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Synthetic dissipation and cascade fluxes in a turbulent quantum gas

Nir Navon, Christoph Eigen, Jinyi Zhang, Raphael Lopes, Alexander L. Gaunt, Kazuya Fujimoto, Makoto Tsubota, Robert P. Smith, Zoran Hadzibabic

Published Jan 15, 2019• 18 citations
Publisher

Abstract

Scale-invariant fluxes are the defining property of turbulent cascades, but their direct measurement is a notorious problem. Here we perform such a measurement for a direct energy cascade in a turbulent quantum gas. Using a time-periodic force, we inject energy at a large lengthscale and generate a cascade in a uniformly-trapped Bose gas. The adjustable trap depth provides a high-momentum cutoff $k_{\textrm{D}}$, which realises a synthetic dissipation scale. This gives us direct access to the particle flux across a momentum shell of radius $k_{\textrm{D}}$, and the tunability of $k_{\textrm{D}}$ allows for a clear demonstration of the zeroth law of turbulence. Moreover, our time-resolved measurements give unique access to the pre-steady-state dynamics, when the cascade front propagates in momentum space.

DOI
10.17863/cam.44144

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Scale-invariant fluxes are the defining property of turbulent cascades, but their direct measurement is a notorious problem. Here we perform such a measurement for a direct energy cascade in a turbulent quantum gas. Using a time-periodic force, we inject energy at a large lengthscale and generate a cascade in a uniformly-trapped Bose gas. The adjustable trap depth provides a high-momentum cutoff kDk_{\textrm{D}}kD​, which realises a synthetic dissipation scale. This gives us direct access to the particle flux across a momentum shell of radius kDk_{\textrm{D}}kD​, and the tunability of kDk_{\textrm{D}}kD​ allows for a clear demonstration of the zeroth law of turbulence. Moreover, our time-resolved measurements give unique access to the pre-steady-state dynamics, when the cascade front propagates in momentum space.
Down and down the energy cascade Injecting energy into a turbulent system at large length scales results in the energy cascading down and eventually dissipating at a characteristic small length scale. In conventional fluids, this small scale is set by fluid viscosity. Navon et al. studied the turbulence energy cascade in a quantum gas, a Bose-Einstein condensate of rubidium-87 atoms held in a uniform trap. Dissipation occurred by atoms escaping from the trap at a scale that could be tuned by varying the height of the trapping potential. Thanks to the flexibility of their setup, the researchers were able to study both the steady state, in which energy is injected at the same rate it is dissipated, and the transient regime preceding the steady state. Science , this issue p. 382
Turbulence Injecting energy into a turbulent system at large length scales results in the energy cascading down and eventually dissipating at a characteristic small length scale. In conventional fluids, this small scale is set by fluid viscosity. Navon et al. studied the turbulence energy cascade in a quantum gas, a Bose-Einstein condensate of rubidium-87 atoms held in a uniform trap. Dissipation occurred by atoms escaping from the trap at a scale that could be tuned by varying the height of the trapping potential. Thanks to the flexibility of their setup, the researchers were able to study both the steady state, in which energy is injected at the same rate it is dissipated, and the transient regime preceding the steady state. Science , this issue p. [382][1] [1]: /lookup/doi/10.1126/science.aau6103
Nonequilibrium equations of state can provide an effective thermodynamic-like description of far-from-equilibrium systems. We experimentally construct such an equation for a direct energy cascade in a turbulent two-dimensional Bose gas. Our homogeneous gas is continuously driven on a large length scale and, with matching dissipation on a small length scale, exhibits a nonthermal but stationary power-law momentum distribution. Our equation of state links the cascade amplitude with the underlying scale-invariant energy flux, and can, for different drive strengths, gas densities, and interaction strengths, be recast into a universal power-law form using scalings consistent with the Gross-Pitaevskii model.
When a Bose-Einstein condensate (BEC) is driven out of equilibrium, density waves interact nonlinearly and trigger turbulent cascades. In a turbulent BEC, energy is transferred toward small scales by a direct cascade, whereas the number of particles displays an inverse cascade toward large scales. In this work, we study analytically and numerically the direct and inverse cascades in wave-turbulent BECs. We analytically derive the Kolmogorov-Zakharov spectra, including the log correction to the direct cascade scaling and the universal prefactor constants for both cascades. We test and corroborate our predictions using high-resolution numerical simulations of the forced-dissipated Gross-Pitaevskii model in a periodic box and the corresponding wave-kinetic equation. Theoretical predictions and data are in excellent agreement, without adjustable parameters. Moreover, in order to connect with experiments, we test and validate our theoretical predictions using the Gross-Pitaevskii model with a confining cubic trap. Our results explain previous experimental observations and suggest new settings for future studies.
In turbulence phenomena, including the quantum turbulence in superfluids, an energy flux flows from large to small length scales, composing a cascade of energy. A universal characteristic of turbulent flows is the existence of a range of scales where the energy flux is scale-invariant: this interval of scales is often referred to as inertial region. This property is fundamental as, for instance, in turbulence of classical fluids it characterizes the behavior of statistical features such as spectra and structure functions. Here we show that also in decaying quantum turbulence generated in trapped Bose-Einstein condensates (BECs), intervals of momentum space where the energy flux is constant can be identified. Indeed, we present a procedure to measure the energy flux using both the energy spectrum and the continuity equation. A range of scales where the flux is constant is then determined employing two distinct protocols and in the same range, the momentum distribution measured is consistent with previous work. The successful 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 trapped atomic superfluids, and also analogies with the related processes that take place in ordinary fluids.
We realize a turbulent cascade of wave excitations in a homogeneous 2D Bose gas and probe on all relevant time and length scales how it builds up from small to large momenta, until the system reaches a steady state with matching energy injection and dissipation. This all-scales view directly reveals the two theoretically expected cornerstones of turbulence formation-the emergence of statistical momentum-space isotropy under anisotropic forcing and the spatiotemporal scaling of the momentum distribution at times before any energy is dissipated.