Probable Observation of Quantized Vortex Lines Through Solid He Under DC Rotation

We report how one can detect quantized vortices in superfluids contained in cylindrical vessels in well-designed torsional oscillator (TO) experiments under DC rotation. We show the case of an artificial 3D superfluid (Fukuda et al. in Phys. Rev. B 71:212502, 2005) which is made of Kosterlits-Thouless 2D He film condensed on a porous glass substrate with a 3D connected surface of well-controlled pore size. We understand the TO experimental results with an extra energy dissipation peak under DC rotation by considering the circular quantized superflow around each of the vortex lines and interaction with thermally excited 2D vortices as discussed in Fukuda et al. (Phys. Rev. B 71:212502, 2005) and in Nemirovskii and Sonin (Phys. Rev. B 76: 224507, 2007). We discuss here the case of hcp solid ⁴He (see for ex. Balibar and Caupin in J. Phys., Condens. Matter 20:173201-1-19, 2008) and show the evidence of observation of the vortex lines penetration below a supersolid transition temperature (Kubota et al. in J. Low Temp. Phys. 158:572, 2010), T _c , where macroscopic phase coherence is realized. It is found at exactly the same temperature T _c , below which the hysteresis occurrs (Shimizu et ail. in arXiv:0903.1326, 2009). For hcp solid ⁴He we have reported the vortex fluid (VF) state onset temperature (Penzev et al. in Phys. Rev. Lett. 101:065301, 2008) T _o =??500 mK, by detailed drive velocity, V _ac dependence study using TO technique. The TO response of the hcp ⁴He is characterized by the energy dissipation peaked at T _p near 100 mK similar to the behavior in a KT transition. The real supersolid (SS) state occurs at T _c below T _p and much lower than T _o . Our observation of the evidence of vortex lines penetration just below T _c together with the VF state properties gives support for the idea that hcp ⁴He shares some features with the ??new type of superconductors??, where the vortex state, involving the VF as well as various vortex solid states, has been commonly discussed (Fisher et al. in Phys. Rev. B 43(1):130, 1991; Leggett in Quantum Liquids, Oxford University Press, London, 2006).     

BCC vs. HCP—The Effect of Crystal Symmetry on the High Temperature Mobility of Solid 4He

We report results of torsional oscillator (TO) experiments on solid ⁴He at temperatures above 1?K. We have previously found that single crystals, once disordered, show some mobility (decoupled mass) even at these rather high temperatures. The decoupled mass fraction with single crystals is typically 20–30%. In the present work we performed similar measurements on polycrystalline solid samples. The decoupled mass with polycrystals is much smaller, ～1%, similar to what is observed by other groups. In particular, we compared the properties of samples grown with the TO’s rotation axis at different orientations with respect to gravity. We found that the decoupled mass fraction of bcc samples is independent of the angle between the rotation axis and gravity. In contrast, hcp samples showed a significant difference in the fraction of decoupled mass as the angle between the rotation axis and gravity was varied between zero and 85 degrees. Dislocation dynamics in the solid offers one possible explanation of this anisotropy.     

Elastic Properties of Solid 4He

The shear modulus of solid ⁴He increases below 200 mK, with the same dependence on temperature, amplitude and ³He concentration as the frequency changes recently seen in torsional oscillator (TO) experiments. These have been interpreted as mass decoupling in a supersolid but the shear modulus behavior has a natural explanation in terms of dislocations. This paper summarizes early ultrasonic and elastic experiments which established the basic properties of dislocations in solid helium. It then describes the results of our experiments on the low temperature shear modulus of solid helium. The modulus changes can be explained in terms of dislocations which are mobile above 200 mK but are pinned by ³He impurities at low temperature. The changes we observe when we anneal or stress our crystals confirm that defects are involved. They also make it clear that the shear modulus measured at the lowest temperatures is the intrinsic value—it is the high temperature modulus which is reduced by defects. By measuring the shear modulus at different frequencies, we show that the amplitude dependence depends on stress in the crystal, rather than reflecting a superfluid-like critical velocity. The shear modulus changes shift to lower temperatures as the frequency decreases, showing that they arise from a crossover in a thermally activated relaxation process rather than from a true phase transition. The activation energy for this process is about 0.7 K but a wide distribution of energies is needed to fit the broad crossover. Although the shear modulus behavior can be explained in terms of dislocations, it is clearly related to the TO behavior. However, we made measurements on hcp ³He which show essentially the same modulus stiffening but there is no corresponding TO anomaly. This implies that the TO frequency changes are not simply due to mechanical stiffening of the oscillator—they only occur in the Bose solid. We conclude by pointing out some of the open questions involving the elastic and TO behavior of solid helium.     

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.     

