Reconnection dynamics for quantized vortices

By analyzing trajectories of solid hydrogen tracers in superfluid ⁴He, we identify tens of thousands of individual reconnection events between quantized vortices. We characterize the dynamics by the minimum separation distance δ(t) between the two reconnecting vortices both before and after the events. Applying dimensional arguments, this separation has been predicted to behave asymptotically as δ(t)≈A(κ|t−t₀|)^1/2, where κ=h/m is the quantum of circulation. The major finding of the experiments and their analysis is strong support for this asymptotic form with κ as the dominant controlling feature, although there are significant event to event fluctuations. At the three-parameter level the dynamics may be about equally well-fit by two modified expressions: (a) an arbitrary power-law expression of the form δ(t)=B|t−t₀|^α and (b) a correction-factor expression δ(t)=A(κ|t−t₀|)^1/2(1+c|t−t₀|). The measured frequency distribution of α is peaked at the predicted value α=0.5, although the half-height values are α=0.35 and 0.80 and there is marked variation in all fitted quantities. Accepting (b) the amplitude A has mean values of 1.24±0.01 and half height values of 0.8 and 1.6 while the c distribution is peaked close to c=0 with a half-height range of −0.9 s⁻¹ to 1.5 s⁻¹. In light of possible physical interpretations we regard the correction-factor expression (b), which attributes the observed deviations from the predicted asymptotic form to fluctuations in the local environment and in boundary conditions, as best describing our experimental data. The observed dynamics appear statistically time-reversible, which suggests that an effective equilibrium has been established in quantum turbulence on the time scales (≤0.25 s) investigated. We discuss the impact of reconnection on velocity statistics in quantum turbulence and, as regards classical turbulence, we argue that forms analogous to (b) could well provide an alternative interpretation of the observed deviations from Kolmogorov scaling exponents of the longitudinal structure functions.     

Three-dimensional growth of quantized vortices in rotating superfluid

This work studies numerically three-dimensional growth of small quantized vortices from the vessel boundary in rotating superfluid.     

Pinning of quantized vortices in helium drops by dopant atoms and molecules

Using a density functional method, we investigate the properties of liquid 4He droplets doped with atoms (Ne and Xe) and molecules ( SF6 and hydrogen cyanide). We consider the case of droplets having a quantized vortex pinned to the dopant. A liquid-drop formula is proposed that accurately describes the total energy of the complex and allows one to extrapolate the density functional results to large N. For a given impurity, we find that the formation of a dopant+vortex+(4)He(N) complex is energetically favored below a critical size N(cr). Our results support the possibility to observe quantized vortices in helium droplets by means of spectroscopic techniques.     

Dirac monopoles on quantized vortices in superfluid3He

Quantized vortices in superfluid³He display a variety of novel structures that have previously not been observed in any other quantum fluids; their basic experimental manifestations and theoretical features have been reviewed by Hakonen, Lounasmaa and Simola [Physica B160 (1989) 1] and by Salomaa and Volovik [Rev. Mod. Phys.59 (1987) 533]. In order not to repeat these reviews in this paper, here we choose to discuss the theme of the possible pointlike orderparameter singularities, monopoles, that can occur on the quantized³He vortices. Such monopoles, mathematical analogues of the magnetic Dirac monopole, may exist in superfluid³He due to the complicated order-parameter structure, which makes it possible to have several different types of quantized vortex lines and phase boundaries between the superfluid states. Analogs of Dirac monopoles, and also monopoles with 1/2 and 1/4 of the magnetic charge of the Dirac monopole, may exist at the points of intersections of quantized vortex lines and phase boundaries — or along vortices if they change their quantum state; several examples are discussed.Invited talk at the International Conference on Macroscopic Quantum Phenomena, Smolenice Castle, Czechoslovakia, September 18–22, 1989.I want to thank G. E. Volovik for a useful discussion and O. V. Lounasmaa for encouragement. This research has been supported through the Award for the Advancement of European Science by the Körber-Stiftung (Hamburg, FRG) and by the Academy of Finland.     

Exactly quantized dynamics of classical incommensurate sliders

We report peculiar velocity quantization phenomena in the classical motion of an idealized 1D solid lubricant, consisting of a harmonic chain interposed between two periodic sliders. The ratio upsilon(c.m.)/upsilon(ext) of the chain center-of-mass velocity to the externally imposed relative velocity of the sliders stays pinned to exact "plateau" values for wide ranges of parameters, such as slider corrugation amplitudes, external velocity, chain stiffness, and dissipation, and is strictly determined by the commensurability ratios alone. The phenomenon is explained by one slider rigidly dragging the kinks that the chain forms with the other slider. Possible consequences of these results for some real systems are discussed.     

