The hydrodynamics of rotating superfluids. II. Finite temperature,dissipative theory

The hydrodynamics of rotating superfluids at finite temperature is formulated, accounting for the elastic properties of the vortex lattice. This theory, which is a generalization of previous work at zero temperature, includes normal fluid motions and dissipation and is used here to investigate the transverse and longitudinal normal modes of the system. Mutual friction, arising microscopically from collisions between the vortex lines and the excitations comprising the normal fluid, leads to a profound change in the nature of the two transverse modes allowed at finite temperatures. One such mode, similar to the Tkachenko mode in zero-temperature theory, is associated with the motion of the total mass current and is damped by first viscosity but unaffected by mutual friction. The other mode, associated with the relative motion of the normal and superfluid-vortex components, is highly damped by mutual friction and cannot propagate at angles greater than a critical angle _c measured from the rotation axis.     

Vortex Lattice Dynamics in a Dilute Gas BEC

We study the dynamics of large vortex lattices in a dilute gas Bose–Einstein condensate. Rapidly rotating condensates are created that contain vortex lattices with up to 300 vortices. The condensates are held in a parabolic trapping potential, and rotation rates exceeding 99% of the radial trapping frequency are achieved. By locally suppressing the density while maintaining the phase singularities, we create vortex aggregates. To illustrate the underlying Coriolis force driven dynamics, oscillation frequencies of the vortex aggregate area are measured. A related technique also enables us to excite and directly image Tkachenko modes in a vortex lattice. These modes provide evidence for the shear strength that a vortex lattice in a superfluid can support.     

Dynamics of Quantized Vortices in a Rotating Cylindrical Vessel

Recently Finne et al. found a transition to the turbulent state in rotating superfluid ³He-B which is insensitive to the fluid velocity, but rather controlled by temperature. They reported that at low temperatures a few seed vortices, injected into a vortex-free region, developed through a transient turbulent state to a vortex array. The experimental observations were consistent with the numerical simulation of dynamics of quantized vortices. However, we do not understand well how the seed vortex follows the above scenario and, especially, how the turbulent vortices change to a vortex array. Although the previous numerical simulation was done for a rotating cubic vessel, we study here the vortex dynamics in a rotating cylindrical vessel which is more suitable for the comparison with the experiments. We developed a numerical method for calculating the vortex dynamics in a cylindrical vessel and investigated the vortex dynamics after a vortex seed loop was injected into a vortex-free region. The numerical result shows that the seed vortex becomes unstable, especially near the cylindrical side wall, and develops into turbulent vortices. After that a vortex array appears in the central region, collecting the vortices from the surrounding tangle. PACS numbers: 67.40.Vs, 47.32.Cc, 47.37.+q.     

Detection of Vortices in Superfluid 4He in the T=0 Limit Using Charged Vortex Rings

We describe the experiment to study the dynamics of quantized vortex lines in superfluid ⁴ at temperatures down to 40 mK, i.e., without any normal component. This was achieved by monitoring the trapping of small charged vortex rings by an array of rectilinear vortex lines produced by rotating the cryostat. The design of the experimental cell is described. Our first observations of the response of the superfluid in the T =0 limit to starting and stopping rotation are presented.     

Evidence for nonsingular vorticity in the Helsinki experiments on rotating3He-A

A theory for the satellite peak in the NMR spectrum of rotating superfluid³He-A is presented. Absorption at the frequency of the satellite is produced by spin waves localized on vortex lines. The peak frequency and total intensity of the satellite are calculated for both singular and nonsingular vortices; a comparison with the Helsinki NMR data strongly suggests that nonsingular vorticity was actually observed in the experiments.     

NMR Spectroscopy of the Double-Quantum Vortex in Superfluid 3He-A

The double-quantum vortex line with continuous (singularity-free) structure is the most common linear defect in rotating ³He-A. Its well-known experimental signature is a frequency-shifted satellite peak in the NMR spectrum. It arises from the absorption of spin wave excitations, which are localized bound states in the dipole-unlocked soft vortex core. In first approximation, the intensity of the satellite peak is proportional to the number of vortex lines. With increased measuring resolution we have found that the absorption contribution of individual vortex lines is not identical, but displays small variations, which depend on the non-uniform global orbital ^ l texture. We attribute the effect to small texture-dependent changes in the ^ l distribution in the soft core, which modify the attractive potential of the localized spin waves. This property can be used for studying the global order parameter texture in the rotating container.     

An ultrasonic study of sodium ion diffusion in polycrystalline ββ″ alumina

Measurements are presented of the relative attenuation of longitudinal stress waves, of frequencies from 10 to 70 MHz, in polycrystalline samples of $\text{β}\text{β″}$ alumina at temperatures from 80 to 300 K. The attenuation is found to exhibit a series of internal friction peaks, attributed to an interaction with Na⁺ ion diffusion. The activation energy for the diffusion of Na⁺ in the sample is estimated to be 0.142 eV. Shear wave attenuation is also found to rise at temperatures of ～300 K in samples contaminated by water absorption.     

