Route to turbulence in a trapped Bose-Einstein condensate
J. A. Seman, E. A. L. Henn, R. F. Shiozaki, G. Roati, F. J. Poveda-Cuevas, K. M. F. Magalhães, V. I. Yukalov, M. Tsubota, M. Kobayashi, K. Kasamatsu, V. S. Bagnato
We have studied a Bose-Einstein condensate of $^{87}Rb$ atoms under an oscillatory excitation. For a fixed frequency of excitation, we have explored how the values of amplitude and time of excitation must be combined in order to produce quantum turbulence in the condensate. Depending on the combination of these parameters different behaviors are observed in the sample. For the lowest values of time and amplitude of excitation, we observe a bending of the main axis of the cloud. Increasing the amplitude of excitation we observe an increasing number of vortices. The vortex state can evolve into the turbulent regime if the parameters of excitation are driven up to a certain set of combinations. If the value of the parameters of these combinations is exceeded, all vorticity disappears and the condensate enters into a different regime which we have identified as the granular phase. Our results are summarized in a diagram of amplitude versus time of excitation in which the different structures can be identified. We also present numerical simulations of the Gross-Pitaevskii equation which support our observations.
We numerically study spin turbulence in a two-dimensional trapped spin-1 spinor Bose--Einstein condensate, focusing on the energy spectrum. The spin turbulence in the trapped system is generated by instability of the helical structure of the spin density vector in the initial state. Our numerical calculation finds that in the trapped system the spectrum of the spin-dependent interaction energy in the ferromagnetic case exhibits a -7/3 power law, which was confirmed in a uniform system by our previous study. The relation between the -7/3 power law and the motion of the spin density vector is di…
The decay of multicharged vortices in trapped Bose-Einstein condensates may lead to a disordered vortex state consistent with the Vinen regime of turbulence, characterized by an absence of large-scale flow and an incompressible kinetic energy spectrum $E\propto k^{-1}$. In this work, we study numerically the dynamics of a three-dimensional harmonically trapped Bose-Einstein condensate excited to a Vinen regime of turbulence through the decay of two doubly-charged vortices. First, we study the momentum distribution and observe the emergence of a power-law behavior $n(k)\propto k^{-3}$ consisten…
We investigate a procedure to generate turbulence in a trapped Bose-Einstein condensate which takes advantage of the decay of multicharged vortices. We show that the resulting singly-charged vortices twist around each other, intertwined in the shape of helical Kelvin waves, which collide and undergo vortex reconnections, creating a disordered vortex state. By examining the velocity statistics, the energy spectrum, the correlation functions and the temporal decay, and comparing these properties with the properties of ordinary turbulence and observations in superfluid helium, we conclude that th…
Turbulence, the complicated fluid behavior of nonlinear and statistical nature, arises in many physical systems across various disciplines, from tiny laboratory scales to geophysical and astrophysical ones. The notion of turbulence in the quantum world was conceived long ago by Onsager and Feynman, but the occurrence of turbulence in ultracold gases has been studied in the laboratory only very recently. Albeit new as a field, it already offers new paths and perspectives on the problem of turbulence. Herein we review the general properties of quantum gases at ultralow temperatures paying partic…
In turbulence phenomena, including the quantum turbulence in superfluids, an energy flux flows from large to small length scales, composing a cascade of energy. A universal characteristic of turbulent flows is the existence of a range of scales where the energy flux is scale-invariant: this interval of scales is often referred to as inertial region. This property is fundamental as, for instance, in turbulence of classical fluids it characterizes the behavior of statistical features such as spectra and structure functions. Here we show that also in decaying quantum turbulence generated in trapp…
We investigate two-dimensional turbulence in finite-temperature trapped Bose-Einstein condensates within damped Gross-Pitaevskii theory. Turbulence is produced via circular motion of a Gaussian potential barrier stirring the condensate. We systematically explore a range of stirring parameters and identify three regimes, characterized by the injection of distinct quantum vortex structures into the condensate: (A) periodic vortex dipole injection, (B) irregular injection of a mixture of vortex dipoles and co-rotating vortex clusters, and (C) continuous injection of oblique solitons that decay in…
