We explore the possible regimes of decaying two-dimensional quantum turbulence, and elucidate the nature of spectral energy transport by introducing a dissipative point-vortex model with phenomenological vortex-sound interactions. The model is valid
for a large system with weak dissipation, and also for systems with strong dissipation, and allows us to extract a meaningful and unambiguous spectral energy flux associated with quantum vortex motion. For weak dissipation and large system size we find a regime of hydrodynamic vortex turbulence in which energy is transported to large spatial scales, resembling the phenomenology of the transient inverse cascade observed in decaying turbulence in classical incompressible fluids. For strong dissipation the vortex dynamics are dominated by dipole recombination and exhibit no appreciable spectral transport of energy.
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 systematic
ally 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 $k^{-5/3}$ power law over almost a decade of length or wavenumber ($k$) scales. The kinetic energy spectrum of regime (C) exhibits a clear $k^{-3/2}$ power law associated with an inertial range for weak-wave turbulence, and a $k^{-7/2}$ power law for high wavenumbers. We thus identify distinct regimes of forcing for generating either two-dimensional quantum turbulence or classical weak-wave turbulence that may be realizable experimentally.
Under suitable forcing a fluid exhibits turbulence, with characteristics strongly affected by the fluids confining geometry. Here we study two-dimensional quantum turbulence in a highly oblate Bose-Einstein condensate in an annular trap. As a compres
sible quantum fluid, this system affords a rich phenomenology, allowing coupling between vortex and acoustic energy. Small-scale stirring generates an experimentally observed disordered vortex distribution that evolves into large-scale flow in the form of a persistent current. Numerical simulation of the experiment reveals additional characteristics of two-dimensional quantum turbulence: spontaneous clustering of same-circulation vortices, and an incompressible energy spectrum with $k^{-5/3}$ dependence for low wavenumbers $k$ and $k^{-3}$ dependence for high $k$.
We report experimental observations and numerical simulations of the formation, dynamics, and lifetimes of single and multiply charged quantized vortex dipoles in highly oblate dilute-gas Bose-Einstein condensates (BECs). We nucleate pairs of vortice
s of opposite charge (vortex dipoles) by forcing superfluid flow around a repulsive gaussian obstacle within the BEC. By controlling the flow velocity we determine the critical velocity for the nucleation of a single vortex dipole, with excellent agreement between experimental and numerical results. We present measurements of vortex dipole dynamics, finding that the vortex cores of opposite charge can exist for many seconds and that annihilation is inhibited in our highly oblate trap geometry. For sufficiently rapid flow velocities we find that clusters of like-charge vortices aggregate into long-lived dipolar flow structures.
We perform finite-temperature dynamical simulations of the arrest of a rotating Bose-Einstein condensate by a fixed trap anisotropy, using a Hamiltonian classical-field method. We consider a quasi-two-dimensional condensate containing a single vortex
in equilibrium with a rotating thermal cloud. Introducing an elliptical deformation of the trapping potential leads to the loss of angular momentum from the system. We identify the condensate and the complementary thermal component of the nonequilibrium field, and compare the evolution of their angular momenta and angular velocities. By varying the trap anisotropy we alter the relative efficiencies of the vortex-cloud and cloud-trap coupling. For strong trap anisotropies the angular momentum of the thermal cloud may be entirely depleted before the vortex begins to decay. For weak trap anisotropies, the thermal cloud exhibits a long-lived steady state in which it rotates at an intermediate angular velocity.
We consider a finite-temperature Bose-Einstein condensate in a quasi-two-dimensional trap containing a single precessing vortex. We find that such a configuration arises naturally as an ergodic equilibrium of the projected Gross-Pitaevskii equation,
when constrained to a finite conserved angular momentum. In an isotropic trapping potential the condensation of the classical field into an off-axis vortex state breaks the rotational symmetry of the system. We present a methodology to identify the condensate and the Goldstone mode associated with the broken rotational symmetry in the classical-field model. We also examine the variation in vortex trajectories and thermodynamic parameters of the field as the energy of the microcanonical field simulation is varied.
We describe a method for evolving the projected Gross-Pitaevskii equation (PGPE) for an interacting Bose gas in a harmonic oscillator potential, with the inclusion of a long-range dipolar interaction. The central difficulty in solving this equation i
s the requirement that the field is restricted to a small set of prescribed modes that constitute the low energy c-field region of the system. We present a scheme, using a Hermite-polynomial based spectral representation, that precisely implements this mode restriction and allows an efficient and accurate solution of the dipolar PGPE. We introduce a set of auxiliary oscillator states to perform a Fourier transform necessary to evaluate the dipolar interaction in reciprocal space. We extensively characterize the accuracy of our approach, and derive Ehrenfest equations for the evolution of the angular momentum.
We review phase space techniques based on the Wigner representation that provide an approximate description of dilute ultra-cold Bose gases. In this approach the quantum field evolution can be represented using equations of motion of a similar form t
o the Gross-Pitaevskii equation but with stochastic modifications that include quantum effects in a controlled degree of approximation. These techniques provide a practical quantitative description of both equilibrium and dynamical properties of Bose gas systems. We develo
We present a quantum mechanical treatment of the mechanical stirring of Bose-Einstein condensates using classical field techniques. In our approach the condensate and excited modes are described using a Hamiltonian classical field method in which the
atom number and (rotating frame) energy are strictly conserved. We simulate a T = 0 quasi-2D condensate perturbed by a rotating anisotropic trapping potential. Vacuum fluctuations in the initial state provide an irreducible mechanism for breaking the initial symmetries of the condensate and seeding the subsequent dynamical instability. Highly turbulent motion develops and we quantify the emergence of a rotating thermal component that provides the dissipation necessary for the nucleation and motional-damping of vortices in the condensate. Vortex lattice formation is not observed, rather the vortices assemble into a spatially disordered vortex liquid state. We discuss methods we have developed to identify the condensate in the presence of an irregular distribution of vortices, determine the thermodynamic parameters of the thermal component, and extract damping rates from the classical field trajectories.
We develop a stochastic Gross-Pitaveskii theory suitable for the study of Bose-Einstein condensation in a {em rotating} dilute Bose gas. The theory is used to model the dynamical and equilibrium properties of a rapidly rotating Bose gas quenched thro
ugh the critical point for condensation, as in the experiment of Haljan et al. [Phys. Rev. Lett., 87, 21043 (2001)]. In contrast to stirring a vortex-free condensate, where topological constraints require that vortices enter from the edge of the condensate, we find that phase defects in the initial non-condensed cloud are trapped en masse in the emerging condensate. Bose-stimulated condensate growth proceeds into a disordered vortex configuration. At sufficiently low temperature the vortices then order into a regular Abrikosov lattice in thermal equilibrium with the rotating cloud. We calculate the effect of thermal fluctuations on vortex ordering in the final gas at different temperatures, and find that the BEC transition is accompanied by lattice melting associated with diminishing long range correlations between vortices across the system.