Analysis of spin wave resonances in3He-4He mixtures

Spin waves have been reported previously in³He-⁴He mixtures using cw-NMR. The spin waves form standing waves in the rf coil used to detect the NMR signal. Only one resonance was observed, corresponding to ak vector of 1 mm^–1. This is in sharp contrast to similar experiments performed on pure³He, where several resonances were seen. An explanation is given of why this occurs and the resonant mode is identified. Furthermore, it is explained why there is an extra width to the resonance for temperatures below 2 mK. This is caused by radiation damping of the resonant mode. The theory is found to be in excellent agreement with the data. Further experiments are suggested, including details of an improved rf coil design, which should lead to information about the quasiparticle interaction.     

Cascade Process of Vortex Tangle Dynamics in Superfluid 4He without Mutual Friction

Recently the Kelvin wave cascade process in superfluid ⁴ He at very low temperatures was discussed. In this mechanism, the dynamics of the waves on the vortex lines plays an important role. In order to investigate this mechanism, we study numerically the dynamics of the waves on the reconnected vortex lines using the full Biot-Savart law. This work shows the reconnection of two vortices leads to the waves on the reconnected vortex lines. To discuss the energy of the waves on the vortex lines, the energy spectrum of the superflow before and after the reconnection is calculated.     

The mobility of ions trapped on vortex lines in pure4He and3He-4He solutions

We present measurements of the mobility of positive and negative ions trapped on vortex lines in pure⁴He and dilute³He-⁴He solutions over the temperature range 1.6>T>0.3 K. In pure⁴He below about 0.7 K, we observe several new effects not seen at higher temperatures and not easily explained with existing theories. Most notable are an enhanced broadening of the ion pulse and a rapid increase in the mobility with decreasing temperature. Measurements of the electric field dependence of the drift velocity in pure⁴He at low temperatures show a limiting velocity for sufficiently large fields. This behavior can be explained using a simple resonance theory. The inverse mobility data for solutions show sharp increases at certain critical temperatures, which are interpreted as being associated with the condensation of³He atoms onto the vortex cores. The dependence of the critical temperature on the bulk³He concentration is found to be in good agreement with a simple condensation theory. An extension of arguments used in this theory to lower temperatures leads to the picture of a³He-rich core growing with decreasing temperature, consistent with our lower temperature experimental data.Supported in part by a grant from the National Science Foundation.     

NMR signatures of topological objects in rotating superfluid3He-A

NMR spectrometry can be used to identify different topological objects in the order parameter field of rotating superfluid³He-A. We list their signatures in the cw NMR absorption line shape. Quantized vortex lines, domain walls, and their combination, the vortex sheet, all induce satellite peaks with specific intensities and frequency shifts in the NMR spectrum. Examples of spectra are presented to allow a comparison and to distinguish between different objects.     

Vortex Tangle Polarized by Rotation

Almost all studies of vortex states in superfluid ⁴He have been concerned with either ordered vortex arrays driven by rotation or disordered vortex tangles driven, for example, by thermal counterfiow. In this work we study numerically what happens to vortices in the presence of both effects. We find that a rotating vortex array becomes unstable, exciting Kelvin waves when it is subject to a counterfiow which is parallel to the rotation axis and which is sufficiently large. After the initial growth of the instability, the vortices enter a new, statistically steady, turbulent state, in which the vortex tangle is polarized along the rotational axis. We determine the polarization of the tangle as a function of the rotation frequency and the counterfiow velocity.     

TM Wave Resonant Peaks in Defect-Free Multilayer Structures Containing Nano-Scale Y123 Superconductors

In the ultraviolet region and by using transfer matrix method, the transmission spectra of electromagnetic waves through one-dimensional quasiperiodic photonic structures consisting of high-temperature yttrium barium copper oxide (Y123) superconductor and strontium titanate (STO) dielectric layers are studied theoretically. It is shown that for TE polarization at oblique incidence two band gaps are created, while for TM polarization three band gaps are produced. The edges of both polarizations shift to higher wavelengths by increasing incident angles. Also, for angles of incidence greater than 78^°, the second gap for TM-polarized light is eliminated. It is also found that in the supposed structure the number of PBGs can be modulated by the thickness of dielectric layer, while it is nearly insensitive to the thickness of superconductor layer. Interestingly, for the incidence angles other than normal incidence the structure can exhibit some narrow resonant peaks near wavelengths where the electric permittivity of the superconductor layer changes sign. These resonant peaks are only for TM polarization and not present for TE polarization. This structure can act as a very compact polarization sensitive splitters and defect-free multichannel narrowband tunable filters.     

