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 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.
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov
k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
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 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
β, which we determine numerically.
We theoretically and numerically study spin wave turbulence in spin-1 ferromagnetic spinor Bose-Einstein condensates, finding direct and inverse cascades with power-law behavior. To derive these power exponents analytically, the conventional weak wave turbulence theory is applied to the spin-1 spinor Gross-Pitaevskii equation. Thus we obtain the $\ensuremath{-}7/3$ and $\ensuremath{-}5/3$ power laws in the transverse spin correlation function for the direct and inverse cascades, respectively. To confirm these power laws, numerical calculations are performed that obtain results consistent with these power laws.