Parallel numerical simulations for quantized vortices in Bose--Einstein condensates

We employ the parallel computing technology to study numerically the three-dimensional structure of quantized vortices of Bose--Einstein condensates. For anisotropic cases, the bending process of vortices is described in detail by the decrease of Gross--Pitaevskii energy. A completely straight vortex and the steady and symmetrical multiple-vortex configurations are obtained. We analyse the effect of initial conditions and angular velocity on the number and shape of vortices.     

Splitting times of doubly quantized vortices in dilute bose-einstein condensates

Recently, the splitting of a topologically created doubly quantized vortex into two singly quantized vortices was experimentally investigated in dilute atomic cigar-shaped Bose-Einstein condensates [Y. Shin, Phys. Rev. Lett. 93, 160406 (2004)10.1103/PhysRevLett.93.160406]. In particular, the dependency of the splitting time on the peak particle density was studied. We present results of theoretical simulations which closely mimic the experimental setup. We show that the combination of gravitational sag and time dependency of the trapping potential alone suffices to split the doubly quantized vortex in time scales which are in good agreement with the experiments.     

Velocity statistics distinguish quantum turbulence from classical turbulence

By analyzing trajectories of solid hydrogen tracers, we find that the distributions of velocity in decaying quantum turbulence in superfluid 4He are strongly non-Gaussian with 1/v(3) power-law tails. These features differ from the near-Gaussian statistics of homogenous and isotropic turbulence of classical fluids. We examine the dynamics of many events of reconnection between quantized vortices and show by simple scaling arguments that they produce the observed power-law tails.     

High precision solutions to quantized vortices within Gross–Pitaevskii equation

The dynamics of vortices in Bose–Einstein condensates of dilute cold atoms can be well formulated by Gross–Pitaevskii equation. To better understand the properties of vortices, a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed. Through two-point Padé approximants, these solutions are presented in terms of simple rational functions, which can be used in the simulation of vortex dynamics. The precision of the solutions is sensitive to the connecting parameter and the truncation orders. It can be improved significantly with a reasonable extension in the order of rational functions. The errors of the solutions and the limitation of two-point Padé approximants are discussed. This investigation may shed light on the exact solution to the nonlinear vortex equation.     

Negative-temperature states and large-scale,long-lived vortices in two-dimensional turbulence

We study Onsager's theory of large, coherent vortices in turbulent flows in the approximation of the point-vortex model for two-dimensional Euler hydrodynamics. In the limit of a large number of point vortices with the energy perpair of vortices held fixed, we prove that the entropy defined from the microcanonical distribution as a function of the (pair-specific) energy has its maximum at a finite value and thereafter decreases, yielding the negative-temperature states predicted by Onsager. We furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy, and, under regularity assumptions, obeys the Joyce-Montgomery mean-field equation. We also prove that, under appropriate conditions, the vorticity distribution is the same as that for the canonical distribution, a form of equivalence of ensembles. We establish a large-fluctuation theory for the microcanonical distributions, which is based on a level-3 large-deviations theory for exchangeable distributions. We discuss some implications of that property for the ergodicity requirements to justify Onsager's theory, and also the theoretical foundations of a recent extension to continuous vorticity fields by R. Robert and J. Miller. Although the theory of two-dimensional vortices is of primary interest, our proofs actually apply to a very general class of mean-field models with long-range interactions in arbitrary dimensions.     

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.     

Topological noise scalings in superfluid and classical turbulence

We calculate the topological noise characterizing the direction of line vortices in superfluid and classical turbulence by finding the intersection of line vortices with square surfaces of edge length l_s positioned normal to three orthogonal axes. In the case of homogeneous superfluid turbulence in thermal counterflow, we find that the noise scales as l_s along the two directions normal to the counterflow and as l _s ^3/2 along the direction parallel to it. In homogeneous isotropic superfluid turbulence, at T→0 K, the noise scales as l _s ^7/4 . In homogeneous isotropic classical turbulence, the scaling is l _s ² . We offer possible interpretations of the computed scalings, as well as justification for their differences.     

Dynamics of cascade three-level system interacting with the classical and quantized field

We study the exact solutions of the cascade three-level atom interacting with a single mode classical and quantized field with different initial conditions of the atom. For the semiclassical model, it is found that if the atom is initially in the middle level, the time-dependent populations of the upper and lower levels are always equal. This dynamical symmetry exhibited by the classical field is spoiled on quantization of the field mode. To reveal this non-classical effect, a Euler matrix formalism is developed to solve the dressed states of the cascade Jaynes-Cummings model (JCM). Possible modification of such an effect on the collapse and revival phenomenon is also discussed by taking the quantized field in a coherent state