vortices self-organize into two macroscopic coherent "Onsager vortex" clusters that are stable indefinitely--despite the absence of driving or external dissipation in the dynamics. We demonstrate that this occurs due to a novel physical mechanism--the evaporative heating of the vortices--that results in a negative-temperature phase transition in the vortex degrees of freedom. At the end of our simulations the system is trapped in a nonthermal state. Our computational results provide a pathway to observing Onsager vortex states in a superfluid Bose gas.
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.
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.
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.