We study the emergence of universal scaling in the time-evolving momentum distribution of a harmonically trapped three-dimensional Bose-Einstein condensate, parametrically driven to a turbulent state. We demonstrate that the out-of-equilibrium dynamics post excitation is described by a single function due to nearby nonthermal fixed points. The observed behavior connects the dynamics of a quantum turbulent state to several far-from-equilibrium phenomena. We present a controllable protocol to explore universality in such systems, obtaining the associated scaling exponents. Our experimental results thus offer a promising route to investigate the complex dynamics of the quantum turbulent regime.

Turbulence is characterized by a large number of degrees of freedom, distributed over several length scales, that result in a disordered state of a fluid. The field of quantum turbulence deals with the manifestation of turbulence in quantum fluids, such as liquid helium and ultracold gases. We review, from both experimental and theoretical points of view, advances in quantum turbulence focusing on atomic Bose–Einstein condensates. We also explore the similarities and differences between quantum and classical turbulence. Last, we present challenges and possible directions for the field. We summarize questions that are being asked in recent works, which need to be answered in order to understand fundamental properties of quantum turbulence, and we provide some possible ways of investigating them.

We present a pedagogical introduction to Bose–Einstein condensation in traps with spherical symmetry, namely, the spherical box and the thick shell, sometimes called bubble trap. In order to obtain the critical temperature for Bose–Einstein condensation, we describe how to calculate the cumulative state number and density of states in these geometries, using numerical and analytical (semi-classical) approaches. The differences in the results of both methods are a manifestation of Weyl's theorem, i.e., they reveal how the geometry of the trap (boundary condition) affects the number of the eigenstates counted. Using the same calculation procedure, we analyzed the impact of going from three-dimensions to two-dimensions, as we move from a thick shell to a two-dimensional shell. The temperature range we obtained, for the most commonly used atomic species and reasonable confinement volumes, is compatible with current cold atom experiments, which demonstrates that these trapping potentials may be employed in experiments.

The field of quantum turbulence is related to the manifestation of turbulence in quantum fluids, such as liquid helium and ultracold gases. The concept of turbulence in quantum systems was conceived more than 70 years ago by Onsager and Feynman, but the study of turbulent ultracold gases is very recent. Although it is a young field, it already provides new approaches to the problem of turbulence. The authors review the advances and present status, of both theory and experiments, concerning atomic Bose–Einstein condensates (BECs). The authors present the difficulties in characterizing turbulence in trapped BECs, if compared to classical turbulence or turbulence in liquid helium. The authors summarize the challenges ahead, mostly related to the understanding of fundamental properties of quantum turbulence, including what is being done to investigate them.

The dynamics of nonequilibrium closed quantum systems and their route to thermalization are of fundamental interest to several fields, from cosmology to particle physics. However, a comprehensive description of nonequilibrium phenomena still presents a significant challenge. In this work, we report the observation of distinct stages during the relaxation of the decaying turbulence in trapped Bose-Einstein condensates. Our findings show a direct particle cascade from low to high momenta, a consequence of the energy injection in the system, exhibiting a characteristic universal scaling. This stage is followed by an inverse particle cascade responsible for repopulating the previously depleted condensate. Both cascades can be explained through self-similar solutions provided by wave turbulence theory. These findings provide important insights into the relaxation stages of out-of-equilibrium quantum many-body systems.