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 [3Ð8] 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 [9Ð12] 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
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 . In this case for a non-interacting gas, the single particle wave functions are harmonic oscillator = + 3 wave functions φn(r) with corresponding energies n ω0(n 2 ). The BEC is macroscopic occupation of the lowest energy (n = 0) state,
2 − 3 − 1 r 2 2 2 2 φ0(r) = 2π x e x , (4)
2 where x =/2mω0. In traps, the impact of interaction is to both reduce n0 be- low 100 % and modify (chiefly broaden) the natural single states away from simple harmonic oscillator states. The natural single particle states in the presence of in- teraction can be found analytically making approximations or numerically [28]by diagonalizing the one-body density matrix (OBDM) as discussed below.
2.2 Interacting Bose Systems and the OBDM
2.2.1 One-Body Density Matrix
While many qualitative features found in a Bose gas extend to strongly interacting fluids (e.g. BEC itself), a complete quantitative description of fluids based on the Bose gas formulation is difficult. For example, while the superfluid fraction of a Bose gas at T = 0 K as calculated numerically [29, 30] or analytically [31] is formally ρS/ρ = 1, a Bose gas is strictly not a superfluid since its critical velocity is zero. Some interaction, however small, is required to change the mode structure of the 2 gas qualitatively from single particle-like (ωQ ∝ Q ) to phonon-like (ωQ ∝ Q)so that the critical velocity is finite. In interacting systems, it is more rigorous to begin with quantities such as the OBDM which are well defined at all levels of interaction, strong and weak, and define BEC in terms of the OBDM. We will also see that the JLowTempPhys
OBDM is a more smoothly varying function and easier to model for comparison with experiment than the momentum distribution and the macroscopic occupation of a single state. The OBDM is defined as [23, 32, 33], ˆ † ˆ ρ1(r1,r2) ≡ Ψ (r2)Ψ(r1) , (5)
ˆ where Ψ(r1) is the field operator which removes a particle from the system at point ˆ † r1 and Ψ (r2) is the operator which creates a particle at point r2.Theρ1(r1,r2) is the probability amplitude that we can remove a particle at r1 and add one at r2 in the system. The expectation value is the usual finite temperature thermal average. In the coordinate representation this gives [11],
1 − ρ (r ,r ) = dRR|e βHΨˆ †(r )Ψ(rˆ )|R, (6) 1 1 2 Z 2 1
−1 where Z is the partition function, H is the Hamiltonian, β = (kB T) and |R is an N particle state. To define BEC, the next step is to introduce single particle states φi(r) with a state index i (e.g. a momentum or energy index). In terms of these states the field operator Ψ(ˆ r) is
ˆ Ψ(r)= aiφi(r) (7) i where ai is the operator which annihilates a particle from state φi(r). The number = † of particles in state φi(r) is Ni ai ai .Theφi(r) can be defined as the “natural” single particle orbitals [32], the states which diagonalize the OBDM, i.e.
= ∗ ρ1(r1,r2) φi (r2)φi(r1)Ni. (8) i
The φi(r) can be found, for example, by diagonalizing ρ1(r1,r2) numerically in the form,
∗ = dr1 dr2φi (r1)ρ1(r1,r2)φi(r2) Ni (9) where φi(r2) are obtained as the eigenvectors and Ni as the eigenvectors that di- agonalize ρ1(r1,r2). Natural orbitals can always be defined by an equation of the form of Eq. (8). Equation (8) is the off-diagonal generalization of the expression for = = ∗ the density ρ(r) ρ1(r, r) i φi (r)φi (r) introduced by Kohn and Sham [34]in their density functional theory of Fermi systems in which Ni = 1 for occupied states. Other definitions of the φi(r) are possible [35], but Eq. (8) is regarded [23, 32]asthe “natural” choice. When there is BEC, the number N0 in the lowest energy state is macroscopic. In this case Eq. (8) becomes JLowTempPhys
= ∗ + ∗ ρ1(r1,r2) N0φ0 (r2)φ0(r1) Niφi (r2)φi(r1) i = ∗ + ∗ N0φ0 (r2)φ0(r1) ρ1 (r1,r2). (10)
Equation (10) may be taken as the definition of BEC in an interacting system. The condensate fraction, N0/N, can be calculated numerically for arbitrary interaction and geometry by diagonalizing the OBDM in the form of Eq. (9). The sum in the ∗ second term is over the i = 0 states and ρ1 (r1,r2) is the OBDM arising from the states above the condensate.