After over a decade of experiments generating and studying the physics of quantized vortices in atomic gas Bose-Einstein condensates, research is beginning to focus on the roles of vortices in quantum turbulence, as well as other measures of quantum turbulence in atomic condensates. Such research directions have the potential to uncover new insights into quantum turbulence, vortices and superfluidity, and also explore the similarities and differences between quantum and classical turbulence in entirely new settings. Here we present a critical assessment of theoretical and experimental studies …
In a recent experiment, Kwon et. al (arXiv:1403.4658 [cond-mat.quant-gas]) generated a disordered state of quantum vortices by translating an oblate Bose-Einstein condensate past a laser-induced obstacle and studying the subsequent decay of vortex number. Using mean-field simulations of the Gross-Pitaevskii equation, we shed light on the various stages of the observed dynamics. We find that the flow of the superfluid past the obstacle leads initially to the formation of a classical-like wake, which later becomes disordered. Following removal of the obstacle, the vortex number decays due to vor…
We present experimental observations and numerical simulations of nonequilibrium spatial structures in a trapped Bose-Einstein condensate subject to oscillatory perturbations. In experiment, first, there appear collective excitations, followed by quantum vortices. Increasing the amount of the injected energy leads to the formation of vortex tangles representing quantum turbulence. We study what happens after the regime of quantum turbulence, with increasing further the amount of injected energy. In such a strongly nonequilibrium Bose-condensed system of trapped atoms, vortices become destroyed…
Generation of different nonequilibrium states in trapped Bose-Einstein condensates is studied by numerically solving nonlinear Schrodinger equation. Inducing nonequilibrium states by shaking the trap, the following states are created: weak nonequilibrium, the state of vortex germs, the state of vortex rings, the state of straight vortex lines, the state of deformed vortices, vortex turbulence, grain turbulence, and wave turbulence. A characterization of nonequilibrium states is advanced by introducing effective temperature, Fresnel number, and Mach number.
The generation of isotropic vortex configurations in trapped atomic Bose-Einstein condensates offers a platform to elucidate quantum turbulence on mesoscopic scales. We demonstrate that a laser-induced obstacle moving in a figure-eight path within the condensate provides a simple and effective means to generate an isotropic three-dimensional vortex tangle due to its minimal net transfer of angular momentum to the condensate. Our characterisation of vortex structures and their isotropy is based on projected vortex lengths and velocity statistics obtained numerically via the Gross-Pitaevskii equ…
We have studied computationally the collision dynamics of spin-2 Bose--Einstein condensates initially confined in a triple-well trap. Depending on the intra- and inter-component relative phases of the initial state spinor wave function, the collision of the three condensate segments produces one of many possible vortex-antivortex lattices after which the system transitions to quantum turbulence. We find that the emerging vortex lattice structures can be described in terms of multi-wave interference. We show that the three-segment collisions can be used to systematically produce staggered vorte…
The process of exciting the gas of trapped bosons from an equilibrium initial state to strongly nonequilibrium states is described as a procedure of symmetry restoration caused by external perturbations. Initially, the trapped gas is cooled down to such low temperatures, when practically all atoms are in Bose-Einstein condensed state, which implies the broken global gauge symmetry. Excitations are realized either by imposing external alternating fields, modulating the trapping potential and shaking the cloud of trapped atoms, or it can be done by varying atomic interactions by means of Feshbac…
We present a route towards the quantum simulation of exotic quantum magnetism in ion traps by exploiting dual relations between different spin models. Our strategy allows one to start from Hamiltonians that can be realized with current technology, while properties of an exotic dual model are inferred from measurements of non-local, string-order-like, operators. The latter can be achieved from fluorescence, or from certain spectroscopic measurements, both of which can be combined with finite-size scaling by controlling the number of ions in the dual quantum simulator. We apply this concept to p…
