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arXiv:cond-mat/0505730 · Physical Review Letters

Emergence and Decay of Turbulence in Stirred Atomic Bose-Einstein Condensates

Nicholas G. Parker, Charles S. Adams

Published Sep 29, 2005• 126 citations
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Abstract

We show that "weak" elliptical deformation of an atomic Bose-Einstein condensate rotating at close to the quadrupole instability frequency leads to turbulence with a Kolmogorov energy spectrum. The turbulent state is produced by energy transfer to condensate fragments that are ejected by the quadrupole instability. This energy transfer is driven by breaking the twofold rotational symmetry of the condensate. Subsequently, vortex-sound interactions damp the turbulent state leading to the crystallization of a vortex lattice.

DOI
10.1103/physrevlett.95.145301

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We show that "weak" elliptical deformation of an atomic Bose-Einstein condensate rotating at close to the quadrupole instability frequency leads to turbulence with a Kolmogorov energy spectrum. The turbulent state is produced by energy transfer to condensate fragments that are ejected by the quadrupole instability. This energy transfer is driven by breaking the twofold rotational symmetry of the condensate. Subsequently, vortex-sound interactions damp the turbulent state leading to the crystallization of a vortex lattice.
We show that the formation of a vortex lattice in a weakly interacting Bose condensed gas can be modeled with the nonlinear Schrödinger equation for both T=0 and finite temperatures without the need for an explicit damping term. Applying a weak rotating anisotropic harmonic potential, we find numerically that the turbulent dynamics of the field produces an effective dissipation of the vortex motion and leads to the formation of a lattice. For T=0, this turbulent dynamics is triggered by a rotational dynamic instability of the condensate. For finite temperatures, noise is present at the start of the simulation and allows the formation of a vortex lattice at a lower rotation frequency, the Landau frequency. These two regimes have different vortex dynamics. We show that the multimode interpretation of the classical field is essential.
We introduce a mechanism to develop a turbulent flow in a spinor Bose-Einstein condensate, consisting in the stirring of a single line vortex by means of an external magnetic field. We find that density and velocity fluctuations have white-noise power spectra at large frequencies and that the energy spectrum obeys Kolmogorov 5/3 law in the turbulent region. As the stirring is turned off, the flow decays to an agitated nonequilibrium state that shows an energy bottleneck crossover at small length scales. We demonstrate our findings by numerically solving two-state spinor coupled three-dimensional Gross-Pitaevskii equations. We suggest that this mechanism may be experimentally implemented in spinor ultracold gases confined by optical traps.
Binary Bose-Einstein condensates of two distinct atomic species are studied, with interesting effects being verified due to the imbalance of atomic masses. We study the formation of vortices and associated turbulence flows created by stirring two mass-imbalanced coupled Bose-Einstein condensates in the miscible regime. We consider the mixture in the laboratory frame, confined in a pancake-like harmonic trap slightly perturbed elliptically by a time-dependent periodic potential, which introduces turbulent dynamics with final stable patterns of vortices, induced by the periodicity of the perturbation. Derived rotational frequencies are shown to be associated with the observed vortex patterns in the asymptotic regime, with a correspondence between the stirring procedure and a direct time-independent approach being indicated. It was also confirmed that a larger pattern of vortices is supported by the higher mass condensed species at the same frequency. For the experimentally accessible mixtures $^{85}\mathrm{Rb}\text{\ensuremath{-}}^{133}\mathrm{Cs}$ and $^{85}\mathrm{Rb}\text{\ensuremath{-}}^{87}\mathrm{Rb}$, we numerically solve the equations of motion and analyze the system dynamics using a range of measures including energy spectra. In the transient turbulent regime, spectral analysis shows evidence for the characteristic ${k}^{\ensuremath{-}5/3}$ Kolmogorov classical law associated with turbulence, modified by the universal ${k}^{\ensuremath{-}3}$ vortex core scaling in the ultraviolet regime.
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 (kkk) 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.
We investigate the scaling of the energy cascade in a harmonically trapped, turbulent, rotating Bose-Einstein condensate in two dimensions. We achieve turbulence by injecting a localized perturbation into the condensate and gradually increasing its rotation frequency from an initial value to a maximum. The main characteristics of the resulting turbulent state depend on the initial conditions, rotation frequency, and ramp-up time. We analyze the energy and the fluxes of kinetic energy by considering initial profiles without vortices and with vortex lattices. In the case without initial vortices, we find the presence of Kolmogorov-like scaling (k−5/3) of the incompressible kinetic energy in the inertial range. However, with initial vortex lattices, the energy spectrum follows Vinen scaling (k−1) at transient iterations. For cases with high rotating frequencies, Kolmogorov-like scaling emerges at longer durations. We observe positive kinetic energy fluxes with both initial states across all final frequencies, indicating a forward cascade of incompressible and compressible kinetic energy.
We present the simulation results of merging harmonically confined rotating Bose–Einstein condensates in two dimensions. Merging of the condensate is triggered by positioning the rotation axis at the trap minima and moving both condensates toward each other while slowly ramping their rotation frequency. We analyze the dynamics of the merged condensate by letting them evolve under a single harmonic trap. We systematically investigate the formation of solitonic and vortex structures in the final, unified condensate, considering both nonrotating and rotating initial states. In both cases, merging leads to the formation of solitons that decay into vortex pairs through snake instability, and subsequently, these pairs annihilate. Soliton formation and decay-induced phase excitations generate sound waves, more pronounced when the merging time is short. We witness no sound wave generation at sufficiently longer merging times that finally leads to the condensate reaching its ground state. With rotation, we notice off-axis merging (where the rotation axes are not aligned), leading to the distortion and weakening of soliton formation. The incompressible kinetic energy spectrum exhibits a Kolmogorov-like cascade [E(k)∼k−5/3] in the initial stage for merging condensates rotating above a critical frequency and a Vinen-like cascade [E(k)∼k−1] at a later time for all cases. Our findings hold potential significance for atomic interferometry, continuous atomic lasers, and quantum sensing applications.