2.2.2 OBDM and Momentum Distribution
The general expression for the atomic momentum distribution in terms of the OBDM is
1 − · − n(k) = dr dr ρ (r ,r )e ik (r1 r2). (11) N(2π)3 1 2 1 1 2 This momentum distribution is normalized to unity (to one particle). The general expression for n(k) follows from the natural orbitals expressed in momentum space,
1 −ik·r φi(k) = dre φi(r). (12) (2π)3/2
The momentum distribution or momentum density is obtained from the φi(k) as,
N n(k) = i φ (k) 2 (13) N i i in the same way that the particle density in r space normalized to unity drρ(r) = 1 is obtained from the φi(r) as,
1 1 N ρ(r) = ρ (r, r) = Ψˆ †(r)Ψ(ˆ r) = i φ (r) 2. (14) N 1 N N i i
On substituting Eq. (12)forφi(k) into Eq. (13) and using ρ1(r1, r2) given by Eq. (8), we obtain the Eq. (11)forn(k) valid for finite sized or non-uniform systems. In these expressions we have retained vector notation since below we will want to distinguish 3D from 1D momentum distributions. In a uniform fluid with translational symmetry in which momentum is a good quantum number, the φi(r) are the plane wave states φk(r) given by Eq. (1) indepen- dent of the level of interaction. On substituting the state φk(r) given by Eq. (1)into Eq. (8), the OBDM is
1 · − ρ (r ,r ) = N eik (r1 r2). (15) 1 1 2 V k k JLowTempPhys
The OBDM for a uniform system depends only on the separation r = r1 − r2, ρ1(r1,r2) = ρ1(r). It is convenient to introduce a OBDM, n(r),
1 N · · · n(r) ≡ ρ (r) = k eik r = dkn(k)eik r = eik r , (16) n 1 N k which is normalized so that n(r = 0) = 1. For a uniform fluid the OBDM n(r) is clearly the Fourier transform of the momentum distribution n(k).Itisalsotheex- pectation value of eik·r and n(r) could be calculated as an equilibrium expectation value of eik·r although this does not appear to have been tried yet. The eik·r is the translation operator that translates a particle a distance r = r1 − r2 from point r1 to r2. The corresponding momentum distribution is normalized to unity.
2.3 BEC and Superfluidity
When there is BEC, we have seen in Eq. (10) that the OBDM may be written as, ≡ ∗ + ∗ ρ1(r1, r2) Ψ0 (r2)Ψ0(r1) ρ1 (r1, r2), (17) ∗ ≡ ∗ ∗ where Ψ0 (r2)Ψ0(r1) N0φ0 (r2)φ0(r1) and ρ1 (r1, r2) is the OBDM arising from the states above the condensate. The macroscopic field Ψ0(r), N 1/2 Ψ (r) = N φ (r) = N φ (r) eiϕ = 0 eiϕ, (18) 0 0 0 0 0 V is the condensate order parameter. The condensate√ field Ψ0(r) introduces coherence into the system. It has a magnitude proportional to N0 and a phase ϕ. The last equal- ity in Eq. (18) is valid for a uniform fluid and in this case Ψ0(r) has long range coher- ence or long range order (LRO). The condensate density is ρ0 = N0/V. The conden- sate velocity can be obtained from the standard quantum mechanics expression for the ∗ ∗ current arising from the field, j =[Ψ0 vΨ0 + (vΨ0) Ψ0]/2 = ρ0(/m)ϕ = ρ0vs , giving the standard expression for the velocity,
vs = (/m)ϕ. (19)
This is the condensate velocity but is generally interpreted as the superfluid veloc- ity [23, 36]. The physical picture used is that the fluid in the condensate interacts with the remainder of the fluid, and much of the fluid can be dragged along by the condensate so that much of the fluid has velocity vs . Similarly, at T → 0Kthesu- perfluid fraction ρs/ρ goes to unity (the whole fluid dragged along with condensate). 4 Remarkably, the superfluid fraction ρs/ρ in uniform bulk liquid He under pressure near the solidification line (p = 25.3 bar) goes to unity at T → 0 K while the con- densate fraction remains at n0 = N0/N = 3%. The temperature dependence of ρs/ρ is well predicted up to 1.5 K by the Landau theory [37]. In the Landau theory, the whole fluid is in the ground (superfluid) state at T = 0Ki.e.