Turbulence is characterized by a large number of degrees of freedom, distributed over several length scales, that result into 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. Lastly, 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. arXiv:1903.12215v1 [cond-mat.quant-gas]

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 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.

In turbulence phenomena, including the quantum turbulence in superfluids, an energy flux flows from large to small length scales, composing a cascade of energy. It is a well-known fact that for multi-scale energy flow, dissipation can be scale-dependent. In particular, the existence of a range of scales where there is no energy accumulation, the inertial range, is an indication of universal behavior in turbulence. There are intrinsic difficulties associated with the measurement of the energy flux during the time evolution of turbulence. Here we present a procedure to measure the energy flux during the time evolution of turbulence in a sample of a trapped Bose-Einstein condensate. The energy flux is evaluated using the energy spectrum and the continuity equation. We identified intervals of momentum where the flux is constant using two different procedures. The identification of a region with constant flux in turbulent BECs is a manifestation of the universal character of turbulence in these quantum systems. These measurements pave the way for studies of energy conservation and dissipation in a scale-dependent manner in trapped atomic superfluids, and also analogies with the related processes that take place in ordinary fluids.