Quantum turbulence deals with the phenomenon of turbulence in quantum fluids, such as superfluid helium and trapped Bose-Einstein condensates (BECs). Although much progress has been made in understanding quantum turbulence, several fundamental questions remain to be answered. In this work, we investigated the entropy of a trapped BEC in several regimes, including equilibrium, small excitations, the onset of turbulence, and a turbulent state. We considered the time evolution when the system is perturbed and let to evolve after the external excitation is turned off. We derived an expression for …
We theoretically study the nonlinear dynamics of the instability of counter-superflow in two miscible Bose-Einstein condensates. The condensates become unstable when the relative velocity exceeds a critical value, which is called counter-superflow instability. We reveal that the counter-superflow instability can lead to quantum turbulence by numerically solving the coupled Gross-Pitaevskii equations. The modes amplified by the instability grow into solitons and decay into quantized vortices.Eventually, the vortices become tangled and quantum turbulence of two superfluids. We show that this pro…
The field of quantum turbulence is related to the manifestation of turbulence in quantum fluids, such as liquid helium and ultracold gases. The concept of turbulence in quantum systems was conceived more than 70 years ago by Onsager and Feynman, but the study of turbulent ultracold gases is very recent. Although it is a young field, it already provides new approaches to the problem of turbulence. We review the advances and present status, of both theory and experiments, concerning atomic Bose-Einstein condensates (BECs). We present the difficulties of characterizing turbulence in trapped BECs,…
Rotating properties of Bose-Einstein condensates in synthetic magnetic field are studied by numerically solving the Gross-Pitaevskii equation and compared with condensates confined in the rotating trap. It seems that it is more difficult to add large angular momentum to condensates spined up by the synthetic magnetic field than by the rotating trap. However, strengthening the repulsive interaction between atoms is an effective and realizable route to overcome this problem and can at least generate vortex-lattice-like structures. In addition, the validity of the Feynman rule for condensates in …
In this work we present numerical study of a trapped Bose-Einstein condensate perturbed by an alternating potential. The relevant physical situation has been recently realized in experiment, where the trapped condensate of $^{87}$Rb, being strongly perturbed, exhibits the set of spatial structures. Firstly, regular vortices are detected. Further, increasing either the excitation amplitude or modulation time results in the transition to quantum vortex turbulence, followed by a granular state. Numerical simulation of the nonequilibrium Bose-condensed system is based on the solution of the time-d…
We show theoretically that elongated vortex-antivortex dipoles can be created controllably in trapped Bose-Einstein condensates, using known experimental techniques. Vortex dipoles of sufficient length are unstable and cascade into slow vortex rings which ultimately decay via sound emission. This instability of antiparallel vortex line elements, which self-generates Kelvin waves on vortex loops and in trapped atomic gases, may play a role in bridging the Kelvin-wave and Kolmogorov-Richardson cascades of quantum turbulence.
We have calculated the low-lying elementary excitations of three-dimensional few-vortex arrays in trapped Bose-Einstein condensates. The number of different Kelvin-Tkachenko vortex wave branches matches the number of vortices in the condensate. The lowest odd-parity modes exhibit superfluid gyroscopic vortex motion. Experimentally, these modes could be excited and observed individually or in connection with the formation and decay of quantum turbulence.
We theoretically study the development of quantum turbulence from two counter-propagating superfluids of miscible Bose-Einstein condensates by numerically solving the coupled Gross-Pitaevskii equations. When the relative velocity exceeds a critical value, the counter-superflow becomes unstable and quantized vortices are nucleated, which leads to isotropic quantum turbulence consisting of two superflows. It is shown that the binary turbulence can be realized experimentally in a trapped system.