ρS/ρ = 1. At low temperature, low energy modes are thermally excited and some of the fluid is excited out of the ground state by these excitations. In a uni- form liquid, these modes are long wavelength phonon modes. At higher temperature, JLowTempPhys phonon-roton modes at higher wave vectors in the roton region (Q 1.95 Å−1)are also thermally excited. While the temperature dependence of ρS/ρ is well reproduced by the Landau theory up to approximately 1.5 K, the specific heat, cV , is remarkably well predicted up to 2.0 K by excitation of the P-R modes [38, 39]. At higher tem- peratures, especially near Tλ = 2.17 K, critical fluctuations play a dominant role in 4 determining ρS/ρ and cV . For liquid He confined in porous media, the ρS/ρ and cV are well predicted by the Landau theory involving excitation of the layer modes rather than the phonon-roton mode [40Ð43]. The layer mode is dominant in porous media because the layer mode has a lower energy than the P-R mode in the roton region. Essentially, in Landau theory, the superfluid fraction is reduced by excitation of the fluid with the lowest energy modes, whatever their origin, making the most important contribution. At the same time, ρS/ρ, and its temperature dependence can be calculated ac- curately using path integral Monte Carlo (PIMC) methods in both 3D and 2D, in- dependently of BEC and of the Landau criterion for superfluidity [11, 12, 44]. The PIMC values agree very well with experiment up to Tλ. In these calculations, a fluid in a rotating container is considered. The superfluid fraction, ρS/ρ, is defined as that fraction of the fluid that does not rotate with the container [11, 30, 45], the same defi- nition as used in experiment. The non-rotating fraction is expressed in terms of a free energy change of the rotating fluid that can be calculated by PIMC methods. While these PIMC ρS/ρ values are quite independent of BEC, the same PIMC formulation can be used to evaluate the OBDM and n0 which agree equally well with experiment as we will see directly below. In 3D uniform liquid 4He, superflow is generally regarded as a consequence of BEC and LRO. In contrast, in 2D the onset of a superfluid density is associated with the onset of binding of pairs of vortices [9]. However, in 2D the superfluid state is also associated [11, 44] with the onset of long range order (LRO) in the OBDM, albeit al- gebraically decaying LRO. That is, in the superfluid phase the OBDM in Eq. (17) has ∗ a macroscopic first term ρ1(r) = ρ1(0, r) = Ψ0 (r)Ψ0(0) which decays slowly [12] over long range in r (typically 100 Å). It is not infinitely long as it is for genuine BEC. This is denoted algebraic LRO. As discussed below, in measurements we can observe the LRO over a length r ≈ 5 Å only. Thus LRO in 2D can be observed [5] with neutrons as readily as LRO (BEC) in 3D. We have observed [5] the 2D LRO in liquid helium films [5] on the walls of porous media. If the “BEC” in solid helium is 2D, it can be observed in the usual way with neutrons.
3 Observing BEC and the OBDM
The one-body density matrix (OBDM) and the momentum distribution can be ob- served directly in neutron scattering measurements. To show this we consider a neu- tron with initial energy Ei and momentum ki scattering from the sample and exiting with final energy and momentum Ef and kf , respectively. The momentum, Q, and energy, ω, transferred from the neutron to the sample is
Q = (ki − kf ), ω = Ei − Ef , (20) JLowTempPhys respectively. We consider high momentum transfer, Q, in which the momentum transferred is so high that the atoms respond as if nearly independent and collective response is not observed. In this limit the incoherent dynamic structure factor (DSF),
1 iωt Si(Q,ω)= dte Si(Q,t), (21) 2π 1 − · · S (Q,t)= e iQ rl (t)eiQ rl(0) (22) i N l
− · is observed where the sum over l is over the N atoms in the system. The e iQ rl(t) is in the Heisenberg representation. Si(Q, t) may be readily evaluated for a Bose gas using the plane wave r|k states in Eq. (1) to write the statistical average in Eq. (22). This gives
· 1 −i(ω + k Q )t −βE −iω t −ik v t S (Q,t)= e R m e k = e R e Q R (23) 0 Z k ˆ where we have introduced the notation, kQ = k · Q and
Q2 Q ωR = ,vR = . (24) 2m m
The ωR and vR are referred to as the free atom recoil frequency and velocity, re- spectively. They are the actual frequency and velocity of the recoiling, struck atom ˆ only if the initial momentum k of the struck atom is zero. In Eq. (23), kQ = k · Q is the initial momentum k of the struck atom along the scattering wave vector Q.Fora spherically symmetric, uniform liquid the direction of Q is unimportant, However, in a single crystal of solid, the direction of Q relative to crystal axes is important. Given the expression for S0(Q,t) at high Q it is useful to introduce the length variable s and wave vector variable y defined by,
