Direct and Inverse Cascades in Turbulent Bose-Einstein Condensates

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.

Figure 1

Wave action spectra for the direct cascade compensated by theoretical prediction [Eq. (12)]. Data obtained by GPE at two different resolutions and by WKE, respectively. Inset: corresponding energy fluxes normalized by their values measured in inertial range.

Figure 2

Wave action spectra for the inverse cascade compensated by theoretical prediction [Eq. (10)]. Data obtained by GPE at two different resolutions and by WKE, respectively. Inset: corresponding particle fluxes normalized by their values measured in inertial range.

Figure 3

GPE simulations of turbulent trapped BEC. (a) Two-dimensional cut of a typical simulation (arbitrary units and scales). The trapping potential is displayed in red and density fluctuations in blue. (b) Wave action spectra normalized by their values at the respective forcing scales $k_f$ for the direct and the inverse cascades. The solid lines display the theoretical predictions [Eqs. (10) and (12)]. The vertical lines show the wave number $k_{\mathrm{trap}}=2\pi /L_{\mathrm{trap}}$ for both of the cascades, with $L_{\mathrm{trap}}$ the trap size.