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Quantum Turbulence In Two Dimensional Bose-Einstein Condensates

Bo Zhang

Published Jan 1, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.21220/s2-yn2e-fw96

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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.
Quantum vortex structures and energy cascades are examined for two-dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice algorithm, is employed to simulate the Bose-Einstein condensate governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincaré recurrence time. It is demonstrated that such short recurrence times are destroyed by stronger nonlinear interaction. The similar loss of Poincaré recurrence is also seen in the 3D GP equation. Various initial conditions are considered in an attempt to discern if 2D QT exhibits inverse cascades as is seen in 2D classical turbulence (CT). In our simulation parameter regimes, no dual cascade spectra were observed for 2D QT-unlike that seen in 2D CT.
Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) using a special unitary evolution algorithm. The qubit lattice gas (QLG) algorithm, is employed to simulate the weakly-coupled Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a very short Poincare recurrence time. This short recurrence time is destroyed as the nonlinear interaction energy is increased. Energy cascades for 2D QT are considered to examine whether 2D QT exhibits the inverse cascades of 2D classical turbulence. In the parameter regime considered, the spectra analysis reveals no such dual cascades---dual cascades being a hallmark of 2D classical turbulence.
A quantum lattice gas algorithm, based on interleaved unitary collide-stream operators, is used to study quantum turbulence of the ground state wave function of a Bose-Einstein condensate (BEC). The Gross-Pitaevskii equation is a Hamiltonian system for a compressible, inviscid quantum fluid. From simulations on a 57603 grid it was observed that a multi-cascade existed for the incompressible kinetic energy spectrum with universal features: the large spatial scales exhibit a classical Kolmogorov k -5/3 spectrum while the very small scales exhibit a quantum Kelvin wave cascade k-3 spectrum. Under certain conditions one can explicitly determine the Poincare recurrence of initial conditions as well as the intermittent destruction of the Kelvin wave cascade.
The time evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) gas is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-stream operators, a qubit lattice gas algorithm is devised, which on taking moments, recovers the Gross-Pitaevskii (GP) equation under diffusion ordering (time scales as length(2)). Unexpectedly, there is a class of initial states whose Poincaré recurrence time is extremely short and which, as the grid resolution is increased, scales with diffusion ordering (and not as length(3)). The spectral results of J. Yepez et al. [Phys. Rev. Lett. 103, 084501 (2009).] for quantum turbulence are revised and it is found that it is the compressible kinetic energy spectrum that exhibits three distinct spectral regions: a small-k classical-like Kolmogorov k(-5/3), a steep semiclassical cascade region, and a large-k quantum vortex spectrum k(-3). For most evolution times the incompressible kinetic energy spectrum exhibits a somewhat robust quantum vortex spectrum of k(-3) for an extended range in k with a k(-3.4) spectrum for intermediate k. For linear vortices of winding number 1 there is an intermittent loss of the quantum vortex cascade with its signature seen in the time evolution of the kinetic energy E(kin)(t), the loss of the quantum vortex k(-3) spectrum in the incompressible kinetic energy spectrum as well as the minimalization of the vortex core isosurfaces that would totally inhibit any Kelvin wave vortex cascade. In the time intervals around these intermittencies the incompressible kinetic energy also exhibits a multicascade spectrum.
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−5/3k^{-5/3}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.
A unitary qubit lattice algorithm, which scales almost perfectly to the full number of cores available (e.g., 216000 cores on a CRAY XT5), is used to examine quantum turbulence and its interrelationship to classical turbulence with production runs on grids up to 57603. The maximal grids achievable by conventional CFD for quantum turbulence is just 20483, and artificial dissipation had to be introduced. Our unitary algorithms preserve the Hamiltonian structure of the Gross-Pitaevskii equation which describes quantum turbulence in a zero-temperature Bose-Einstein condensate (BEC). As a result, parameter regimes have been uncovered which exhibit very short Poincare recurrence time, as well as a strong triple cascade structure in the kinetic energy spectrum. Moreover, a detailed examination of the incompressible kinetic energy spectrum has revealed for the first time within a turbulence simulation the κ−17/5 quantum Kelvin wave cascade. By generalizing the unitary entanglement operators on the 2 qubits, a finite temperature BEC system is examined. These unitary qubit lattice algorithms are directly applicable to quantum computers as they become available.