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Bose-Einstein Condensation Measurements and Superflow in Condensed Helium

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Bose-Einstein Condensation Measurements and Superflow in Condensed Helium JLowTempPhys DOI 10.1007/s10909-013-0855-0 Bose-Einstein Condensation Measurements and Superflow in Condensed Helium H.R. Glyde Received: 9 November 2012 / Accepted: 18 January 2013 © Springer Science+Business Media New York 2013 Abstract We review the formulation and measurement of Bose-Einstein condensa- tion (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 uni- form and non-uniform systems. The connection between the phase coherence created by BEC and superflow is made. Recent measurements show that the condensate frac- tion 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 4 set at n0 ≤ 0.3 %. Opportunities to observe BEC directly in liquid He confined in porous media, where BEC is localized to patches by disorder, and in amorphous solid helium is discussed. Keywords Solid · Liquid · Helium · BEC · OBDM · Neutron scattering 1 Introduction This article is part of a series motivated by reports of possible superfluidity in solid helium, initially in 2004 [1, 2]. Other articles in this series survey extensively the experiments and theory in this new field and the current status of the field. In this article we focus tightly on recent measurements [3–8] of Bose-Einstein conden- sation (BEC) and of the atomic momentum distribution n(k) in solid and liquid 4He. H.R. Glyde () Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2593, USA e-mail: [email protected] JLowTempPhys In 3D, superflow arises from BEC. In a uniform fluid, when there is BEC, the one- body density matrix (OBDM), n(r), has a long range tail of magnitude n0 = N0/N, the BEC condensate fraction. This long range component in the OBDM introduces coherence and order in the system which enables superflow. The Fourier transform of this OBDM, the momentum distribution, has delta function component, n0δ(k), which can be observed with neutrons. In 2D, the onset of superflow is associated with the onset of algebraically decaying long range order [9–12] in the OBDM. This decaying long range order has also been observed [13] (in liquid 4He films) using neutrons. The drive to observe BEC in solid helium is motivated chiefly by the drive to verify the existence of superflow. Since BEC has not been observed in the solid, we devote much of the article to recent measurements in liquid 4He as a function pressure and at pressures near the solidification pressure which provide some indication of the possible magnitude and behavior of the condensate fraction in the solid. To date, the condensate fraction and atomic momentum distribution in helium have been uniquely measured using neutron scattering techniques. The use of neutrons to observe the condensate fraction was first proposed by Miller et al. [14] in 1962 and Hohenberg and Platzmann [15] in 1964. Essentially, if an incoming neutron trans- fers a high momentum and high energy to the system, the energy imparted to the struck atom (the nucleus of the atom) is so high that the struck atom responds as if nearly independent. That is, the energy transferred is large compared to the po- tential energy arising from interaction with neighbors, too high for the system to respond collectively. In this high momentum transfer limit, the observed neutron en- ergy transfer is Doppler broadened by the atomic momentum distribution, n(k),of the individual atom and hence n(k) and the fraction n0 in the k = 0 state can be determined. 4 The first measurement of n0 in liquid He appears to have been reported [16] in 1968. Precision improved significantly with access to higher incidence energy neutrons and much improved spectrometer performance. The history of measure- ment is reviewed by Glyde and Svensson [17], by Silver and Sokol [18], Sokol [19], Glyde [20] and others. Hence we focus on our recent measurements made since 2000 at the ISIS Facility, Rutherford Appleton Laboratory and at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory. In Sect. 2.1, we introduce BEC beginning with the Bose gas. In Sect. 2.2,the OBDM is defined and the momentum distribution n(k) and n0 are expressed in terms of the OBDM. The OBDM is well defined for all levels of interaction and for non- uniform and finite sized systems. For uniform systems, n(k) is the Fourier transform of the OBDM. In Sect. 3 we see that