s = vRt, y = (ω − ωR)/vR, (25) and write S0(Q,t)in the form iωRt −ikQs J0(s) ≡ e S0(Q,t)= e . (26)
The s = vRt is the distance traveled by a freely recoiling struck atom, initially at rest, a time t after the scattering. The S0(Q,t) and J0(s) are actually valid for an inter- acting system if we assume that the struck atom does not interact with its neighbors subsequent to the scattering. This assumption is referred to as the Impulse Approxi- mation (IA), i.e. −ikQs JIA(s) = e = n(s). (27) If we compare Eqs. (16) and (27) we see that the intermediate scattering function in ˆ the IA, JIA(s), is the OBDM of the system for lengths s = r · Q parallel to Q. Thus, when we measure JIA(s) we determine the OBDM. JLowTempPhys
The dynamic structure factor, Eq. (21), expressed in the y = (ω −ωR)/vR variable is
1 iys J(Q,y)= vRSi(Q,ω)= dse J(Q,s). (28) 2π In the IA, this is 1 J (y) = δ(y − k ) = N δ(y − k ) IA Q N k Q k
= dkn(k)δ(y − kQ). (29)
JIA(y) is denoted the longitudinal momentum distribution. It is the momentum dis- tribution integrated over the two dimensions perpendicular to Q ina3Dsystem. The exact expression for an interacting system including the interaction of the recoiling struck atom with its neighbors in the final state is (p. 325, Ref. [20]), s −i ds kQ(s ) J(Q,s)= Tse 0
= JIA(s)R(Q, s) (30) where Ts means a time ordered product with shorter distances s = vRt (shorter times) ordered to the right side in the product. In Eq. (30), the atom is struck at the origin (s = 0) at t = 0 and kQ = kQ(s = 0) is the initial momentum of the atom parallel to Q. As the recoiling atom moves and interacts with its neighbors, its momentum at distance s, kQ(s)+Q, can be different from kQ(0)+Q. The interactions between the recoiling, struck atom and its neighbors which change the momentum of the recoiling atom are denoted final state (FS) interactions. If we ignore FS interactions so that kQ(s) = kQ at all s, then Eq. (30) reduces to the IA given by Eq. (27). To include FS effects, we can write J(Q,s)= JIA(s)R(Q,s) which defines a FS function, R(Q,s). Expressions for JIA(s) and R(Q,s) can be obtained by making a cumulant expansion of both JIA(s) in Eq. (27) and J(Q,s)in Eq. (30) and identify- ing which terms belong to R(Q,s). These expansions in powers of s are rigorously valid when J(Q,s) is short range in s, i.e. when there is no condensate. We assume that this short range behavior remains valid when a long range, constant condensate ∗ term n0 is added to n(s) = JIA(s) in Eq. (27). With a condensate, n(s) = n0 + n (s) and the cumulant expansion is used for the “above the condensate” term n∗(s) only.
4 Models of the OBDM
We have seen that the intermediate dynamic structure factor (DSF) in the impulse ap- −ik s proximation (IA), JIA(s) =e Q in Eq. (27), expressed in the length variable s is ik·r ˆ exactly the OBDM, n(r) =e .InJIA(s) the length s = r · Q is a 1D variable par- allel to the scattering vector Qˆ rather than the vector r that appears in n(r) but other- wise the two are identical i.e. JIA(s) = n(s). Similarly, we saw that the full observed J(Q,s) can be written as a product J(Q,s)= JIA(s)R(Q, s) where R(Q,s) is the JLowTempPhys
Final State function that incorporates interactions of the recoiling struck atom with its neighbors in the final state. To analyse data we construct models of the OBDM and the FS function to fit to the data. In a neutron scattering experiment at high Q and energy transfer, the corresponding times t and lengths s = vRt are short. We draw on this physical feature and the nature of the OBDM to construct our models. In the absence of BEC, JIA(s) can be expanded in a cumulant expansion [46], 2 4 6 ∗ ∗ α s α s α s J (s) = n (s) = exp − 2 + 4 − 6 +··· , (31) IA 2! 4! 6! where = 2 α2 kQ , = 4 − 2 2 α4 kQ 3 kQ , (32) = 6 − 4 2 + 2 3 α6 kQ 15 kQ kQ 30 kQ , etc. are the cumulants. This expansion will converge well if s is small or if ∗ JIA (s) (and n(k)) are close to a Gaussian so that higher order cumulants (e.g. α4) are small. Similarly, we can expand the full J(Q,s)= n∗(k)R(Q, s) in cumulants. The terms in J(Q,s) that are not in n∗(s) given by Eq. (31) belong to R(Q,s) and we can obtain an expansion for R(Q,s) in this way. Up to s6 this gives iβ s3 β s4 iβ s5 β s6 R(Q,s) = exp 3 + 4 − 5 − 6 +··· . (33) 3! 4! 5! 6!
