Vortices and turbulence in trapped atomic condensates

Hidden vortices and Feynman rule in Bose-Einstein condensates with density-dependent gauge potential

In this paper, we numerically investigate the vortex nucleation in a Bose-Einstein condensate (BEC) trapped in a double-well potential and subjected to a density-dependent gauge potential. A rotating Bose-Einstein condensate, when confined in a double-well potential, not only gives rise to visible vortices but also produces hidden vortices. We have empirically developed Feynman's rule for the number of vortices versus angular momentum in Bose-Einstein condensates in the presence of density-dependent gauge potentials. The variation of the average angular momentum with the number of vortices is also sensitive to the nature of the nonlinear rotation due to the density-dependent gauge potentials. The empirical result agrees well with the numerical simulations and the connection is verified by means of curve-fitting analysis. The modified Feynman rule is further confirmed for the BECs confined in harmonic and toroidal traps. In addition, we show the nucleation of vortices in double-well and toroidally confined Bose-Einstein condensates solely through nonlinear rotations (without any trap rotation) arising through the density-dependent gauge potential.

Temperature-Dependent Decay of Quasi-Two-Dimensional Vortices across the BCS-BEC Crossover

We systematically study the decay of quasi-two-dimensional vortices in an oblate strongly interacting Fermi gas over a wide interaction range and observe that, as the system temperature is lowered, the vortex lifetime increases in the Bose-Einstein condensate (BEC) regime but decreases at unitarity and in the Bardeen-Cooper-Schrieffer (BCS) regime. The observations can be qualitatively captured by a phenomenological model simply involving diffusion and two-body collisional loss, in which the vortex lifetime is mostly determined by the slower process of the two. In particular, the counterintuitive vortex decay in the BCS regime can be interpreted by considering the competition between the temperature dependence of the vortex annihilation rate and that of unpaired fermions. Our results suggest a competing mechanism for the complex vortex decay dynamics in the BCS-BEC crossover for the fermionic superfluids.

Observing the loss and revival of long-range phase coherence through disorder quenches

Relaxation of quantum systems is a central problem in nonequilibrium physics. In contrast to classical systems, the underlying quantum dynamics results not only from atomic interactions but also from the long-range coherence of the many-body wave function. Experimentally, nonequilibrium states of quantum fluids are usually created using moving objects or laser potentials, directly perturbing and detecting the system's density. However, the fate of long-range phase coherence for hydrodynamic motion of disordered quantum systems is less explored, especially in three dimensions. Here, we unravel how the density and phase coherence of a Bose-Einstein condensate of ⁶Li₂ molecules respond upon quenching on or off an optical speckle potential. We find that, as the disorder is switched on, long-range phase coherence breaks down one order of magnitude faster than the density of the quantum gas responds. After removing it, the system needs two orders of magnitude longer times to reestablish quantum coherence, compared to the density response. We compare our results with numerical simulations of the Gross-Pitaevskii equation on large three-dimensional grids, finding an overall good agreement. Our results shed light on the importance of long-range coherence and possibly long-lived phase excitations for the relaxation of nonequilibrium quantum many-body systems.

Incoherent localized structures and hidden coherent solitons from the gravitational instability of the Schrödinger-Poisson equation

The long-term behavior of a modulationally unstable conservative nonintegrable system is known to be characterized by the soliton turbulence self-organization process. We consider this problem in the presence of a long-range interaction in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for the gravitational interaction. By increasing the amount of nonlinearity, the system self-organizes into a large-scale incoherent localized structure that contains "hidden" coherent soliton states: The solitons can hardly be identified in the usual spatial or spectral domains, but their existence can be unveiled in the phase-space representation (spectrogram). We develop a theoretical approach that provides the coupled description of the coherent soliton component [governed by the Schrödinger-Poisson equation (SPE)] and of the incoherent structure [governed by a wave turbulence Vlasov-Poisson equation (WT-VPE)]. We demonstrate theoretically and numerically that the incoherent structure introduces an effective trapping potential that stabilizes the hidden coherent soliton and we show that the incoherent structure belongs to a family of stationary solutions of the WT-VPE. The analysis reveals that the incoherent structure evolves in the strongly nonlinear regime and that it is characterized by a compactly supported spectral shape. By relating the analytical properties of the hidden soliton to those of the stationary incoherent structure, we clarify the quantum-to-classical (i.e., SPE-to-VPE) correspondence in the limit ℏ/m→0: The hidden soliton appears as the latest residual quantum correction preceding the classical limit described by the VPE. This study is of potential interest for self-gravitating Boson models of fuzzy dark matter. Although we focus our paper on the Schrödinger-Poisson equation, we show that the regime of hidden solitons stabilized by an incoherent structure is general for long-range wave systems featured by an algebraic decay of the interacting potential. This work should stimulate nonlinear optics experiments in highly nonlocal nonlinear (thermal) media that mimic the long-range nature of gravitational interactions.