Simultaneous Measurement of Torsional Oscillation and Ultrasound Propagation in Solid 4He at Low Temperatures

Preliminary results of simultaneous measurements of 10 MHz longitudinal ultrasound propagation in solid ⁴He loaded onto torsional oscillator (TO) are reported. Temperature dependence of sound velocity and attenuation and that of amplitude and frequency of TO are measured. The properties of dislocation lines present in the solid samples are extracted from the ultrasound data and are compared with the shifts in TO frequency below 100 mK.     

Disorder, Supersolidity, and Quantum Plasticity in Solid Helium 4

Several years after Kim and Chan’s discovery of an anomaly in the rotation properties of solid helium (Kim and Chan in Nature 427:225, 2004; Science 305:1941, 2004), the interpretation of the observed phenomena as a manifestation of supersolidity remains controversial. J.?Beamish and his collaborators have shown that the rotation anomaly is accompanied by an elastic anomaly (Day and Beamish in Nature 450:853, 2007; Day et?al. in Phys. Rev. Lett. 104:075302, 2010; Syshchenko et?al. in Phys. Rev. Lett. 104:195301, 2010): when the rotational inertia apparently increases, the shear modulus decreases. This softening is due to the appearance, in the solid, of a large reversible plasticity that is a consequence of the evaporation of ³He impurities from dislocations that become mobile. This plasticity is called “quantum plasticity” because the dislocations move by quantum tunneling in the low temperature limit. Since the main evidence for supersolidity comes from torsional oscillator (TO) experiments, and since the TO period depends on both the inertia and the stiffness of solid ⁴He, it is not totally clear if supersolidity really induces a change in inertia or if it is the disappearance of quantum plasticity that mimics supersolidity in TO experiments. In order to distinguish between supersolidity and quantum plasticity, we have studied the rotational and the acoustic properties of solid ⁴He samples with a variable amount of disorder and of ³He impurities. Of particular interest is the comparison of single crystals to polycrystals but the whole problem is not yet solved. This short review article is an opportunity to discuss several questions regarding the exact role of disorder in supersolidity and in quantum plasticity.     

Tunneling Defect Nanoclusters in hcp 4He Crystals: Alternative to Supersolidity

A?simple model based on the concept of resonant tunneling clusters of lattice defects is used to explain the low temperature anomalies of hcp ⁴He crystals (mass decoupling from a torsional oscillator, shear modulus anomaly, dissipation peaks, heat capacity peak). Mass decoupling is a result of an internal Josephson effect: mass supercurrent inside phase coherent tunneling clusters. Quantitative results are in reasonable agreement with experiments.     

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.     

Observation of Non-Classical Rotational Inertia in Bulk Solid 4He

In recent torsional oscillator experiments by Kim and Chan (KC), a decrease of rotational inertia has been observed in solid ⁴He in porous materials (Kim, E., Chan, M.H.W. in Nature 427:225, 2004; J. Low Temp. Phys. 138:859, 2005) and in a bulk annular channel (Kim, E., Chan, M.H.W. in Science 305:1941, 2004). This observation strongly suggests the existence of “non-classical rotational inertia” (NCRI), i.e. superflow, in solid ⁴He. In order to study such a possible “supersolid” phase, we perform torsional oscillator experiments for cylindrical solid ⁴He samples. We have observed decreases in rotational inertia below 200 mK for two solid samples (pressures P=4.1 and 3.0 MPa). The observed NCRI fraction at 70 mK is 0.14%, which is about 1/3 of the fraction observed in the annulus by KC. Our observation is the first experimental confirmation of the possible supersolid finding by KC.     

Switching of Dislocations in Solid Helium Between Pinned and Unpinned States in the Torsional Oscillator Experiments

The anomalous temperature dependence of the period of a torsional oscillator (TO) containing solid helium can be explained by the dislocation-vibration model. Namely, the dislocations in solid helium vibrate in phase with the torsional oscillation and the TO-period change varies as the square of the average dislocation pinning length which becomes shorter at lower temperature due to pinning of dislocations by ³He impurity atoms. This dislocation-vibration model is further developed by introducing distribution functions for the network pinning lengths for dislocation segments and by considering the mechanisms of pinning of dislocations by and unpinning of dislocations from ³He impurity atoms. Experimental data, such as the TO-period change at different ³He concentrations, the rim-velocity dependence of the TO period, the hysteresis of the TO period with respect to the rim velocity and to the temperature, are analyzed based on the dislocation-vibration model. Above all, the hysteresis is caused by switching of dislocations between pinned and unpinned states.     