Expressions for the coefficients βn are: = β3 a3/λQ, = 2 β4 a4/(λQ) , (34) = 3 + β5 a52/(λQ) a54/λQ, = 4 + 2 β6 a62/(λQ) a64/(λQ) ,
2 2 4 where the anm are independent of Q and λ = /m = 1.0443 meV Å in liquid He. In fits to data we have found a4, a54 and a62 negligible and have omitted these terms. Expressions for the an and anm appear in Refs. [46Ð48]. When there is BEC in a uniform liquid, we have a macroscopic fraction n0 = N /N in the k = 0 state, φ (r) = √1 , and the OBDM Eqs. (10), (16), (17) and (27) 0 0 V for s = r · Qˆ is,
Nk ∗ n(s) = n + eikQs = n + A n (s). (35) 0 N 0 1 k We assume that we can add the long range condensate term to n(s) as in Eq. (35) and continue to use the cumulant expansion for the short range components n∗(s) JLowTempPhys
∗ Fig. 1 LHS: The One body density matrix (OBDM), n(s),givenbyEq.(37), and n (s) of the Bose condensed liquid at SVP. Also shown is the FS function R(Q,s) at Q = 27.5Å.RHS: The Final State (FS) function R(Q,y) at SVP and 24 bar. R(Q,y) broadens with increasing pressure (from Ref. [6]) (Color figure online) and R(Q,s) in the product J(Q,s)= n(s)R(s).InEq.(35) we have introduced a normalizing constant A1, which is less than unity so that we can use a normalized n∗(r) = dkn∗(k)eik·r for the atoms above the condensate, n∗(r = 0) = 1. The cor- ∗ responding momentum contribution is n(k) = n0δ(k) + A1n (k). In a Bose condensed fluid there is a coupling between the single particle excita- tions and density excitations [49]. This coupling leads to a small term n0f(k) in n(k) that couples (or contains both) the k = 0 and (k = 0) states so that (p. 363, Ref. [20]), ∗ n(k) = n0 δ(k) + f(k) + A1n (k). (36)
The corresponding OBDM n(r), the Fourier transform of n(k),forr parallel to Qˆ ,s= r · Qˆ ,is ∗ n(s) = n0 1 + f(s) + A1n (s). (37) This means that the simple separation of n(r) into a BEC term plus an “above the condensate” term is not always valid. The A1 is chosen so that n(r) is normalized including f(r), i.e., n(s = 0) = n0[1 + f(s= 0)]+A1 = 1. The model OBDM that we fit to experiment consists of Eq. (37)forn(s) and the cumulant expansion for ∗ ∗ n (s). The fitting parameters are n0, the cumulants α2, α4, α6 in n (s) and a3, a52 6 and a64 and in R(Q,s) (out to s ). In Fig. 1 we show n(s), n∗(s) and R(Q,s) for liquid 4He under its own satu- ∗ rated vapor pressure (SVP)(p ∼ 0) at T 0.05 K where n0 = 7.25 %. The n (s) is approximately a Gaussian. This n∗(s) represents the normal, uncondensed part of the OBDM and is short ranged in s. The corresponding momentum distribution is approximately given by n(k) of Eq. (2), a Gaussian. There is some deviation from a Gaussian arising from non-zero higher order terms in n∗(s) of Eq. (31). The n(s) includes the term n0[1 + f(s)] which is long range in s representing the ODLRO in n(s).Thirdly,R(Q,s) is a simple function in s space which decreases uniformly with increasing s and goes to zero in this case at s 4 Å. The FS function serves to cut off the OBDM so that the condensate contribution to n(k) is seen as a broadened delta JLowTempPhys function in J(Q,y). Since the R(Q,s) is short ranged we can observe the conden- sate contribution out to s 4 Å only. As a result, as noted above, we cannot readily distinguish between full ODLRO (3D) and algebraically decaying LRO (in 2D).
