Wave turbulence and vortices in Bose-Einstein condensation

We report a numerical study of turbulence and Bose-Einstein condensation within the two-dimmensional GrossPitaevski model with repulsive interaction. In presence of weak forcing localized around some wave number in the Fourier space, we observe three qualitatively different evolution stages. At the initial stage a thermodynamic energy equipartition spectrum forms at both smaller and larger scales with respect to the forcing scale. This agrees with predictions of the the four-wave kinetic equation of the Wave Turbulence (WT) theory. At the second stage, WT breaks down at large scales and the interactions become strongly nonlinear. Here, we observe formation of a gas of quantum vortices whose number decreases due to an annihilation process helped by the acoustic component. This process leads to formation of a coherent-phase Bose-Einstein condensate. After such a coherent-phase condensate forms, evolution enters a third stage characterised by three-wave interactions of acoustic waves that can be described again using the WT theory. 1 Background and motivation For dilute gases with large energy occupation numbers the Bose-Einstein condensation (BEC) can be described by the Gross-Pitaevsky (GP) equation [1, 2]: iΨt +∆Ψ− |Ψ|Ψ = γ, (1) where Ψ is the condensate “wave function” (i.e. the c-number part of the boson annihilation field) and γ is an operator which models possible forcing and dissipation mechanisms which will be discussed later. Renewed interest to the nonlinear dynamics described by GP equation is related to relatively recent experimental discoveries of BEC [3, 4, 5]. GP equation also describes light behaviour in media with Kerr nonlinearities. In the nonlinear optics context it is usually called the Nonlinear Schoedinger (NLS) equation. It is presently understood, in both the nonlinear optics and BEC contexts, that the nonlinear dynamics described by GP equation is typically chaotic and often non-equilibrium [6, 7, 8, 9]. Thus, it is best characterised as “turbulence” emphasizing its resemblance to the classical Navier-Stokes (NS) turbulence. On the other hand, the GP model has an advantage over NS because it has a weakly nonlinear limit in which the stochastic field evolution can be represented as a large set of weakly interactive dispersive waves. A systematic statistical closure is possible for such systems and the corresponding theory is called Wave Turbulence (WT) [10]. For small perturbations about the zero state in the GP model, WT closure predicts that the main nonlinear process will be four-wave resonant interaction. This closure was used in [6, 8, 9] to describe the initial stage of BEC. In the present paper we will examine this description numerically. We report that our numerics agree with the predicted by WT spectra at the initial evolution stage.