Strong and weak wave turbulence regimes in Bose-Einstein condensates

When a turbulent Bose-Einstein condensate 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 study numerically wave turbulence in a three-dimensional Bose-Einstein condensate in forced and dissipated 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 very different 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.

💡 Research Summary

In this work the authors investigate wave‑turbulence regimes in a three‑dimensional Bose‑Einstein condensate (BEC) driven far from equilibrium by a small‑scale forcing that generates an inverse cascade of particles. The dynamics are modeled with the Gross‑Pitaevskii equation (GPE) on a periodic cubic domain (L = 2π) using a pseudo‑spectral code at 512³ resolution. A narrow band of high‑k forcing and low‑k plus high‑k dissipation create a statistically steady state with a constant particle flux Q₀. By varying the forcing amplitude f₀, the authors span more than five orders of magnitude in Q₀ and observe three distinct regimes.

At low flux the spectrum follows the weak‑wave‑turbulence (WWT) Kolmogorov‑Zakharov prediction nₖ = C|Q₀|^{1/3}k^{‑7/3} with C≈7.58×10⁻², and the spatio‑temporal Fourier transform (STFT) shows a narrow ridge along the linear dispersion ωₖ = αk², confirming δ ω(k) ≪ ωₖ.

When the flux is increased, a critical‑balance (CB) window appears around the healing‑length scale k ≈ 1/ξ. In this window the linear and nonlinear time scales become comparable (δ ω ≈ ωₖ) and the spectrum steepens to k^{‑4}. The CB regime is an intermediate state; the crossover wavenumber between KZ and CB scales as |Q₀|^{1/5}.

For the strongest forcing the CB balance breaks down. A coherent condensate forms at k ≈ 0, and the remaining excitations behave as Bogoliubov sound waves with dispersion ω_B(k)=β|ψ₀|² ± √(2αβ|ψ₀|²k²+α²k⁴). The high‑k part of the spectrum follows a thermal Bogoliubov scaling nₖ ∝ |Q₀|^{1/2}k^{‑2}, while the low‑k condensate contribution decays roughly as k^{‑7}. Importantly, incompressible kinetic energy associated with vortices remains a minor fraction of the total energy, indicating that vortex tangles do not dominate this strong‑turbulence state.

Using the measured spectra the authors derive a non‑equilibrium equation of state linking condensate density to particle flux, |ψ₀|² ∝ |Q₀|^{2/5}. This relation provides a thermodynamic‑like description of the inverse‑cascade BEC far from equilibrium.

Overall, the paper demonstrates a clear pathway from weak wave turbulence through a critical‑balance intermediate regime to a condensate‑dominated strong turbulence, highlighting the marginal role of vortices and establishing new scaling laws for the inverse particle cascade in three‑dimensional Bose‑Einstein condensates.