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arXiv:2606.18059 · Brazilian Journal of Physics

Universal Scaling and Relaxation in Decaying Turbulence of Bose Gases

Arnol Daniel Garcia Orozco, Michelle A. Moreno-Armijos, Sarah Sab, A. R. Fritsch, Vanderlei Salvador Bagnato

Published Oct 10, 2026• 0 citations
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Abstract

Abstract Understanding the relaxation dynamics of isolated quantum systems remains a central challenge in nonequilibrium physics. In this short review, we discuss recent experimental and theoretical advances in universal dynamics of turbulent Bose gases, with particular emphasis on nonthermal fixed points and wave turbulence in our experimental system at the São Carlos Institute of Physics of the University of São Paulo. We highlight the emergence of self-similar scaling behavior and its connection to particle and energy transport across momentum scales. In addition, we discuss an approach in which a differential equation has a universal scaling solution, providing a practical framework for extracting universal exponents from limited regions of the momentum distribution. Furthermore, we recap experimental observations of direct and inverse cascades in a three–dimensional trapped Bose–Einstein condensate, revealing distinct relaxation stages governed by universal scaling laws. These results demonstrate that turbulence plays a key role in far–from–equilibrium dynamics, offering a unified perspective on transport processes and thermalization in quantum many–body systems.

DOI
10.1007/s13538-026-02155-y

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Abstract Understanding the relaxation dynamics of isolated quantum systems remains a central challenge in nonequilibrium physics. In this short review, we discuss recent experimental and theoretical advances in universal dynamics of turbulent Bose gases, with particular emphasis on nonthermal fixed points and wave turbulence in our experimental system at the São Carlos Institute of Physics of the University of São Paulo. We highlight the emergence of self-similar scaling behavior and its connection to particle and energy transport across momentum scales. In addition, we discuss an approach in which a differential equation has a universal scaling solution, providing a practical framework for extracting universal exponents from limited regions of the momentum distribution. Furthermore, we recap experimental observations of direct and inverse cascades in a three–dimensional trapped Bose–Einstein condensate, revealing distinct relaxation stages governed by universal scaling laws. These results demonstrate that turbulence plays a key role in far–from–equilibrium dynamics, offering a unified perspective on transport processes and thermalization in quantum many–body systems.
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.
Investigating the initial conditions that lead many-body quantum systems to an out-of-equilibrium state is fundamental for understanding their thermalization dynamics. In this work we observe the relaxation for two regimes of excitation that can drive the turbulent Bose-Einstein condensate into two distinct final states, and are defined by the amount of energy injected into the system. The subcritical regime is characterized by a lower injection of energy, which can lead to an inverse particle cascade and, consequently, to the BEC mode repopulation during the relaxation process. The supercritical regime is marked by a higher energy injection, that may lead to the BEC dissolution and a final thermal state. In both cases we observe relaxation stages that exhibit the same key features: a direct cascade, a non-thermal fixed point with the same exponents, a prethermalization region and, finally, the thermalization of the system. In the final thermalization stage, universal scaling is observed for both regimes, even though their final states are completely different. By analyzing the coherence length of our turbulent cloud, we clearly visualize the recovery and the loss of the coherence for the subcritical and supercritical regimes after relaxation. These results indicate that the evolution of turbulence occurs independent of its initial conditions and of the final state achieved.
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.
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.
Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution toward thermalization. Notably, the concept of nonthermal fixed points (NTFPs)-responsible for the emergence of spatiotemporal universal scaling in far-from-equilibrium systems-has played a crucial role in both theoretical and experimental investigations. In this work, we introduce a differential equation that has the universal scaling associated with NTFPs as a solution. The advantage of working with a differential equation, rather than only with its solution, is that we can extract several insightful properties not necessarily present in the solution alone. How the differential equation is derived allows physical interpretation of the universal exponents in terms of the time dependence of the amplitude of the distributions and their momentum scaling. Employing two limiting cases of the equation, we determined the universal exponents related to the scaling using the distributions near just two momentum values. We established a solid agreement with previous investigations by validating this approach with three distinct physical systems. This consistency highlights the universal nature of scaling due to NTFPs and emphasizes the predictive capabilities of the proposed differential equation. Moreover, under specific conditions, the equation predicts a power-law related to the ratio of the two universal exponents, leading to implications concerning particle and energy transport. This suggests that the observed power-laws in far-from-equilibrium turbulent fluids could be related to the universal scaling due to NTFPs, potentially offering insights into the study of turbulence.