Turbulent fluid dynamics typically involves excitations on many different length scales. Classical incompressible fluids can be cleanly represented in Fourier space enabling spectral analysis of energy cascades and other turbulence phenomena. In quantum fluids, additional phase information and singular behavior near vortex cores thwarts the direct extension of standard spectral techniques. We develop a formal and numerical spectral analysis for $\text{U}(1)$ symmetry-breaking quantum fluids suitable for analyzing turbulent flows, with specific application to the Gross--Pitaevskii fluid. Our analysis builds naturally on the canonical approach to spectral analysis of velocity fields in compressible quantum fluids and establishes a clear correspondence between energy spectral densities, power spectral densities, and autocorrelation functions, applicable to energy residing in velocity, quantum pressure, interaction, and potential energy of the fluid. Our formulation includes all quantum phase information and also enables arbitrary resolution spectral analysis, a valuable feature for numerical analysis. A central vortex in a trapped planar Bose--Einstein condensate provides an analytically tractable example with spectral features of interest in both the infrared and ultraviolet regimes. Sampled distributions modeling the dipole gas, plasma, and clustered regimes exhibit velocity correlation length increasing with vortex energy, consistent with known qualitative behavior across the vortex clustering transition. The spectral analysis of compressible quantum fluids presented here offers a rigorous tool for analyzing quantum features of superfluid turbulence in atomic or polariton condensates.
Turbulent fluid dynamics typically involves excitations on many different length scales. Classical incompressible fluids can be cleanly represented in Fourier space enabling spectral analysis of energy cascades and other turbulence phenomena. In quantum fluids, additional phase information and singular behavior near vortex cores thwarts the direct extension of standard spectral techniques. We develop a formal and numerical spectral analysis for U(1) symmetry-breaking quantum fluids suitable for analyzing turbulent flows, with specific application to the Gross--Pitaevskii fluid. Our analysis builds naturally on the canonical approach to spectral analysis of velocity fields in compressible quantum fluids and establishes a clear correspondence between energy spectral densities, power spectral densities, and autocorrelation functions, applicable to energy residing in velocity, quantum pressure, interaction, and potential energy of the fluid. Our formulation includes all quantum phase information and also enables arbitrary resolution spectral analysis, a valuable feature for numerical analysis. A central vortex in a trapped planar Bose--Einstein condensate provides an analytically tractable example with spectral features of interest in both the infrared and ultraviolet regimes. Sampled distributions modeling the dipole gas, plasma, and clustered regimes exhibit velocity correlation length increasing with vortex energy, consistent with known qualitative behavior across the vortex clustering transition. The spectral analysis of compressible quantum fluids presented here offers a rigorous tool for analyzing quantum features of superfluid turbulence in atomic or polariton condensates.
We explore the energy spectra and associated fluxes of turbulent two-dimensional quantum droplets subjected to a rotating paddling potential which is removed after a few oscillation periods. A systematic analysis on the impact of the characteristics (height and velocity) of the rotating potential and the droplet atom number reveals the emergence of different dynamical response regimes. These are classified by utilizing the second-order sign correlation function and the ratio of incompressible versus compressible kinetic energies. They involve, vortex configurations ranging from vortex dipoles to vortex clusters and randomly distributed vortex-antivortex pairs. The incompressible kinetic energy spectrum features Kolmogorov (k−5/3) and Vinen like (k−1) scaling in the infrared regime, while a k−3 decay in the ultraviolet captures the presence of vortices. The compressible spectrum shows k−3/2 scaling within the infrared and k power law in the case of enhanced sound-wave emission suggesting thermalization. Significant distortions are observed in the droplet periphery in the presence of a harmonic trap. A direct energy cascade (from large to small length scales) is mainly identified through the flux. Our findings offer insights into the turbulent response of exotic phases-of-matter, featuring quantum fluctuations, and may inspire investigations aiming to unravel self-similar nonequilibrium dynamics.
