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arXiv:1209.2447 · Physical Review A

Classical and quantum regimes of two-dimensional turbulence in trapped Bose-Einstein condensates

Matthew T. Reeves, Brian P. Anderson, Ashton S. Bradley

Published Nov 21, 2012• 80 citations
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Abstract

We investigate two-dimensional turbulence in finite-temperature trapped Bose-Einstein condensates within damped Gross-Pitaevskii theory. Turbulence is produced via circular motion of a Gaussian potential barrier stirring the condensate. We systematically explore a range of stirring parameters and identify three regimes, characterized by the injection of distinct quantum vortex structures into the condensate: (A) periodic vortex dipole injection, (B) irregular injection of a mixture of vortex dipoles and co-rotating vortex clusters, and (C) continuous injection of oblique solitons that decay into vortex dipoles. Spectral analysis of the kinetic energy associated with vortices reveals that regime (B) can intermittently exhibit a Kolmogorov ${k}^{\ensuremath{-}5/3}$ power law over almost a decade of length or wave-number ($k$) scales. The kinetic energy spectrum of regime (C) exhibits a clear ${k}^{\ensuremath{-}3/2}$ power law associated with an inertial range for weak-wave turbulence and a ${k}^{\ensuremath{-}7/2}$ power law for high wave numbers. We thus identify distinct regimes of forcing for generating either two-dimensional quantum turbulence or classical weak-wave turbulence that may be realizable experimentally.

DOI
10.1103/physreva.86.053621

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We investigate two-dimensional turbulence in finite-temperature trapped Bose-Einstein condensates within damped Gross-Pitaevskii theory. Turbulence is produced via circular motion of a Gaussian potential barrier stirring the condensate. We systematically explore a range of stirring parameters and identify three regimes, characterized by the injection of distinct quantum vortex structures into the condensate: (A) periodic vortex dipole injection, (B) irregular injection of a mixture of vortex dipoles and co-rotating vortex clusters, and (C) continuous injection of oblique solitons that decay into vortex dipoles. Spectral analysis of the kinetic energy associated with vortices reveals that regime (B) can intermittently exhibit a Kolmogorov k\ensuremath−5/3{k}^{\ensuremath{-}5/3}k\ensuremath−5/3 power law over almost a decade of length or wave-number (kkk) scales. The kinetic energy spectrum of regime (C) exhibits a clear k\ensuremath−3/2{k}^{\ensuremath{-}3/2}k\ensuremath−3/2 power law associated with an inertial range for weak-wave turbulence and a k\ensuremath−7/2{k}^{\ensuremath{-}7/2}k\ensuremath−7/2 power law for high wave numbers. We thus identify distinct regimes of forcing for generating either two-dimensional quantum turbulence or classical weak-wave turbulence that may be realizable experimentally.
We study two-dimensional quantum turbulence in miscible binary Bose-Einstein condensates in either a harmonic trap or a steep-wall trap through the numerical simulations of the Gross-Pitaevskii equations. The turbulence is generated through a Gaussian stirring potential. When the condensates have unequal intracomponent coupling strengths or asymmetric trap frequencies, the turbulent condensates undergo a dramatic decay dynamics to an interlaced array of vortex-antidark structures, a quasiequilibrium state, of like-signed vortices with an extended size of the vortex core. The time of formation of this state is shortened when the parameter asymmetry of the intracomponent couplings or the trap frequencies is enhanced. The corresponding spectrum of the incompressible kinetic energy exhibits two noteworthy features: (i) a ${k}^{\ensuremath{-}3}$ power law around the range of the wave number determined by the spin healing length (the size of the extended vortex core) and (ii) a flat region around the range of the wave number determined by the density healing length. The latter is associated with the small scale phase fluctuation relegated outside the Thomas-Fermi radius and is more prominent as the strength of intercomponent interaction approaches the strength of intracomponent interaction. We also study the impact of the intercomponent interaction to the cluster formation of like-signed vortices in an elliptical steep-wall trap, finding that the intercomponent coupling gives rise to the decay of the clustered configuration.
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov k−5/3k^{-5/3}k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
We investigate the dynamics of turbulent flow in a two-dimensional trapped Bose-Einstein condensate by solving the Gross-Pitaevskii equation numerically. The development of the quantum turbulence is activated by the disruption of an initially embedded vortex quadrupole. By calculating the incompressible kinetic-energy spectrum of the superflow, we conclude that this quantum turbulent state is characterized by the Kolmogorov-Saffman scaling law in the wave-number space. Our study predicts the coexistence of two subinertial ranges responsible for the energy cascade and enstrophy cascade in this prototype of two-dimensional quantum turbulence.
We theoretically propose a new method of making quantum turbulence from many dark solitons in atomic Bose-Einstein condensates. We solve numerically the two-dimensional Gross-Pitaevskii equation. We set initially many solitons so that they can form a square grid. A dark soliton is known to be stable in one-dimensional systems, but unstable in two- or three-dimensional systems and decay to vortices. Our simulation shows that these solitons decay to a lot of vortices which move around in the system and eventually lead to two-dimensional quantum turbulence. The probability distribution function of the superfluid velocity obeys a Gaussian distribution in the low-velocity region and a power-law distribution in the high-velocity region. The decay of the total number of vortices obeys a power-law for a relatively long period. This scenario may be experimentally realized through interference of Bose-Einstein condensates in a trap potential.
We show the generation of two-dimensional quantum turbulence through simulations of a giant vortex decay in a trapped Bose-Einstein condensate. While evaluating the incompressible kinetic energy spectra of the quantum fluid described by the Gross-Pitaevskii equation, a bilinear form in a log-log plot is verified. A characteristic scaling behavior for small momenta shows resemblance to the Kolmogorov k−5/3k^{-5/3}k−5/3 law, while for large momenta it reassures the universal behavior of the core-size k−3k^{-3}k−3 power-law. This indicates a mechanism of energy transportation consistent with an inverse cascade. The feasibility of the described physical system with the currently available experimental techniques to create giant vortices opens up a new route to explore quantum turbulence.
We report the experimental generation of two-dimensional quantum turbulence in dilute-gas Bose-Einstein condensates by stirring the condensate with a laser beam, and the decay of the turbulent state to a large-scale flow in the form of a persistent current in a toroidal trap geometry. From numerical simulations of the stirring process, we characterize the dynamics of the quantized vortices in the condensate, and analyze the kinetic energy spectrum. Our observations are consistent with basic features of two-dimensional turbulence in classical incompressible fluids.