Bose-Einstein Condensation in Liquid 4He in Vycor

We present neutron scattering measurements of Bose-Einstein condensation (BEC) in liquid ⁴He confined in Vycor. The data show clear evidence of a condensate in Vycor with a condensate fraction comparable to that of bulk superfluid ⁴He, approximately 7.5% at low temperature and SVP. The temperature dependence of n₀(T) is also similar to that in the bulk with critical temperature for BEC, T_BEC, in the range 1.80BEC<2.05 K. The data are not accurate enough to show whether T_BEC for Vycor is the same or greater than the depressed critical temperature for superfluidity, T_c=1.95 K.     

Neutron scattering from superfluid helium at large momentum transfers

The inelastic scattering of thermal neutrons from liquid helium at 1.2 K has been measured for wave-vector transfers in the range 6.0 Q 12.0 »^–1. Oscillations in the widths of the scattered-neutron distributions as a function of Q are observed to persist at least up to 12.0 »^–1. A simple physical model suggests that these oscillations, which reflect the presence of final-state interactions and interference effects, have their origin in similar oscillations which occur in the He-He scattering cross section. Detailed measurements at Q = 10.0»^–1 show no evidence of a well-defined condensate peak, and calculations suggest that Q values in the range 50–100 »^–1 or greater may be required to observe a separate condensate component in the line shape. A method is developed for eliminating a major part of the effects of final-state interactions from the Doppler part of the distributions, and the corrected data at Q = 10.0 »^–1 are analyzed to determine the momentum distribution in liquid helium at 1.2 K. The resulting distribution is found to be non-Gaussian with a well-defined high-momentum tail. The result is compared with the momentum distribution obtained recently by Mook from thermal neutron time-of-flight data near Q = 15 »^–1 and with theoretical calculations of McMillan and of Kalos.     

Thermodynamics of a Trapped Bose Condensate with Negative Scattering Length

We study the Bose–Einstein condensation (BEC) for a system of ⁷Li atoms, which have negative scattering length (attractive interaction), confined in a harmonic potential. Within the Bogoliubov and Popov approximations, we numerically calculate the density profile for both condensate and non-condensate fractions and the spectrum of elementary excitations. In particular, we analyze the temperature and number-of-boson dependence of these quantities and evaluate the BEC transition temperature T _BEC. We calculate the loss rate for inelastic two- and three-body collisions. We find that the total loss rate is strongly dependent on the density profile of the condensate, but this density profile does not appreciably change by increasing the thermal fraction. Moreover, we study, using the quasi-classical Popov approximation, the temperature dependence of the critical number N _c of condensed bosons, for which there is the collapse of the condensate. There are different regimes as a function of the total number N of atoms. For N<N _c the condensate is always metastable but for N>N _c the condensate is metastable only for temperatures that exceed a critical value T _c.     

The Wall Potential in the Pores of Vycor Glass Experienced by 3He Quasiparticles in a Dilute 3He-4He Solution

We have studied the Fermi fluid, ³He in ³He-⁴He solution, confined within the narrow channels of Vycor glass. Continuous wave NMR signals were observed in two almost identical coils, one containing bulk solution and the other the solution in Vycor glass. Three different solutions (0.1, 1.6, and 6%) have been studied over the temperature range 3mK to above 1K. The measurements relate to two exclusion effects of ³He from Vycor glass. One concerns potential energy, and we demonstrate the partial exclusion of ³He independent of concentration as the temperature is lowered from high temperatures down to around 200mK, thus giving an experimental measure of the van der Waals forces between the wall and the helium. The second effect is the additional quantum exclusion which arises from the influence of quantum wire confinement on the kinetic energy, as the temperature is progressively lowered below 50mK in the very dilute (0.1%) solution.     

Condensate Fraction in Liquid 4He at Zero Temperature

We present results of the one-body density matrix ρ ₁(r) and the condensate fraction n ₀ of liquid ⁴He calculated at zero temperature by means of the Path Integral Ground State Monte Carlo method. This technique allows to generate a highly accurate approximation for the ground state wave function Ψ ₀ in a totally model-independent way, that depends only on the Hamiltonian of the system and on the symmetry properties of Ψ ₀. With this unbiased estimation of ρ ₁(r), we obtain precise results for the condensate fraction n ₀ and the kinetic energy K of the system. The dependence of n ₀ with the pressure shows an excellent agreement of our results with recent experimental measurements. Above the melting pressure, overpressurized liquid ⁴He shows a small condensate fraction that has dropped to 0.8% at the highest pressure of p=87?bar.     