Sound emission and annihilations in a programmable quantum vortex collider

In quantum fluids, the quantization of circulation forbids the diffusion of a vortex swirling flow seen in classical viscous fluids. Yet, accelerating quantum vortices may lose their energy into acoustic radiations^1,2, similar to the way electric charges decelerate on emitting photons. The dissipation of vortex energy underlies central problems in quantum hydrodynamics³, such as the decay of quantum turbulence, highly relevant to systems as varied as neutron stars, superfluid helium and atomic condensates^4,5. A deep understanding of the elementary mechanisms behind irreversible vortex dynamics has been a goal for decades^3,6, but it is complicated by the shortage of conclusive experimental signatures⁷. Here we address this challenge by realizing a programmable vortex collider in a planar, homogeneous atomic Fermi superfluid with tunable inter-particle interactions. We create on-demand vortex configurations and monitor their evolution, taking advantage of the accessible time and length scales of ultracold Fermi gases^8,9. Engineering collisions within and between vortex-antivortex pairs allows us to decouple relaxation of the vortex energy due to sound emission and that due to interactions with normal fluid (that is, mutual friction). We directly visualize how the annihilation of vortex dipoles radiates a sound pulse. Further, our few-vortex experiments extending across different superfluid regimes reveal non-universal dissipative dynamics, suggesting that fermionic quasiparticles localized inside the vortex core contribute significantly to dissipation, thereby opening the route to exploring new pathways for quantum turbulence decay, vortex by vortex.

Deep-learning-based quantum vortex detection in atomic Bose–Einstein condensates

Quantum vortices naturally emerge in rotating Bose–Einstein condensates (BECs) and, similarly to their classical counterparts, allow the study of a range of interesting out-of-equilibrium phenomena, such as turbulence and chaos. However, the study of such phenomena requires the determination of the precise location of each vortex within a BEC, which becomes challenging when either only the density of the condensate is available or sources of noise are present, as is typically the case in experimental settings. Here, we introduce a machine-learning-based vortex detector motivated by state-of-the-art object detection methods that can accurately locate vortices in simulated BEC density images. Our model allows for robust and real-time detection in noisy and non-equilibrium configurations. Furthermore, the network can distinguish between vortices and anti-vortices if the phase profile of the condensate is also available. We anticipate that our vortex detector will be advantageous for both experimental and theoretical studies of the static and dynamic properties of vortex configurations in BECs.

Abrupt Transition between Three-Dimensional and Two-Dimensional Quantum Turbulence

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

Characterizing coherent structures in Bose-Einstein condensates through dynamic-mode decomposition

We use the dynamic mode decomposition (DMD) methodology to study weakly turbulent flows in two-dimensional Bose-Einstein condensates modeled by a Gross-Pitaevskii equation subject to band-limited stochastic forcing. The forcing is balanced by the removal of energy at both ends of the energy spectrum through phenomenological hypoviscosity and hyperviscosity terms. Using different combinations of these parameters, we simulate three different regimes corresponding to weak-wave turbulence, and high- and low-frequency saturation. By extracting and ranking the primary DMD modes carrying the bulk of the energy, we are able to characterize the different regimes. In particular, the proposed DMD mode projection is able to seamlessly extract the vortices present in the condensate. This is achieved despite the fact that we do not use any phase information of the condensate as it is usually not directly available in realistic atomic BEC scenarios. Being model independent, this DMD methodology should be portable to other models and experiments involving complex flows. The DMD implementation could be used to elucidate different types of turbulent regimes as well as identifying and pinpointing the existence of delicate and hidden coherent structures within complex flows.

Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links

Quantized vortices in a complex wave field described by a defocusing nonlinear Schrödinger equation with a space-varying dispersion coefficient are studied theoretically and compared to vortices in the Gross-Pitaevskii model with external potential. A discrete variant of the equation is used to demonstrate numerically that vortex knots in three-dimensional arrays of oscillators coupled by specially tuned weak links can exist for as long times as many as tens of typical vortex turnover periods.

