A central concept in the modern understanding of turbulence is the existence of cascades of excitations from large to small length scales, or vice versa. This concept was introduced in 1941 by Kolmogorov and Obukhov1,2, and such cascades have since been observed in various systems, including interplanetary plasmas3, supernovae4, ocean waves5 and financial markets6. Despite much progress, a quantitative understanding of turbulence remains a challenge, owing to the interplay between many length scales that makes theoretical simulations of realistic experimental conditions difficult. Here we observe the emergence of a turbulent cascade in a weakly interacting homogeneous Bose gas—a quantum fluid that can be theoretically described on all relevant length scales. We prepare a Bose–Einstein condensate in an optical box7, drive it out of equilibrium with an oscillating force that pumps energy into the system at the largest length scale, study its nonlinear response to the periodic drive, and observe a gradual development of a cascade characterized by an isotropic power-law distribution in momentum space. We numerically model our experiments using the Gross–Pitaevskii equation and find excellent agreement with the measurements. Our experiments establish the uniform Bose gas as a promising new medium for investigating many aspects of turbulence, including the interplay between vortex and wave turbulence, and the relative importance of quantum and classical effects.
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We thank G. V. Shlyapnikov, B. Svistunov, S. Stringari, N. R. Cooper, J. Dalibard, M. J. Davis, R. J. Fletcher, M. W. Zwierlein, K. Fujimoto and M. Tsubota for discussions, and C. Eigen for experimental assistance. This work was supported by AFOSR, ARO, DARPA OLE, EPSRC (Grant No. EP/N011759/1) and ERC (QBox). The GeForce GTX TITAN X used for the numerical simulations was donated by the NVIDIA Corporation. N.N. and A.L.G. acknowledge support from Trinity College, Cambridge; R.P.S. acknowledges support from the Royal Society.
The authors declare no competing financial interests.
Extended data figures and tables
a, b, Comparison of n1D(kz) obtained using TOF expansion (solid lines) and Bragg spectroscopy (points), in the case of the initial, quasi-pure BEC (a) and the turbulent gas (b). The red dashed line in a corresponds to the Heisenberg-limited momentum distribution. All distributions are normalized to unity (), without any adjustable parameters. Source data
a, In-plane momentum distribution for various shaking times ts. b, Ratio of the compressible- (c) to incompressible-flow (i) components of the fluid-dynamical kinetic energy, with the colours corresponding to the shaking times in a. The simulation parameters for both panels are N = 8 × 104, shaking frequency ω/(2π) = 9 Hz and shaking amplitude ΔU = μ. Source data
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Navon, N., Gaunt, A., Smith, R. et al. Emergence of a turbulent cascade in a quantum gas. Nature 539, 72–75 (2016). https://doi.org/10.1038/nature20114
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