the intermediate dynamic structure factor (DSF) in the impulse approximation (IA) is the OBDM. Models for the OBDM and Final State function suitable to fitting to the observed DSF are set out in Sect. 4. Mea- 4 surements of n(k), n0, and the FS function in liquid He as a function of pressure are discussed in Sect. 5. Determinations of n(k) and attempts to observe an n0 in solid 4He are presented in Sect. 6. Future opportunities for observing BEC in he- lium in porous media (in disorder) and other non-uniform systems are discussed in Sect. 7. JLowTempPhys 2 BEC and One-Body Density Matrix 2.1 BEC in a Bose Gas 2.1.1 Uniform Gas In 1924, Bose [21] introduced Bose statistics in a paper translated into German by Einstein. Einstein immediately recognized [22] that in a gas of particles obeying Bose statistics, a macroscopic fraction of the particles will condense into one single particle state at low temperature, denoted Bose-Einstein condensation (BEC). Specifically below a critical temperature Tc, in a three dimensional (3D) system, all of the particles can no longer be accommodated in the finite momentum states and condensation of a macroscopic fraction into the lowest energy single particle state begins. At T = 0K, all the Bosons occupy the lowest energy single particle quantum state, usually chosen as the zero of energy. At the time, BEC was regarded as an artifact of a Bose gas. However, today we know that BEC is widely observed in interacting systems. The macroscopic occupation of a single state introduces phase coherence in the system which is the origin of superflow and superconductivity [23]. In a uniform gas of N Bosons in a volume V (uniform density n = N/V), the single particle wave functions are plane waves, 1 ik·r φk(r) = √ e . (1) V = 2 2 The quantum number k is the particle momentum ( k) with state energy k ( 2m )k . If the gas were classical the distribution of the particles over the momentum states would be a Gaussian, the Maxwell-Boltzmann distribution, − k2 ∗ −3/2 2α n (k) =[2πα2] e 2 , (2) 2 where α2 =kα is the mean square of k along one dimension, α. For a quantum Bose gas the number of Bosons in discrete state k is given by the Bose distribution [β( −μ)] −1 −1 Nk =[e k − 1] where β = (kB T) , kB is Boltzmann’s constant, T is the temperature and μ is the chemical potential. At T = Tc, μ goes to zero since we are beginning to occupy the zero energy state. At temperatures T below Tc, a macro- scopic fraction, N0/N, of the Bosons occupy the zero momentum (k = 0) state. The condensate fraction, defined as n0 = N0/N,atT<Tc is given by T 3/2 n0 = 1 − (3) Tc and at T = 0K,n0 = 1. The normalized, discrete (Nk/N) and the normalized, con- 3 tinuous momentum distributions n(k) are related by Nk/N =[(2π) /V]n(k) with = = k Nk/N dkn(k) 1. When n0 = 1 in a uniform Bose gas, all Bosons have a well-defined momen- tum. From the uncertainty principle, their position is completely undetermined. When there is interaction between the Bosons, particularly hard core repulsion, the Bosons JLowTempPhys localize each other in space. Their positions become partially determined and n0 is rapidly reduced below one by the uncertainty principle. Beginning with the remark- able paper of Bogoliubov [24], there have been many, extensive calculations both analytic and numerical of the reduction of n0 by interaction in Bose gases, liquids and solids [11]. In calculations [25] the density/interaction is often characterized by the parameter na3 where as before n = N/V is the density and a is a scattering length or the hard core diameter of the interatomic potential. Liquid 4He can be quite well described [26] by a system of interacting hard spheres of diameter a with na3 = 0.21. 4 In liquid He under its own saturated vapor pressure n0 is reduced [27]to7.25% by interaction. It is also interesting that in strongly interacting liquid 4He, the mo- mentum distribution over the k>0 states is much better represented by a Gaussian as in Eq. (2) than by the Bose distribution function. 2.1.2 Confined Gas As shown first in 1995, Bose-Einstein Condensation in gases of alkali atoms in mag- netic traps can be spectacularly created. Since the gases are dilute, n 1, conden- sation fractions of n0 100 % are possible. In contrast to the uniform gas or liquid, the density in the trap is not uniform but is rather dictated by the external trapping = 1 2 potential. Typically the trapping potential is harmonic, V(r) 2 ω0r .
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