5 Measurements in Liquid 4He
Figure 2 shows the phase diagram of bulk liquid and solid including the superfluid phases, a phase yet to be verified in the solid case. In the superfluid regions, we anticipate BEC. Figure 3 shows the observed dynamic structure factor J(Q,y)at Q = 27.5Å−1 of liquid 4He at pressures p = 12 and p = 24 bar. Shown is J(Q,y) at low tempera- ture in the superfluid phase and at a temperature above Tλ in the normal phase at the two pressures. We see additional intensity in J(Q,y) at y = 0 at low tempera- ture arising from the condensate fraction n0 in the superfluid. There is an additional term n0R(Q,y) in J(Q,y) at low T when there is BEC that is not there in the nor- mal phase above Tλ. This term contributes chiefly at y = 0, following the shape of R(Q,y) shown in Fig. 1. Particularly, the width of the additional peak in J(Q,y) at y = 0 arising from n0 is set by the width of R(Q,y).IntheIA,R(Q,y) = δ(y).The oscillations in R(Q,y) at larger y can be seen faintly in the low T data. Particularly, we may see visually that n0 decreases with increasing pressure. To obtain the condensate fraction n0, the OBDM and the FS function, we fit the model J(Q,s)= n(s) R(Q,s) with n(s) given by Eq. (37) and R(Q,s) by Eq. (33) to the data. The n0,theαn in n(s) and the βn in R(Q,y) are treated as free fitting parameters. A fit of the model to J(Q,y) at 24 bars and Q = 24 Å−1 is shown in Fig. 4. By fitting the J(Q,y) at several Q values for 22 ≤ Q ≤ 29 Å−1 (and at 5 pressures) we can obtain reasonably consistent values for all the parameters. Particularly, fitting the model to data as a function of Q enables us to distinguish the ∗ αn parameters in n (s) which are independent of Q from the parameters βn in the FS function which decrease with increasing Q.
Fig. 2 Phase diagram of bulk helium (Color figure online) JLowTempPhys
Fig. 3 Observed dynamic structure factor of liquid 4He, J(Q,y), at wave vector Q = 27.5 Å folded with the instrument resolution. Shown is J(Q,y)versus y, the energy transfer in momentum units, at low temperature in the Bose condensed phase (blue diamonds) and in the normal liquid phase (red circles)at pressures p = 12 and 24 bar. The difference between J(Q,y) at y 0 in the Bose condensed and normal liquid phases, which arises chiefly from BEC, clearly decreases with increasing pressure (from Ref. [6]) (Color figure online)
Fig. 4 Observed J(Q,y)(open circles) at pressure p = 24 bar and temperature T = 0.055 K showing a fit (solid line)ofthe model J(Q,y)given by Eqs. (28)and(30) to the data. Both the observed and fitted J(Q,y)include the MARI instrument resolution function shown by the dotted line. A scale in the y variable and in the neutron energy transfer (ω − ωR),where 2 2 ωR = Q /2m = 1.0443Q2/2 = 300.8 meV is the free atom recoil energy, is shown. A condensate fraction, n0 = 3.2 %, provides the best fit (from Ref. [6]) (Color figure online)
The condensate fraction n0 at low temperature versus pressure is shown in Fig. 5. The n0 decreases from n0 = 7.25 ± 0.75 % at SVP to n0 = 3.0 ± 0.75 % at the solidification pressure (p = 25.3 bar). Monte Carlo calculations clearly agree well with the observed n0. Figure 6 shows the same values of n0 plotted versus the liquid density rather than the pressure. Previous values observed by Snow et al. [52]are JLowTempPhys
Fig. 5 Condensate fraction, n0, at low temperature in liquid 4He versus pressure. The solid circles are observed values from Ref. [6]. The lines are calculated values, PIGS by Rota and Boronat [50]andDMCby Moroni and Boninsegni [51]. At SVP, values calculated by Boninsegni et al. [12](triangle) and observed previously [48]are also shown (from Ref. [6]) (Color figure online)
Fig. 6 Condensate fraction, n0, at low temperature versus density. Solid circles are values from Ref. [6] and Diallo et al. [5]andthecrosses are earlier values observed by Snow et al. [52] (from Ref. [6]) (Color figure online)
also shown in Fig. 6 as are observed values of n0 in the solid phase. Clearly, on the basis of n0 in the liquid as a function of density, a small n0 in the denser solid is expected. The n0(T ) as a function temperature at SVP is shown in Fig. 7.Then0(T ) clearly rises steeply at temperatures immediately below Tλ = 2.17 K and then increases much less rapidly below T 1.5 K. The observed n0(T ) can be fitted by T γ n0(T ) = n0(0) 1 − , (38) Tλ where n0(0) = 7.25 ± 0.75 % and γ = 5.5 ± 1.0. The exponent γ is much larger than the Bose gas limit for which γ = 3/2. The n0(T ) at 24 bars from Ref. [8]isshown in Fig. 8. The function Eq. (38) has again been fitted to the data and a best fit gives n0 = 2.8 ± 0.20 % and γ = 13 ± 2. At p = 24 bar n0(T ) rises very rapidly below Tλ = 1.86 K but flattens off to its maximum value at T = 1 K. The rapid rise below Tλ requires a large γ in Eq. (38) to fit the data. Essentially, the large interaction between atoms at p = 24 bar limits n0. This upper limit of n0 is reached relatively quickly JLowTempPhys
