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arXiv:2405.01537 · Physical Review Letters

Observation of an Inverse Turbulent-Wave Cascade in a Driven Quantum Gas

Andrey Karailiev, Martin Gazo, Maciej Gałka, Christoph Eigen, Tanish Satoor, Zoran Hadzibabic

Published Dec 9, 2024• 12 citations
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Abstract

We observe an inverse turbulent-wave cascade, from small to large length scales, in a driven homogeneous 2D Bose gas. Starting with an equilibrium condensate, we drive the gas isotropically on a length scale much smaller than its size, and observe a nonthermal population of modes with wavelengths larger than the drive one. At long drive times, the gas exhibits a steady nonthermal momentum distribution. At length scales increasing from the drive one to the system size, this distribution features in turn: (i) a power-law spectrum with an exponent close to the analytical result for a particle cascade in weak-wave turbulence, and (ii) a spectrum reminiscent of a nonthermal fixed point associated with universal coarsening in an isolated 2D gas. In further experiments, based on anisotropic driving, we reveal the complete qualitative picture of how the steady-state cascade forms.

DOI
10.1103/physrevlett.133.243402

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We observe an inverse turbulent-wave cascade, from small to large length scales, in a driven homogeneous 2D Bose gas. Starting with an equilibrium condensate, we drive the gas isotropically on a length scale much smaller than its size, and observe a nonthermal population of modes with wavelengths larger than the drive one. At long drive times, the gas exhibits a steady nonthermal momentum distribution. At length scales increasing from the drive one to the system size, this distribution features in turn: (i) a power-law spectrum with an exponent close to the analytical result for a particle cascade in weak-wave turbulence, and (ii) a spectrum reminiscent of a nonthermal fixed point associated with universal coarsening in an isolated 2D gas. In further experiments, based on anisotropic driving, we reveal the complete qualitative picture of how the steady-state cascade forms.
Relaxation and condensation of an isolated low-energy Bose gas provide an ideal setting for the study of the universal features of far-from-equilibrium many-body dynamics and the emergence of long-range order. Conceptually, the emergence of such order involves two steps: the formation of local coherence, on a system-specific microscopic lengthscale, and the spreading of coherence, over lengthscales much larger than any microscopic scale. The latter is understood in terms of universal phase-ordering kinetics, or coarsening, characterized by an algebraic growth of the coherence length. Here, for a homogeneous Bose gas with tunable interactions, we show that the former also has a universal description, within the framework of weak wave turbulence (WWT). Specifically, the initial transport of particles to low momenta corresponds to an inverse turbulent cascade that is, in agreement with the WWT theory, characterized by a power-law momentum distribution, with exponent γ=2.4(1)γ= 2.4(1)γ=2.4(1), and transport times ∝(na)−2{\propto} (na)^{-2}∝(na)−2, where nnn is the gas density and aaa the sss-wave scattering length.
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.
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.
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.
We investigate the emergence and evolution of turbulence in Bose-Einstein condensates driven far from equilibrium through a controlled energy injection. By applying a timeperiodic modulation for a fixed duration, we trigger nonequilibrium dynamics that lead to the formation of a turbulent state characterized by a complex distribution of excitations and density fluctuations. Our study focuses on the post-excitation relaxation process, where we observe a transient broadening of the momentum distribution: a signature of a direct energy cascade followed by an inverse particle cascade due to the nonlinear intra-scales energy transfer. Through systematic analysis based on the nonthermal fixed points and weak wave turbulence theories, we identify universal scaling laws governing the repopulation dynamics of the condensate. These findings advance our understanding of turbulent relaxation pathways in superfluids while enhancing the understanding of driven-dissipative quantum systems.
When a turbulent Bose-Einstein condensate (BEC) is driven out-of-equilibrium at a scale much smaller than the system size, nonlinear wave interactions transfer particles towards large scales in an inverse cascade process. In this work, we numerically study wave turbulence in a three-dimensional BEC in forced and dissipative inverse cascade settings. We observe that when the forcing rate increases, thereby increasing the particle flux, the turbulence spectrum gradually transitions from the weak-wave Kolmogorov-Zakharov cascade to a critical balance state characterized by a range of scales with balanced linear and nonlinear dynamic timescales. Further forcing increases lead to a coherent condensate component superimposed with Bogoliubov-type acoustic turbulence. The role of vortices in such a strongly forced state is marginal, which makes this new state distinct from the strongly turbulent state composed of a tangle of quantized vortex lines. We then use our predictions and numerical data to formulate a new out-of-equilibrium equation of state for the 3D inverse cascade.