Mutual-friction-driven turbulent statistics in the hydrodynamic regime of superfluid ^{3}He-B

It is well known that the turbulence that evolves from the tangles of vortices in quantum fluids at scales larger than the typical quantized vortex spacing ℓ has a close resemblance with classical turbulence. The temperature-dependent mutual friction parameter α(T) drives the turbulent statistics in the hydrodynamic regime of quantum fluids that involves a self-similar cascade of energy. From a simple theoretical analysis, here we show that superfluid ^{3}He-B in the presence of mutual damping exhibits a k^{-5/3} Kolmogorov energy spectrum in the entire inertial range ℓ<r<L at temperature T≲0.2T_{c}, while at T≳0.2T_{c} dissipation begins to dominate larger eddies exhibiting a k^{-3} spectrum toward the energy pumping scale L. At T≈0.35T_{c}, eddies of all size, being highly affected by damping, exhibit a k^{-3} spectrum in the entire inertial range. The consistency of this result with the predictions of recent direct numerical simulations indicates that the present theoretical framework is applicable in quantifying the hydrodynamic regime of quantum turbulence.

Quantum Ferrofluid Turbulence

We study the elementary characteristics of turbulence in a quantum ferrofluid through the context of a dipolar Bose gas condensing from a highly nonequilibrium thermal state. Our simulations reveal that the dipolar interactions drive the emergence of polarized turbulence and density corrugations. The superfluid vortex lines and density fluctuations adopt a columnar or stratified configuration, depending on the sign of the dipolar interactions, with the vortices tending to form in the low-density regions to minimize kinetic energy. When the interactions are dominantly dipolar, the decay of the vortex line length is enhanced, closely following a t^{-3/2} behavior. This system poses exciting prospects for realizing stratified quantum turbulence and new levels of generating and controlling turbulence using magnetic fields.

Turbulence in a matter-wave supersolid

Quantum turbulence associated with wave and vortex dynamics is numerically investigated for a two-dimensional trapped atomic Rydberg-dressed Bose-Einstein condensate (BEC). When the coupling constant of the soft-core interaction is over a critical value, the superfluid (SF) system can transition into a hexagonal supersolid (SS) state. Based on the Gross-Pitaevskii equation approach, we have discovered a new characteristic k^-13/3 scaling law for wave turbulence in the SS state, that coexists with the waveaction k^-1/3 and energy k^-1 cascades commonly existing in a SF BEC. The new k^-13/3 scaling law implies that the SS system exhibits a negative, minus-one power energy dispersion (E ~ k^-1) at the wavevector consistent with the radius of the SS droplet. For vortex turbulence, in addition to the presence of the Kolmogorov energy k^-5/3 and Saffman enstrophy k^-4 cascades, it is found that large amount of independent vortices and antivortices pinned to the interior of the oscillating SS results in a strong k^-1 scaling at the wavevector consistent with the SS lattice constant.

From coherent shocklets to giant collective incoherent shock waves in nonlocal turbulent flows

Understanding turbulent flows arising from random dispersive waves that interact strongly through nonlinearities is a challenging issue in physics. Here we report the observation of a characteristic transition: strengthening the nonlocal character of the nonlinear response drives the system from a fully turbulent regime, featuring a sea of coherent small-scale dispersive shock waves (shocklets) towards the unexpected emergence of a giant collective incoherent shock wave. The front of such global incoherent shock carries most of the stochastic fluctuations and is responsible for a peculiar folding of the local spectrum. Nonlinear optics experiments performed in a solution of graphene nano-flakes clearly highlight this remarkable transition. Our observations shed new light on the role of long-range interactions in strongly nonlinear wave systems operating far from thermodynamic equilibrium, which reveals analogies with, for example, gravitational systems, and establishes a new scenario that can be common to many turbulent flows in photonic quantum fluids, hydrodynamics and Bose–Einstein condensates.

Understanding turbulent flows arising from random dispersive waves that interact through nonlinearities is a challenging issue in physics. Here, the authors model and observe experimentally in a nonlinear optics set-up the transition between a sea of small-scale shocklets and a giant collective shock wave.

Vortex gyroscope imaging of planar superfluids

We propose a robust imaging technique that makes it possible to distinguish vortices from antivortices in quasi-two-dimensional Bose-Einstein condensates from a single image of the density of the atoms. Tilting the planar condensate prior to standard absorption imaging excites a generalized gyroscopic mode of the condensate, revealing the sign and location of each vortex. This technique is anticipated to enable experimental measurement of the incompressible kinetic energy spectrum of the condensate and the observation of a negative-temperature phase transition of the vortex gas, driven by two-dimensional superfluid turbulence.

Introduction to quantum turbulence

The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose-Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article introduces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics.