Quantum turbulence in a trapped Bose-Einstein condensate

Abstract

We study quantum turbulence in trapped Bose-Einstein condensates by numerically solving the Gross-Pitaevskii equation. Combining rotations around two axes, we successfully induce quantum turbulent state in which quantized vortices are not crystallized but tangled. The obtained spectrum of the incompressible kinetic energy is consistent with the Kolmogorov law, the most important statistical law in turbulence. The study of turbulence has a very long history, going back at least to Leonardo da Vinci, and understanding and controlling turbulence are great dreams of science and technology. Classical turbulence (CT) exhibits highly complicated configurations of eddies. Many studies have been devoted to the dynamical and statistical properties of CT after Kolmogorov's pioneering work [1,2] on flow at very high Reynolds number, namely, fully developed CT. The characteristic behavior of CT has been believed to be sustained by the Richardson cascade of eddies from large to small scales. However, in CT, there is no universal way to identify each eddy, because they continue to nucleate, diffuse, and disappear. As a result, many aspects of CT are still not perfectly understood.Turbulence is also possible in superfluids, such as the superfluid phases of 4 He and 3 He. Such quantum turbulence (QT) consists of definite topological defects known as quantized vortices and has recently attracted interest as a way to better understand turbulence [3].Superfluid 4 He has been extensively studied, in particular with relation to quantized vortices [4]. Below the lambda temperature T λ = 2.17 K, liquid 4 He enters the superfluid state through Bose-Einstein condensation. The hydrodynamics of superfluid 4 He is strongly influenced by quantum effects; any rotational motion is sustained by quantized vortices with quantized circulation κ = /m, where m is the particle mass. There are two typical cooperative phenomena of quantized vortices. One is a vortex lattice under rotation in which straight quantized vortices form a triangular lattice along the rotation axis [5]. The other is a vortex tangle in QT in which vortices become tangled in a flow [6,7].QT has been studied as a problem in low temperature physics since its discovery some 50 years ago. Its study has recently entered a new stage beyond low temperature physics. One of the main motivations of recent studies is to investigate the relationship between QT and CT. Some similarities between the two types of turbulence have been experimentally observed in superfluid 4 He [8, 9] and 3 He [10,11], and have been theoretically confirmed by numerical simulations of the quantized vortex-filament model [12] and a model using the Gross-Pitaevskii (GP) equation [13,14,15,16]. In particular, we have successfully obtained the Kolmogorov law for QT, which is one of the most important statistical laws in CT [17] by a numerical simulation of the GP equation [14,15].The similarity between QT and CT means that QT is an ideal prototype to study the statistics and vortex dynamics of turbulence, because QT ...

Numerical Studies of Quantum Turbulence

“…[126], where the decaying QT is generated by the initial state with random phase and the Kolmogorov law is confirmed. Furthermore, Kobayashi et al generated the stationary steady state QT using the dissipative GP equation and obtained the Kolmogorov law [127,128]. Recently, QT was investigated in terms of a non-thermal fixed point, and the Kolmogorov law was also observed [129,130].…”

“…Originally, this study focused on superfluid helium, but this result is important for understanding QT in ultracold atomic BECs. After this study, some theoretical works studied QT in ultracold atomic BECs, finding the same Kolmogorov −5/3 law [125,126,127,128,129,130].…”

Quantum turbulence—from superfluid helium to atomic Bose–Einstein condensates

“…Kobayashi and Tsubota proposed an easy, powerful method to make a steady QT in a trapped BEC, by using precession [74]. The dynamics of the wavefunction are described by the GP equation with dissipation.…”

Macroscopic Excitations in Confined Bose–Einstein Condensates, Searching for Quantum Turbulence

“…We point out that there are already experimental realizations of quantum turbulence 9,10,11 as well of studies of the route to turbulence. 5,12,13 In this work, through the Gross-Pitaevskii (GP) model 14,15 of an ultracold quantum gas, and assuming that arbitrary phase-imprinted states can be generated 16,17,18,19 we present here a survey of macroscopic excitations that lead a Bose-Einstein condensate (BEC), confined in an external harmonic potential, to stationary agitated or chaotic states that, under appropriate conditions, may be considered to pass through turbulent transients or remain as such. Although the details of the model and of the different excitations that we explore are presented below, we want to advance here a general result that we find may be of relevance in a more complete study.…”

“…Most notorious is the cascade of energy 1 through vortices or eddies, in the classical case, or by a tangle of vortices 2 in the quantum version. Among the relevant questions one must deal with, specially in the quantum regime, is the initiation or generation of such a complex state 3,4,5,6 and its characterization. 7,8 These issues motivate the study presented in this article.…”