Quantum turbulence indicators in dipolar Bose-Einstein condensed fluids, following emissions of vortex-antivortex pairs generated by a circularly moving detuned laser, are being provided by numerical simulations of the corresponding quasi-two-dimensional Gross-Pitaevskii formalism with repulsive contact interactions combined with tunable dipole-dipole strength. The critical velocities of two variants of a circularly moving obstacle are determined and analyzed for vortex-antivortex nucleation in the form of regular and cluster emissions. The turbulent dynamical behavior is predicted to follow closely the initial emission of vortex-antivortex pairs, relying on the expected Kolmogorov classical scaling law, which is verified by the spectral analysis of the incompressible part of the kinetic energy. Within our aim to provide further support in the investigations to date of quantum turbulence, which have been focused on nondipolar Bose-Einstein condensates, we emphasize the role of dipole-dipole interactions in the fluid dynamics.
We review turbulence in ultracold quantum gases, using the scalar contact-interaction Bose-Einstein condensate as the reference system for quantized circulation, compressibility, vortices, sound, and cascades. We focus on the quantitative diagnostics that connect helium and classical phenomenology to microscopic wave-function dynamics: incompressible and compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. These diagnostics distinguish equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence, and show why power laws alone are insufficient evidence for a cascade. We survey experiments on two-dimensional Onsager clustering, three-dimensional vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. We close by briefly placing the contact-interaction scalar superfluid system in a broader landscape of nonlocal, multicomponent, fermionic, and driven-dissipative quantum fluids, where turbulence concepts can be tested for universality.
We present a quantitative analytical framework for calculating two-point velocity correlations in compressible quantum fluids, focusing on two key classes of superfluid excitations: vortices and rarefaction pulses. We employ two complementary approaches. First, we introduce a new Ansatz for vortex cores in planar quantum fluids that enhances analytic integrability. This Ansatz yields closed-form expressions for power spectra and velocity correlations for general vortex distributions. Using it, we identify distinct signatures of short- and long-range velocity correlations corresponding to vortex dipoles and vortex pairs, respectively. Second, we analyze the fast rarefaction pulse regime of the Jones-Roberts soliton. By applying the asymptotic high-velocity wave function, we derive analytic expressions for the velocity power spectrum and correlation function, capturing the soliton characteristic length scale. We validate our analytical results for the homogeneous system against numerical treatment of a large trapped system, finding close quantitative agreement. Our findings provide the first analytic treatment of velocity correlations for Jones-Roberts solitons in quantum fluids of light [M. Baker-Rasooli et al., Phys. Rev. Lett. 134, 233401 (2025)] and establish a foundation for characterizing vortices and solitons in compressible quantum fluids.
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.
We examine the energy cascades and quantum vortex structures in two-dimensional quantum turbulence through a special unitary time evolution algorithm. An early attempt at using the Lattice Boltzmann Method proved successful in correctly representing some features of the Nonlinear Schrodinger System (NLS), such as the phase shift following the one-dimensional soliton-soliton collision, as well as the two-dimentional modulation instability. However, to accurately evaluate NLS, the implicit Euler method is required to resolve the time evolution, which is computationally expensive. A more accurate and efficient method, the Quantum Lattice Gas model is employed to simulate the quantum turbulence governed by the Gross-Pitaevskii equation, an equaiton that describes the evolution of the ground state wave function for a Bose-Einstein condensate (BEC). It is discovered that when the ratio of the internal energy to the kinetic energy is below 0.05, an unexpected short Poincare recurrence occurs independent of the initial profile of the wave function. It is demonstrated that this short recurrence is destroyed as the internal energy is strengthened. to compare the two-dimensional quantum turbulence with its classical counterpart, the incompressible energy spectra of quantum turbulence is analyzed. However, the result reveals no sign of dual cascades which is a hallmark of the classical incompressible two-dimensional fluid (inverse energy cascade to large scales with a direct cascade of enstrophy to small scales). It is the spectra of the compressible energy that can exhibits multiple cascades, but this is strongly dependent on the initial condition.