We present precision neutron scattering measurements of the Bose-Einstein condensate (BEC) fraction ${n}_{0}(T)$ and the atomic momentum distribution ${n}^{\ensuremath{\star}}(\mathbf{k})$ of liquid ${}^{4}$He as a function of temperature at pressure $p=24$ bars. Both the temperature dependence of ${n}_{0}(T)$ and of the width of ${n}^{\ensuremath{\star}}(\mathbf{k})$ are determined. The ${n}_{0}(T)$ can be represented by ${n}_{0}(T)={n}_{0}(0)[1\ensuremath{-}{(T/{T}_{\ensuremath{\lambda}})}^{\ensuremath{\gamma}}]$ with a small ${n}_{0}(0)=2.80\ifmmode\pm\else\textpm\fi{}0.20$$%$ and large $\ensuremath{\gamma}=13\ifmmode\pm\else\textpm\fi{}2$ for $T<{T}_{\ensuremath{\lambda}}$, indicating a strong interaction. The onset of BEC is accompanied by a significant narrowing of the ${n}^{\ensuremath{\star}}(\mathbf{k})$. The narrowing accounts for 65$%$ of the drop in kinetic energy below ${T}_{\ensuremath{\lambda}}$ and reveals an important coupling between BEC and $k>0$ states. The experimental results are well reproduced by path integral Monte Carlo calculations.
We present precision neutron scattering measurements of the Bose-Einstein condensate (BEC) fraction n0(T) and the atomic momentum distribution n\ensuremath⋆(k) of liquid 4He as a function of temperature at pressure p=24 bars. Both the temperature dependence of n0(T) and of the width of n\ensuremath⋆(k) are determined. The n0(T) can be represented by n0(T)=n0(0)[1\ensuremath−(T/T\ensuremathλ)\ensuremathγ] with a small {n}_{0}(0)=2.80\ifmmode\pm\else\textpm\fi{}0.20$$% and large \ensuremathγ=13\ifmmode±\else\textpm\fi2 for T<T\ensuremathλ, indicating a strong interaction. The onset of BEC is accompanied by a significant narrowing of the n\ensuremath⋆(k). The narrowing accounts for 65 of the drop in kinetic energy below T\ensuremathλ and reveals an important coupling between BEC and k>0 states. The experimental results are well reproduced by path integral Monte Carlo calculations.
Bose-Einstein Condensation and Kinetic energy of liquid 3He-4He Mixtures
We present neutron scattering measurements of the momentum distribution of liquid 3He-4He mixtures. The experiments were performed at wavevectors Q, 26 ≤Q≤ 29 {Å−1}, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with 3He concentrations x between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction n0 and the momentum distributions of 3He and 4He atoms. We find that n0 increases somewhat above the pure 4He value when 3He is added; e.g from n0=(7.25±0.75) at x= 0 to (11±3) at x= 15-20%. This agrees with predictions but is less than the only previous measurement. We find a 4He kinetic energy K4 for pure 4He that agrees with previous determinations. K4 decreases somewhat with increasing 3He concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The 3He response is not well reproduced by a Fermi gas momentum distribution, n(k). Rather an n(k) having a small step height at the Fermi surface and a substantial high momentum tail characteristic of a strongly interacting Fermi liquid provides a good fit. This n(k) is consistent with calculated n(k). Thus agreement between theory and experiment is obtained comparing n(k) in contrast to earlier findings based on comparing calculated and observed 3He kinetic energies.
We present neutron scattering measurements of the atomic momentum distribution n(k) in solid helium under a pressure p=41bar (molar volume ${V}_{m}=20.01\ifmmode\pm\else\textpm\fi{}0.02\text{ }\text{ }{\mathrm{cm}}^{3}/\mathrm{mol}$) and at temperatures between 80 and 500 mK. The aim is to determine whether there is Bose-Einstein condensation (BEC) below the critical temperature, Tc=200mK, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the k=0 state below Tc, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at T=80mK and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at T=120mK, consistent with zero. The shape of n(k) also does not change on crossing Tc within measurement precision.
We present neutron-scattering measurements of the momentum distribution of liquid $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ mixtures. The experiments were performed at wave vectors Q, $26\ensuremath{\leqslant}Q\ensuremath{\leqslant}29\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with 3He concentrations x between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction n0 and the momentum distributions of 3He and 4He atoms. We find that n0 increases somewhat above the pure 4He value when 3He is added; e.g., from ${n}_{0}=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ at x=0 to $(11\ifmmode\pm\else\textpm\fi{}3)%$ at x=15−−20. This agrees with predictions but is less than the only previous measurement. We find a 4He kinetic energy K4 for pure 4He that agrees with previous determinations. K4 decreases somewhat with increasing 3He concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The 3He response is not well reproduced by a Fermi-gas momentum distribution, n(k). Rather an n(k) having a small step height at the Fermi surface and a substantial high-momentum tail characteristic of a strongly interacting Fermi liquid provides a good fit. This n(k) is consistent with calculated n(k). Thus agreement between theory and experiment is obtained comparing n(k) in contrast to earlier findings based on comparing calculated and observed 3He kinetic energies.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
We present path integral Monte Carlo (PIMC) calculations of the superfluid fraction, ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}$, and the one-body density matrix (OBDM) [Bose-Einstein condensation (BEC)] of liquid 4He confined in nanopores. Liquid 4He in nanopores represents a dense Bose liquid at reduced dimension and in disorder. The goal is to determine the effective dimensions of the liquid in the pores. It is to test whether observed properties, such as a very low onset temperature for superflow, Tc, can be predicted by a standard, static PIMC ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}$. We simulate a cylinder of liquid of diameter dL surrounded by 5 \AA{} of inert solid 4He in a nanopore of diameter d; d=dL+10A˚. We find a PIMC ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$ and OBDM that scales as a 1D fluid Luttinger liquid at extremely small liquid pore diameters only, dL=6A˚. At this dL, the liquid fills the pore in a 1D line at the center of the pore and there is no PIMC superflow. In the range $8\ensuremath{\le}{d}_{L}\ensuremath{\le}22\phantom{\rule{0.28em}{0ex}}\AA{}$ the PIMC ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$ scales as a 2D liquid. In this dL range the liquid fills the pores in 2D-like cylindrical layers. The crossover from no superflow at d=16A˚ to superflow at $d\ensuremath{\ge}18\phantom{\rule{0.28em}{0ex}}\AA{}$ agrees with experiment. There is a crossover to 3D scaling at larger ${d}_{L}\ensuremath{\simeq}22\phantom{\rule{0.28em}{0ex}}\AA{}$. In the range $8\ensuremath{\le}{d}_{L}\ensuremath{\le}22\phantom{\rule{0.28em}{0ex}}\AA{}$, the Tc predicted using the Kosterlitz-Thouless 2D scaling criterion of the OBDM agrees well with that obtained from ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$. These results suggest that the superflow observed in small pore media is standard static superflow with the low Tc arising from its 2D character. An operational onset temperature, TBEC, for BEC can be defined as the temperature at which there is a crossover from exponential to algebraic decay in the OBDM. This definition leads to a ${T}_{\mathrm{BEC}}\ensuremath{\ge}{T}_{c}$ as observed in larger pore media.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.