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arXiv:2408.15163 · Communications Physics

Turbulence and far-from-equilibrium equation of state of Bogoliubov waves in Bose-Einstein condensates

Ying Zhu, Giorgio Krstulovic, Sergey Nazarenko

Published Jul 8, 2026• 1 citation
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Abstract

Abstract Bogoliubov waves are fundamental excitations of Bose-Einstein condensates. They emerge from a perturbed ground state and interact nonlinearly, triggering turbulent cascades. Here, we study turbulent Bose-Einstein condensates theoretically and numerically using the three-dimensional Gross-Pitaevskii model and its associated wave-kinetic equations. We derive a Kolmogorov-like stationary spectrum for short Bogoliubov waves and find a complete analytical expression for the spectrum in the long-wave acoustic regime. We then use our predictions to explain the far-from-equilibrium equation of state observed in recent turbulent quantum gas experiments, and to suggest future experimental settings.

DOI
10.1038/s42005-026-02740-2

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Abstract Bogoliubov waves are fundamental excitations of Bose-Einstein condensates. They emerge from a perturbed ground state and interact nonlinearly, triggering turbulent cascades. Here, we study turbulent Bose-Einstein condensates theoretically and numerically using the three-dimensional Gross-Pitaevskii model and its associated wave-kinetic equations. We derive a Kolmogorov-like stationary spectrum for short Bogoliubov waves and find a complete analytical expression for the spectrum in the long-wave acoustic regime. We then use our predictions to explain the far-from-equilibrium equation of state observed in recent turbulent quantum gas experiments, and to suggest future experimental settings.
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.
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.
We theoretically and numerically study Bogoliubov-wave turbulence in three-dimensional atomic Bose-Einstein condensates with the Gross-Pitaevskii equation, investigating three spectra for the macroscopic wave function, the density distribution, and the Bogoliubov-wave distribution. In this turbulence, Bogoliubov waves play an important role in the behavior of these spectra, so that we call it Bogoliubov-wave turbulence. In a previous study [Proment et al., Phys. Rev. A 80, 051603(R) (2009)], a $\ensuremath{-}3/2$ power law in the spectrum for the macroscopic wave function was suggested by using weak wave turbulence theory, but we find that another $\ensuremath{-}7/2$ power law appears in both theoretical and numerical calculations. Furthermore, we focus on the spectrum for the density distribution, which can be observed in experiments, discussing the possibility of experimental observation. Through these analytical and numerical calculations, we also demonstrate that the previously neglected condensate dynamics induced by the Bogoliubov waves is remarkably important.
When driven out of equilibrium, a Bose-Einstein condensate develops nonlinearly interacting density waves that trigger a turbulent cascade, transferring energy toward small scales. In this Letter, we investigate the nonstationary evolution of solutions to the two-dimensional Gross-Pitaevskii equation (GPE). Through numerical simulations of both the GPE and the corresponding Wave Kinetic Equation (WKE), we identify self-similar solutions relevant to turbulence in atomic and polariton Bose-Einstein Condensates. These solutions correspond to a new type of non-thermal fixed point and exhibit characteristics of both first and second kind self-similarity. In particular, we show that the dynamics of the propagating front is universal, governed by a dimensionless universal constant βββ, which we determine numerically.
We investigate formation of Bose-Einstein condensates under nonequilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent ${x}^{*}\ensuremath{\approx}1.24$ for the three-dimensional wave action spectrum. We also study the regimes of very weak turbulence when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when the emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.
We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation (GPE) and the associated wave kinetic equation (WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.