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

Abstract

Bogoliubov waves are fundamental excitations of Bose-Einstein Condensates (BECs). They emerge from a perturbed ground state and interact nonlinearly, triggering turbulent cascades. Here, we study turbulent BECs theoretically and numerically using the 3D Gross-Pitaevskii model and its associated wave-kinetic equations. We derive a new 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 BEC equation of state reported in [Dora et al. Nature 620,521 (2023)], and to suggest new experimental settings.

Figures (5)

• Figure 1: Numerical results obtained by simulating WKE and GPE. (a) Dimensionless energy spectra $E(k)/\sqrt{c_sP_0\xi^3}$ vs. dimensionless wave number $k\xi$ with theoretical predictions \ref{['eq:SZspectrum']} and \ref{['eq:KZshort']} superimposed. Run 1 is performed using the WKE \ref{['eq:wke_disp']} and contains both the acoustic (long-wave) and short-wave regimes, whereas run 2 uses the $3D$ acoustic WKE \ref{['eq:wke_acous']}. The GPE runs 3 and 4 are obtained for homogeneous cases with stochastic forcing, and runs 5 and 6 are obtained with a shaken potential trap. All the GPE runs are optimized such that the inertial range extends through the scales of interest. (b) Corresponding energy fluxes of run 1-4 normalized by their values measured in the inertial range. (c) Snapshot of density for GPE run 5 to verify the prediction in the acoustic limit. (d) Snapshot of density for GPE run 5 to verify the prediction in the short-wave limit.
• Figure 2: Equation of State (EoS) regenerated from the experimental data of dogra2023universal. (a) Particle spectrum amplitude $n^{\rm exp}$ .vs. energy flux $\epsilon$, where $n^{\rm exp}$ is taken either as $n^{\rm exp}_{4W}$ in green fitting by the 4-wave prediction \ref{['eq:EoS4']} or $n^{\rm exp}_{3W}$ in magenta fitting by the 3-wave prediction \ref{['eq:EoS3']}. The fitting procedure is indicated by the inset of (a): Each experimental spectrum $n^\textbf{exp}_k$ is compensated by $k^3 \log^{1/3}(\frac{k}{k_{\rm f}})$ and $k^3$, respectively. $n^{\rm exp}_{4W}$ and $n^{\rm exp}_{4W}$ are calculated by averaging the corresponding compensated spectrum on a plateau range. We take $n^{\rm exp}$, $n^{\rm exp}_{4W}$ or $n^{\rm exp}_{3W}$ depending on which is closer to its theoretical prediction, $n_{4W}$ or $n_{3W}$. (b) Non-dimensional particle spectrum amplitude $n^{\rm exp}/N$ .vs. non-dimensional flux $\tilde{\epsilon}$. Theoretical predictions $C_{\rm 4W}\tilde{\epsilon}^{1/3}$ and $C_{\rm 3W}\tilde{\epsilon}^{1/2}$ and the ones replacing $C_{\rm 4W}$ and $C_{\rm 3W}$ by the mean values of the respective constants obtained by fits of the experimental spectra, $\langle C_{\rm 4W}^{\rm exp} \rangle$ and $\langle C_{\rm 3W}^{\rm exp} \rangle$, are superimposed.
• Figure S1: (a) Coordinates system to compute $\Delta$; (b) Wave vector triad.
• Figure S2: Collision integrals in their convergence windows: $I(x)$ is given by Eq. \ref{['eq:IacA']} in (a) and by Eq. \ref{['eq:Ishort']} in (b).
• Figure S3: Normalized spatio-temporal spectral density of $\Psi(\bm{x}, t)$ for the GPE simulations of (a) run 3, and (b) run 4; Frequency broadening $\delta\omega(k)$ (blue points) extracted from the corresponding spatial-temporal density spectra for (c) run 3, and (d) run 4.