Simultaneous Measurements of an Ultrasound and a Torsional Oscillator for 4He in a Nanoporous Glass

Previously, we carried out ultrasonic measurements for liquid ⁴He filled in a nanoporous glass (Gelsil), and observed an increase in the sound velocity due to decoupling of the superfluid component. At zero pressure, the superfluid transition temperature T _C is suppressed to 1.4 K from the bulk lambda point, 2.17 K. This behavior is the same as torsional oscillator measurements by Yamamoto et al. (Phys. Rev. Lett. 93:075302, 2004). However, the pressure dependence of T _C and the temperature dependence of the superfluid fraction are very different from the torsional oscillator measurements. In order to clarify the origin of the difference, we have developed a new technique of simultaneous measurement of an ultrasound and a torsional oscillator, and the system successfully works for a nanoporous glass. Here, we compare decoupling of the superfluid component for ⁴He films between an ultrasound and a torsional oscillator.     

Density and temperature dependence of the momentum distribution in liquid helium 4

Deep inelastic neutron scattering measurements on liquid⁴He have been carried out for temperatures from 0.35 K to 4.2 K and densities from 0.125 to 0.200g/cm³ at a momentum transfer of 23 Å^–1. These measurements are at large enough momentum transfer that deviations from the Impulse Approximation are accurately described by current theories and information on the single particle momentum distribution may be extracted from the measured scattering. The scattering exhibits non-Gausian behavior in both the normal liquid and superfluid phases. A distinct change in the scattering, marked by a reduction in the width and increased deviations from the classical Gaussian shape, occurs at the suerfluid transition. We present a comparison of our experimental results with recent calculations at a variety of temperatures and densities and show that theory and experiment are in excellent agreement. We also present model scattering functions, obtained by correcting for instrumental resolution and final state effects, that represent the scattering in the IA limit. Finally, we present values for the average kinetic energy and the Bose condensate fraction over a broad range densities and temperatures.     

Boson Number in Bose–Einstein Condensate of Liquid Helium Near the λ Point

Bose–Einstein condensation has been well investigated in dilute atomic gases. For liquid helium system, the superfluid component is considered to be a background flow in the Landau theory. We study the relation between Bose–Einstein condensate and superfluid component. The concept of dressed bosons is introduced, which are eigenstates of the total Hamiltonian. The total energy is the sum of the kinetic energy of the center of mass and Galilean invariant terms. Therefore, the energy of the dressed boson has a nonlinear form for their number distribution function. If it is required that the excitation energy of a dressed boson be in agreement with the elementary excitation energy of the Landau theory near 0.0 K, then the functional form of the dressed boson energy can be determined. Because of this functional form, the dressed bosons with zero momentum have no friction against a vessel only in existence of a Bose–Einstein condensate. Consequently, the condensate of the dressed bosons with zero momentum is the superfluid component. The number n ₀ of dressed bosons with zero momentum is calculated. It shows temperature dependence (1−(T/T _λ ))^1/3 near the λ point, where T is the temperature and T _λ is the λ transition temperature.     

Thermodynamics of a Trapped Bose-Condensed Gas

We investigate the thermodynamic behaviour of a Bose gas interacting with repulsive forces and confined in a harmonic anisotropic trap. We develop the formalism of mean field theory for non uniform systems at finite temperature, based on the generalization of Bogoliubov theory for uniform gases. By employing the WKB semiclassical approximation for the excited states we derive systematic results for the temperature dependence of various thermodynamic quantities: condensate fraction, density profiles, thermal energy, specific heat and moment of inertia. Our analysis points out important differences with respect to the thermodynamic behaviour of uniform Bose gases. This is mainly the consequence of a major role played by single particle states at the boundary of the condensate. We find that the thermal depletion of the condensate is strongly enhanced by the presence of repulsive interactions and that the critical temperature is decreased with respect to the predictions of the non-interacting model. Our work points out an important scaling behaviour exhibited by the system in large N limit. Scaling permits to express all the relevant thermodynamic quantities in terms of only two parameters: the reduced temperature t = T/T _c ⁰ and the ratio between the T = 0 value of the chemical potential and the critical temperature T _c ⁰ for Bose-Einstein condensation. Comparisons with first experimental results and ab-initio calculations are presented.     