Fig. 7 Condensate fraction 4 n0(T ) in liquid He at SVP vs. temperature (from Ref. [48]) (Color figure online)
Fig. 8 As Fig. 7 at pressure p = 24 bar (from Ref. [8]) (Color figure online)
below Tλ since temperature is relatively less important in reducing n0 at p = 24 bar. We have to warm the fluid to temperatures close to Tλ before temperature (rather than interaction) becomes the leading factor that reduces n0. Figure 9 shows the parameter α2 that sets the width of the Gaussian component of the momentum distribution n∗(k). This width arises from the atoms in finite k states above the condensate. Shown in Fig. 9 is α2 at low temperature (e.g. T 0.05 K) 2 and in the normal phase above Tλ.Theα2 =kQ , the mean square of k along the direction of Q, drops significantly with decreasing temperature in the Bose condensed phase between Tλ and T 0.05 K. Q can be along any direction in a bulk liquid, e.g. 2 2 kQ = kα in Eq. (2). In contrast, the α2 changes little in the normal phase [11, 48, 53]. For example, the atomic kinetic energy is given by Eq. (40) in the next section. On substituting the momentum distribution n(k) from Eq. (36) into Eq. (40), we JLowTempPhys
Fig. 9 The parameter α2 that describes the one body density matrix (OBDM) of the liquid ∗ above the condensate, n (s) given by Eq. (31), as obtained from fits to data. The α2 determines the Gaussian ∗ ∗ component of n (s) and n (k). It is larger in the normal liquid phase than in the Bose condensed phase (from Ref. [7]) (Color figure online)
obtain 32 K= A α , (39) 2m 1 2 where A1(T ) = 1 − n0(T )[1 + f(s = 0)]. From measurements and calculations of K, we know that α2 changes little in the normal phase where n0 = 0 and A1 = 1. However, unexpectedly, the onset of BEC is associated with a decrease in the mean 2 square momentum kQ =α2 of the atoms above the condensate in addition to BEC itself. The drop in α2 below Tλ isshowninFig.7. Recent measurements and PIMC calculations show that the temperature dependence of α2(T ) below Tλ tracks that of n0 closely [8]. In a Bose gas high occupation of low k states is expected at low temperature when the chemical potential goes to zero and this would reduce α2.Ina 4 strongly interacting system such as liquid He, the drop in α2(T ) below Tλ appears to be coupled to n0(T ). The decrease in α2 with temperature below Tλ, as well as being physically inter- esting, has important implications for the measurement of n0.Then0 is sometimes determined from measurements of the atomic kinetic energy assuming that α2 does not change with temperature. The aim is to determine A1(T ) in Eq. (39) and obtain n0(T ) from A1(T ) as n0(T ) = (1 − A1(T ))/[1 + f(0)], f(s= 0) = 1.28. Measure- ments of K in the normal phase where A1 = 1 are used to determine α2 and the con- densate fraction n0(T ) can be obtained from the temperature dependence of A1(T ) below Tλ, assuming that α2 is independent of temperature. This method is valid only if α2 is indeed independent of temperature. If α2 decreases with decreasing tempera- ture below Tλ,thenthedropinK below Tλ arises from both the decrease in α2 and the onset of BEC. We have found that it is very important to take account of the de- crease of α2 below Tλ. At higher pressure where n0 is small, n0 can be overestimated by a factor of two using the K method and assuming α2 is independent of T . Finally, Fig. 10 shows the momentum distribution n∗(k) in liquid 4He. Shown is the full momentum distribution and its Gaussian component. This shows that the full quantum n∗(k) differs significantly from a Gaussian (a classical Maxwell-Boltzmann distribution) with higher occupation of low momentum states than a classical liquid. This is as may be expected for a cold quantum liquid. Also shown is the change JLowTempPhys
Fig. 10 Top: The 3D ∗ momentum distribution n (k) for the liquid above the condensate at SVP and 24 bar. ∗ Bottom: The 3D n (k) at 24 bar and its Gaussian component (α4 = α6 = 0) (from Ref. [6]) (Color figure online)
in n∗(k) with pressure. The n∗(k) clearly broadens with increasing pressure as the atoms become increasingly localized in space under pressure. This is consistent with ∗ n0 decreasing with pressure. Within precision, we found that the shape of n (k) did not change with pressure.