Measurements of Non-classical Rotational Inertia of Solid 4He at Two Frequencies

We have developed a double resonance compound torsional oscillator for studying non-classical rotational inertia (NCRI) of solid ⁴He at two different frequencies. The torsional oscillator consists of two beryllium copper torsion members, and two masses. The two masses rotate in phase in the first mode and out of phase in the second mode at resonance frequencies, 496 Hz and 1173 Hz, respectively. Samples of solid ⁴He (commercial grade with ³He impurity level less than 1 ppm) are grown with the blocked capillary method at pressures between 27 and 42 bar. Temperature dependences of NCRI signals at two different frequencies are observed in identical solid ⁴He at 37 bar.      

Torsional Oscillator measurements of multiply connected monolayer 4He films under high speed rotation up to 4.0 rev./s

No Heading We are studying the physical phenomena of superfluid thin films adsorbed on multiply connected surface of porous glasses under high speed rotation. The combination of thin superfluid ⁴He films on large pore porous surfaces and high-speed rotation gives rise to a new quantum vortex excitation, where vortex lines are pinned on the center of pores and have the core at the pore. This excitation has been predicted and named as a pore vortex. We are trying to measure the behavior of pore vortices using the Torsional Oscillator (TO) technique and we have succeeded in making measurements up to the high-speed rotation of 4.0 rev./s. Here we report the actual design of the TO set up.PACS numbers: 67.40.Vs, 67.57.Fg, 67.70+n, 67.40.Rp.     

Transverse Ultrasound Measurements in 4He Single Crystals

Recently, Kim and Chan (Science 305:1941, 2004; Phys. Rev. Lett. 97:115302, 2006) have reported an anomalous decoupling transition of solid ⁴He in a torsional oscillator measurement, and interpret their results as evidence for non-classical rotational inertia and a possible supersolid phase of ⁴He. The detailed nature and properties of such a “supersolid” state in ⁴He are still far from being clear, although there are clues from experiments involving ³He impurities, different sample cell geometries, annealing effects and grain boundary flow. Defects produced during crystal growth or deformation (e.g. dislocations) may affect supersolidity, or even produce it, and they are expected to have significant impact on the elastic properties of the solid. The supersolid fraction could also decouple from the lattice and produce a decrease in the transverse sound speed. We have begun the experiments in this laboratory to study such effects, measuring the velocity and attenuation of transverse ultrasound at 10 MHz in ⁴He single crystals grown at constant pressure.      

Measurements of Torsional Oscillations and Thermal Conductivity in Solid 4He

Polycrystalline samples of hcp ⁴He of molar volume 19.5?cm³ with small amount of ³He impurities were grown in an annular container by the blocked-capillary method. Three concentrations of ³He, x ₃, were studied: isotopically purified ⁴He with the estimated x ₃≤10^?10, ‘well-grade’ helium with x ₃～3×10^?7 and a specially prepared mixture with x ₃=2.5×10^?6. The torsional oscillator response and thermal conductivity were investigated before and after annealing. The temperature and width of the torsional anomaly increase with increasing x ₃. Annealing resulted in an increased phonon mean free path but often in little change in the torsional oscillator response. While the magnitude of the torsional anomaly and phonon mean free path can be very different in different samples, no correlation was found between them; this implies that these two properties are controlled by different types of crystal defects. It seems plausible that the mean free path of thermal phonos at ～200?mK is controlled by vibrating dislocations while the magnitude of the frequency shift of torsional oscillations is governed by static defects such as pinned dislocations and grain boundaries.     

Generalized Rotational Susceptibility Studies of Solid 4He

Using a novel SQUID-based torsional oscillator (TO) technique to achieve increased sensitivity and dynamic range, we studied TO’s containing solid ⁴He. Below ～250?mK, the TO resonance frequency f increases and its dissipation D passes through a maximum as first reported by Kim and Chan. To achieve unbiased analysis of such ⁴He rotational dynamics, we implemented a new approach based upon the generalized rotational susceptibility $\chi_{{}^{4}\mathrm{He}}^{ - 1}(\omega,T)$ . Upon cooling, we found that equilibration times within f(T) and D(T) exhibit a complex synchronized ultraslow evolution toward equilibrium indicative of glassy freezing of crystal disorder conformations which strongly influence the rotational dynamics. We explored a more specific $\chi_{{}^{4}\mathrm{He}}^{ -1}(\omega,\tau(T))$ with τ(T) representing a relaxation rate for inertially active microscopic excitations. In such models, the characteristic temperature T ^? at which df/dT and D pass simultaneously through a maximum occurs when the TO angular frequency ω and the relaxation rate are matched: ωτ(T ^?)=1. Then, by introducing the free inertial decay (FID) technique to solid ⁴He TO studies, we carried out a comprehensive map of f(T,V) and D(T,V) where V is the maximum TO rim velocity. These data indicated that the same microscopic excitations controlling the TO motions are generated independently by thermal and mechanical stimulation of the crystal. Moreover, a measure for their relaxation times τ(T,V) diverges smoothly everywhere without exhibiting a critical temperature or velocity, as expected in ωτ=1?models. Finally, following the observations of Day and Beamish, we showed that the combined temperature-velocity dependence of the TO response is indistinguishable from the combined temperature-strain dependence of the ⁴He shear modulus. Together, these observations imply that ultra-slow equilibration of crystal disorder conformations controls the rotational dynamics and, for any given disorder conformation, the anomalous rotational responses of solid ⁴He are associated with generation of the same microscopic excitations as those produced by direct shear strain.     