Momentum Distribution and Final State Effects in Liquid Neon

We report high precision inelastic neutron scattering measurements in liquid Neon at a temperature of 25.8 K and saturated vapour pressure. The data covers a wide range of energy and momentum transfer (2 Å^?1 ? Q ? 13 Å^?1). The atomic momentum distribution, n(p), and final state effects (FSE) can be readily extracted from this intermediate wavevector transfer data provided a suitable method of analysis is used. We find that the momentum distribution in liquid Neon is marginally sharper than a Gaussian and that final state effects contribute predominantly an antisymmetric component to the the dynamic structure factor. The width of n(p) and the kinetic energy are increased by 37% above the classical values due to quantum effects. The experimental results are in agreement with theoretical values obtained by a Path Integral Monte Carlo simulation.     

Confinement Effects on Superfluid Helium

Inelastic neutron scattering has been used to probe the collective excitations of superfluid helium confined to silica aerogel. Confinement in porous aerogel results in a crossover behavior in the temperature dependence of the roton where the energy is relatively temperature independent below 1.9 K and strongly temperature dependent above 1.9 K. As in the bulk liquid, the relationship between the microscopic excitation spectrum and the macroscopic thermodynamics of the confined system are found to be in good agreement.     

Specific heat of3He-4He mixtures in Vycor glass

The specific heat of a ³ He- ⁴ He mixture filling a narrow pore is calculated in the temperature range 0.5–1 K, including the effect of van der Waals forces due to the walls, and is found to be compatible with the enhancement over the bulk liquid value experimentally observed for mixtures in Vycor glass.     

Bose-Einstein Condensation Measurements and Superflow in Condensed Helium

We review the formulation and measurement of Bose-Einstein condensation (BEC) in liquid and solid helium. BEC is defined for a Bose gas and subsequently for interacting systems via the one-body density matrix (OBDM) valid for both uniform and non-uniform systems. The connection between the phase coherence created by BEC and superflow is made. Recent measurements show that the condensate fraction in liquid ⁴He drops from 7.25±0.75 % at saturated vapor pressure (p≈0) to 2.8±0.2 % at pressure p=24 bars near the solidification pressure (p=25.3 bar). Extrapolation to solid densities suggests a condensate fraction in the solid of 1 % or less, assuming a frozen liquid structure such as an amorphous solid. Measurements in the crystalline solid have not been able to detect a condensate with an upper limit set at n ₀≤0.3 %. Opportunities to observe BEC directly in liquid ⁴He confined in porous media, where BEC is localized to patches by disorder, and in amorphous solid helium is discussed.     

Microstructure,chemistry, elastic properties and internal friction of Silceram glass-ceramics

The temperature dependence of both Young's modulus (E) and internal friction (Q ^–1) from room temperature to 700° C has been determined by Förster's forced-resonance method for three Silceram glass-ceramics, produced by the direct controlled cooling of glass melts in the quaternary system CaO-MgO-Al₂O₃-SiO₂. These results are correlated with microstructural and phase chemistry data as well as calculated viscosity against temperature data. In particular, the viscosity of the residual glass is shown to predominate over its volume fraction in deter mining the temperature dependence ofE andQ ^–1 for a given Silceram. A simple model which enables the residual glass content of Silceram glass-ceramics to be estimated from a know ledge of the proportions of silicon, iron and magnesium in the corresponding glass melts is also proposed. Furthermore, the room-temperature bulk modulus (K) and Poisson's ratio of two Silceram glass-ceramics are calculated using experimentalE and shear modulus (G) values obtained using both Förster's method and another forced-vibration technique.     

Momentum distributions and the impulse approximation in helium

The momentum distributionn(p) of atoms in condensed matter may be determined from neutron inelastic scattering measurements. However, scattering at high momentum transfers Q (high enough that the impulse approximation holds) is required. A model of solid helium is used to assess the error introduced inn(p) by assuming the impulse approximation holds at severalQ values. AtQ=20 Å^–1 the error inn(p) is 5–10%. This error can be reduced by a factor of ten using a symmetrization procedure.     

Excitations Beyond the Roton of Liquid 4He in Aerogel

The dynamic structure factor, S(Q, ), for wavevectors, 2.0Q3.6 Å ^–1 of liquid ⁴ He in 95% porous aerogel has been measured by inelastic neutron scattering methods. The aerogel was grown with deuterated materials and the multiple scattering involving the aerogel was negligible. S(Q, ) in the superfluid phase consists of a single peak plus broad intensity at higher energy , as in bulk superfluid ⁴ He. The single peak is identified with the phonon-roton excitation at higher Q. The weight in the peak, Z_Q , and the excitation energy dispersion curve, _Q , has the same basic wavevector dependence as in the bulk. The energy _Q is 2–3% below the bulk value at the end point and the peak is unobservable beyond Q=3 Å ^–1 within the present statistical precision. No peak is observed at T=2.3 K in normal ⁴ He suggesting, as in bulk ⁴ He, that the characteristic excitation at higher Q is associated with the superfluid phase.     