Anomalous Damping of a Low Frequency Vibrating Wire in Superfluid 3He-B due to Vortex Shielding

We have investigated the behaviour of a large vibrating wire resonator in the B-phase of superfluid ³He at zero pressure and at temperatures below 200 μK. The vibrating wire has a low resonant frequency of around 60 Hz. At low velocities the motion of the wire is impeded by its intrinsic (vacuum) damping and by the scattering of thermal quasiparticle excitations. At higher velocities we would normally expect the motion to be further damped by the creation of quasiparticles from pair-breaking. However, for a range of temperatures, as we increase the driving force we observe a sudden decrease in the damping of the wire. This results from a reduction in the thermal damping arising from the presence of quantum vortex lines generated by the wire. These vortex lines Andreev-reflect low energy excitations and thus partially shield the wire from incident thermal quasiparticles.     

Vortex Reconnection and Acoustic Emission by the Numerical Analysis of the Gross-Pitaevskii Equation

Reconnection of quantized vortices and concurrent acoustic emission are studied numerically by analyzing the Gross-Pitaevskii equation. Although vortex reconnection was studied earlier by Koplik and Levine, the present work first investigates in more detail how the singular vortex cores reconnect. Second, acoustic emission, i.e., the emission of the density waves of the condensate, is studied when two vortices annihilate or reconnect. The local drastic change of the condensate density is found to work as an acoustic source to emit the density waves. This phenomenon may be related to the vortex tangle decay observed at mK temperatures and the eddy viscosity which is thought to be an intrinsic dissipative mechanism in a vortex tangle.     

Vortex Mutual Friction in Superfluid 3He

We give a full account of our extensive measurements of vortex mutual friction in rotating superfluid ³He, in both the A- and B-phases. The B-phase results are in qualitative agreement with a theory based on the concept of “spectral flow”; the agreement becomes quantitative if an effective energy gap of 0.63 Δ is used, but the Justification for such a substitution is not clear. The vortex core transition, at first not seen because of metastability and hysteresis, has now been observed. Detailed investigation suggests that the high temperature vortex state is a temperature dependent mixture of at least two vortex types. The A-phase mutual friction is found to be well described by two hydrodynamic coefficients, the orbital viscosity and the orbital inertia. The latter corresponds to an orbital angular momentum per Cooper pair of (0.0015 ± 0.0017 ) ħ, consistent with the prediction of the spectral flow theory. We find that the most uniform l texture is obtained by cooling through T_c while rotating, and then stopping rotation. Detailed investigation of textural memory effects shows that the uniform l-up and l-down textures are associated with opposite directions of rotation. We discuss the various types of texture that may be formed in our experiments. Finally, we compare our mutual friction results with those found in ⁴HeII.     

Collisionless spin waves in normal and superfluid3He

Standing spin-wave modes in liquid³He have been studied by cw NMR at Larmor frequencies of 1, 2, and 4 MHz and pressures of 0, 6.3, and 12.3 bar. The spin waves, which produce peaks in the NMR line, are visible at temperatures below 5 mK at zero pressure. With the assumption of a slightly simplified sample shape and no transverse spin relaxation at the walls, the theory of Leggett fits the spin-wave frequencies in the normal liquid very well, giving a value of the Fermi liquid parameterF ₁ ^a =–0.6±0.2 at zero pressure. The width of some of the peaks is larger than expected from other determinations of the quasiparticle diffusion time _D . This could be due to wall relaxation or to deviations from the assumed sample geometry. In the superfluid A₁ and A phases, where the data cannot be fitted to existing theories, the spin-wave modes are shifted in frequency and suffer additional damping as the temperature is decreased. At still lower temperatures in the B phase an inversion of the spin-wave spectrum from one side of the NMR line to the other is observed, agreeing quantitatively with the predictions of the 1975 theory of Combescot.     

The Dynamics of Vortex Generation in Superfluid 3He-B

A profound change occurs in the stability of quantized vortices in externally applied flow of superfluid ³He-B at temperatures ?0.6?T _c, owing to the rapidly decreasing damping in vortex motion with decreasing temperature. At low damping an evolving vortex may become unstable and generate a new independent vortex loop. This single-vortex instability is the generic precursor of turbulence. We investigate the instability with non-invasive NMR measurements on a rotating cylindrical sample in the intermediate temperature regime (0.3–0.6)?T _c. From comparisons with numerical calculations we interpret that the instability occurs at the container wall, when the vortex end moves along the wall in applied flow.     

Transport Properties of Two-dimensional Snowballs Below the Surface of 4He-II Under Rotation

Kelvin-waves play an important role for the dissipation of quantum turbulence at low temperatures. Here the plasma resonance of two-dimensional (2D) snowballs trapped below the surface of rotating superfluid ⁴He are measured for the first time in order to examine whether 2D snowballs could be a new probe for study of vortex dynamics. Below 200 mK, a positive shift of the resonant frequency f ₁ and linewidth broadening are observed as small variations in the absorption spectra under rotation. Both f ₁ and the linewidth Δf increase linearly with the rotation speed, and the slopes of f ₁ and Δf against the rotation speed have no temperature dependence. The increase of Δf suggests that an additional dissipation is caused by the coupling between the snowballs and vortices. We provide a qualitative explanation for the linear increase of Δf in the context of Kelvin-waves excited by the motion of snowballs.