Acoustic Resonances and Non-linearity in Solid 4He

Direct measurements of the low frequency shear modulus of solid ⁴He show a remarkable increase below 150?mK, a regime where torsional oscillator (TO) experiments show evidence of mass decoupling. Acoustic resonance measurements at higher frequencies confirm the unusual elastic behavior. A striking feature of both TO and shear modulus measurements is their amplitude dependence—the low temperature anomalies are reduced as the drive amplitudes increase. We have studied the amplitude dependence of acoustic resonances in both standard and isotopically pure ⁴He crystals. The resonance peaks shift to lower frequencies as the drive amplitude increases. At high amplitudes the peaks are asymmetric and exhibit both bistability and hysteresis—classic features of nonlinear oscillators. At the lowest amplitudes the peak frequency and shape are independent of drive amplitude and are non-hysteretic. The threshold for nonlinear behavior is lower for the isotopically pure crystal. We also studied the effects of annealing on the acoustic resonances. The nonlinear behavior was qualitatively unchanged but annealing often affected the shape of the resonance peaks, which often had a complicated structure at low temperatures.     

Tkachenko waves

The existence of Tkachenko waves in rotating superfluid⁴He has been confirmed experimentally. Tkachenko waves are displacement waves in the vortex line array that exists in a rotating superfluid. The waves were excited and detected in resonant cavities formed by a stack of closely spaced parallel plates. The dispersion relation, studied as a function of rotation speed, disk spacing (which fixed one component of the wave vector), and temperature, is compared with existing theories and found to be in good agreement at low temperatures. The theories predict peaks in the density of states associated with a lattice of vortices; the resonant responses observed are found to correspond to these peaks. The need for a more complete theory is presented in the light of the behavior of the vortex resonances at elevated temperatures.This work was supported by NSF grant DMR-78-25409.     

The Glassy Response of Solid 4He to Torsional Oscillations

We calculated the glassy response of solid ⁴He to torsional oscillations assuming a phenomenological glass model. Making only a few assumptions about the distribution of glassy relaxation times in a small subsystem of otherwise rigid solid ⁴He, we can account for the magnitude of the observed period shift and concomitant dissipation peak in several torsion oscillator experiments. The implications of the glass model for solid ⁴He are threefold: (1) The dynamics of solid ⁴He is governed by glassy relaxation processes. (2) The distribution of relaxation times varies significantly between different torsion oscillator experiments. (3) The mechanical response of a torsion oscillator does not require a supersolid component to account for the observed anomaly at low temperatures, though we cannot rule out its existence.     

The Superfluid Transition and Vortex Dynamics in Submonolayer Superfluid 4He Films in Porous Glass Under Rotation

A combination of a rotating dilution refrigerator and high-Q torsional oscillator technique has been used to study dynamics of vortices in thin ⁴ He films adsorbed on the porous glass (d=1m pore size). Under rotation an additional dissipation peak with the amplitude proportional to the angular velocity is seen at the middle of the superfluid transition, on the low temperature side of the stationary peak which is present even at =0. We attribute this peak to the 3D Type vortices created in multiply connected ⁴ He film by the rotation. Peak shape of the rotation-induced dissipation could be interpreted as a freezing of the 3D vortices well below T _c     

Hysteretic Response of Torsionally Oscillated Solid 4He and Its Dependence on 3He Impurity Concentration

Measurements on hysteretic response of compound torsional oscillator containing annular-shaped solid ⁴He samples were carried out by varying the oscillator drive amplitude starting from high to low and then back up to the initial high value. Hysteresis in the oscillator frequency and amplitudes were observed only below an onset temperature. The hysteresis onset temperature (T _H ) did not depend on the oscillator frequency, width of the sample annulus, annealing and refreezing after melting. A systematic increase in T _H was observed as the ³He impurity concentration in solid ⁴He samples was increased. The dependence of T _H on ³He impurity concentration followed approximately that of the dissipation peak temperatures. Possible relationships of the observed hysteresis phenomena with models of solid ⁴He dynamics based on freezing of a vortex liquid and dislocation motion are discussed.