The Kinetic Energy of the Condensate in the Presence of an External Applied

Average quantities can be determined in many-body physics by the virial theorem even in cases when knowledge about the interaction among particles is not complete. Starting from the virial theorem for superconductivity we obtain an expression for the magnetization of two periodic systems, the infinitely extended film of arbitrary thickness under a perpendicular applied field and the bulk superconductor. We show that the kinetic energy of the condensate can be directly retrieved from the magnetization for a large κ type II superconductor or near the upper critical field, in the pinning free (reversible) regime. Isofield kinetic energy lines for low- and high-T _c compounds have striking differences near the critical transition that clearly indicate for the latter case the presence of a unique superconducting condensate below and above the mean field critical temperature.      

The Quest for Bose-Einstein Condensation in Solid 4He

Ever since the seminal torsional oscillator (TO) measurements of Kim and Chan which suggested the existence of a phase transition in solid ⁴He, from normal to a ??supersolid?? state below a critical temperature T _c = 200 mK, there has been an unprecedented amount of excitement and research activity aimed at better understanding this phase. Despite much work, this remarkable phase has yet to be independently confirmed by conventional scattering techniques, such as neutron scattering. We have carried out a series of neutron scattering measurements, which we here review, aimed at observing Bose-Einstein condensation (BEC) in solid ⁴He at temperatures below T _c . In bulk liquid ⁴He, the appearance of BEC below T _?? signals the onset of superfluidity. The observation of a condensate fraction in the solid would provide an unambiguous confirmation for ??supersolidity??. Although, our measurements have not yet revealed a non-zero condensate fraction or algebraic off diagonal long-range order n ₀ in solid ⁴He down to 65 mK, i.e. n ₀=(0±0.3)%, our search for BEC and its corollaries continues with improved instrumentation.     

The reflection of low energy phonons at the free surface of liquid4He. I. Theory

The scattering of a low energy phonon at the free surface of liquid⁴He at finite temperature is discussed in relation to the reflection measurements in the accompanying paper (II). If not specularly reflected, the phonon may be inelastically scattered, with the emission or absorption of a single ripplon, or it may be absorbed with the emission of two ripplons. The inelastic scattering produces deviations from ideal specular reflection that depend logarithmically on the angular sensitivity of the measurement. For the experiment described in II, the predicted deviations due to scattering and absorption are roughly equal and too small (5×10^–4) to be measured. In addition, there should be a small broadening (less than 1 ° in angle) of the reflected image of the phonon source due to phonon decay in the bulk liquid. This was calculated from the curvature of the phonon spectrum measured by Rugar and Foster. Phonon decay also determines the distribution of the incident phonon beam with respect to energy. From the known decay rate, the average incident phonon energy in our experiment is calculated to be 0.5 K. We also discuss the attenuation of surface second sound due to the inelastic scattering and absorption of thermal phonons at the surface. We find that two ripplon absorption is the dominant effect. Below 0.7 K, the attenuation due to phonons is probably just small enough for pure ripplon surface sound to exist in a narrow range of low frequencies. To show this, we have recalculated the ripplon lifetime using the 3-ripplon interaction as recently revised by Rocheet al. The results for the ripplon lifetime are displayed in a simple scaling format.     

Quasiparticle Kinetic Equation in a Trapped Bose Gas at Low Temperatures

Recently the authors used the Kadanoff–Baym non-equilibrium Green's function formalism to derive kinetic equation for the non-condensate atoms, in conjunction with a consistent generalization of the Gross–Pitaevskii equation for the Bose condensate wavefunction. This work was limited to high temperatures, where the excited atoms could be described by a Hartree–Fock particle-like spectrum. Following the approach of Kane and Kadanoff in 1965, we present the generalization of our recent work which is valid at low temperatures, where the input single-particle spectrum is now described by the Bogoliubov–Popov approximation. We derive a kinetic equation for the quasiparticle distribution function with collision integrals describing scattering between quasiparticles and the condensate atoms. From the general expression for the collision integral for the scattering between quasiparticle excitations, we find the quasiparticle distribution function corresponding to local equilibrium. This expression includes a quasiparticle chemical potential that controls the non-diffusive equilibrium between condensate atoms and the quasiparticle excitations. We derive a generalized Gross–Pitaevskii equation for the condensate wavefunction that also includes the damping effects due to collisions between atoms in the condensate and the thermally excited quasiparticles. For a uniform Bose gas, our kinetic equation for the thermally excited quasiparticles reduces to that found by Eckern, as well as by Kirkpatrick and Dorfman.