A trapped Bose-Einstein condensate, being strongly perturbed, exhibits several spatial structures. First, there appear quantum vortices. Increasing the amount of the injected energy leads to the formation of vortex tangles representing quantum vortex turbulence. Continuing energy injection makes the system so strongly perturbed that vortices become destroyed and there develops another kind of spatial structures with essentially heterogeneous spatial density. These structures consist of high-density droplets, or grains, surrounded by the regions of low density. The droplets are randomly distrib…
We numerically simulate the free decay of two-dimensional quantum turbulence in a large, homogeneous Bose-Einstein condensate. The large number of vortices, the uniformity of the density profile and the absence of boundaries (where vortices can drift out of the condensate) isolate the annihilation of vortex-antivortex pairs as the only mechanism which reduces the number of vortices, $\Nv$, during the turbulence decay. The results clearly reveal that vortex annihilation is a four-vortex process, confirming the decay law $\Nv \sim t^{-1/3}$ where $t$ is time, which was inferred from experiments …
We present a survey of macroscopic excitations of harmonically confined Bose-Einstein condensates (BEC), described by Gross-Pitaevskii (GP) equation, in search of routes to develop quantum turbulence. These excitations can all be created by phase imprinting techniques on an otherwise equilibrium Bose-Einstein condensate. We analyze two crossed vortices, two parallel anti-vortices, a vortex ring, a vortex with topological charge Q = 2, and a tangle of 4 vortices. Since GP equation is time-reversal invariant, we are careful to distinguish time intervals in which this symmetry is preserved and th…
The vortex nucleation and the emergence of quantum turbulence induced by oscillating magnetic fields, introduced by Henn E A L, et al. 2009 (Phys. Rev. A 79, 043619) and Henn E A L, et al. 2009 (Phys. Rev. Lett. 103, 045301), left a few open questions concerning the basic mechanisms causing those interesting phenomena. Here, we report the experimental observation of the slosh dynamics of a magnetically trapped $^{87}$Rb Bose-Einstein condensate (BEC) under the influence of a time-varying magnetic field. We observed a clear relative displacement in between the condensed and the thermal fraction…
I review the basic physics of ultracold dilute trapped atomic gases, with emphasis on Bose-Einstein condensation and quantized vortices. The hydrodynamic form of the Gross-Pitaevskii equation (a nonlinear Schr{ö}dinger equation) illuminates the role of the density and the quantum-mechanical phase. One unique feature of these experimental systems is the opportunity to study the dynamics of vortices in real time, in contrast to typical experiments on superfluid $^4$He. I discuss three specific examples (precession of single vortices, motion of vortex dipoles, and Tkachenko oscillations of a vort…
The nonlinear dynamics of a Bose-Einstein condensate (BEC) of dipolar excitons trapped in an external confining potential in coupled quantum wells is analysed. It is demonstrated that under typical experimental conditions the dipolar excitons BEC can be described by a generalized Gross-Pitaevskii equation with the local interaction between the excitons, which depends on the exciton distribution function. It is shown that, if the system is pumped at sufficiently high frequencies, a steady turbulent state can be formed.
We propose a scheme for generating two-dimensional turbulence in harmonically trapped atomic condensates with the novelty of controlling the polarization (net rotation) of the turbulence. Our scheme is based on an initial giant (multicharged) vortex which induces a large-scale circular flow. Two thin obstacles, created by blue-detuned laser beams, speed up the decay of the giant vortex into many singly-quantized vortices of the same circulation; at the same time, vortex-antivortex pairs are created by the decaying circular flow past the obstacles. Rotation of the obstacles against the circular…
We study the relaxation dynamics of an isolated zero temperature quasi-two-dimensional superfluid Bose-Einstein condensate (BEC) that is imprinted with a spatially random distribution of quantum vortices. Following a period of vortex annihilation, we find that the remaining vortices self-organise into two macroscopic coherent `Onsager vortex' clusters that are stable indefinitely. We demonstrate that this occurs due to a novel physical mechanism --- the evaporative heating of the vortices --- that results in a negative temperature phase transition in the vortex degrees of freedom. At the end o…
Turbulence in the quantum (superfluid) regime, similarly to its classical counterpart, continues to attract a great deal of scientific inquiry, due to the yet high number of unresolved problems. While turbulent states can be routinely created in degenerate atomic gases, there is no generic scheme to produce turbulence in fluids of light. Under paraxial propagation, light in bulk nonlinear media behaves as a two-dimensional superfluid, described by a nonlinear Schrödinger equation formally equivalent to the Gross-Pitaevskii model of a weakly interacting Bose gas, where photon-photon interaction…