6 Measurements in Solid 4He
The property of solid helium that has been most widely investigated with neutrons is the atomic kinetic energy [11, 53Ð56]. For a spherically or cubic symmetric system, this is,
2 3 2 K= dkn(k)kα , (40) 2m where kα is the momentum along an axis α in the system. The α2 is the width parameter of n(k) discussed above and shown in Fig. 9 for liquid 4He. Since a solid has structure, α2 in principle depends on the orientation of the solid. However, since measurements are usually made on polycrystals, an average of α2 over all directions is observed. Also, there does not appear to be a large orientation dependence of α2. When there is BEC, n(k) = n0δ(k) + A1n(k).Then0δ(k) term does not contribute JLowTempPhys
Fig. 11 Kinetic energy per atom in solid and liquid 4He. The solid symbols are experimental values in the solid from Hilleke et al. [54](solid triangles), Celli et al. [57](solid circle), Blasdell et al. [58](solid squares) and Diallo et al. [3] (solid star). The open circles and squares are calculations in the solid from Whitlock and Panoff [59] and from Ceperley Refs. [11, 58], respectively. The open triangles and diamonds are measurements [7]inthe superfluid and normal liquid respectively (from Ref. [6]) (Color figure online)
to the second moment in Eq. (40) and K reduces to Eq. (39) with A1 = 1 − n0.In thesolidatermf(k) that appears in Eq. (36) has not been identified. Figure 11 shows the K of liquid and solid helium close to the liquid/solid bound- ary (the vertical dashed lines) as a function of molar volume. The volume between the dashed lines is inaccessible since there is a volume drop, V ∼ 1.1cm3/mole, on solidification. In the liquid, on the RHS in Fig. 11,theK is lower in the super- fluid phase than in the normal liquid phase [6] because of the onset of BEC and the decrease of α2 below Tλ.TheK values in the solid phase shown in Fig. 11 are low temperature values. Up to 3Ð4 K, K in the solid changes little with temperature, be- cause the dynamics is dominated by zero point effects at low temperature and there is no phase change with temperature in the solid. The chief point is that there is little change in K on solidification. At a given temperature there is a small increase in K on solidification of the liquid, but it is small. The K appears to depend chiefly on the volume, much less so on the structure. An extensive comparison of K in the solid and liquid especially at higher temperatures has been made by Ceperley and co-workers [53]. This shows that the width α2 of n(k) at low temperature is quite similar in the solid and liquid phases. In the liquid, the n(k) is narrower than a Gaussian. The leading deviation from a ∗ 2 Gaussian can be characterized by the kurtosis of n (s) which is δ = α4/(α2) = 0.4 independent of pressure within precision in the liquid. Similarly, we observed [3] a positive kurtosis of δ = 0.44±0.15 in solid helium at a volume V = 20.87cm3/mole just inside the solid phase. Thus the shape and width of n(s) and n(k) are very similar in solid and liquid 4He. On this basis, we might expect n0 to be similar in liquid helium and amorphous solid helium at the same density, if an amorphous bulk solid could be created. Ex- trapolation of n0 observed in the liquid as shown in Fig. 6 to solid densities suggests that n0 will be less that 1 % in an amorphous solid at the liquid/solid interface. For a perfect crystalline solid, the perfect order drives n0 to zero effectively [44, 60]. In contrast, amorphous solid helium [44] and solid helium containing defects such as vacancies [61] are predicted to have an n0 of 0.3Ð0.5 %. JLowTempPhys
Fig. 12 Observed J(Q,y)at − Q = 28.0Å 1 versus y in hcp solid 4He at molar volume V = 19.8cm3/mol at temperatures 500 mK, 150 mK and65mK(circles). The solid line is a fit of the model OBDM with n0 = 0 to the data at T = 500 mK. The fit at 500 mK is superimposed on the data at 150 mK and 65 mK to show that the data is similar at each temperature. The dashed line is the instrument resolution (from Ref. [5]) (Color figure online)
We made two attempts to measure BEC in solid helium. In the first [4], we mea- sured J(Q,y) in bulk solid 4He cooled using the blocked capillary method over a period of 2Ð3 hours. We were not able to detect any change in shape of J(Q,y) on cooling the sample between T = 500 mK through Tc 200 mK to 120 mK. Particu- larly no additional intensity centered at y = 0 suggesting BEC down to temperatures T = 120 mK was observed. At that time the Tc for superfluidity in commercial grade 4 3 He (0.3 ppm He) was believed to be Tc 200 mK. If there were BEC in three- dimensions or in a 2D film, as discussed above, we would expected to see additional intensity in J(Q,y)at y = 0 (a constant or nearly constant term in the OBDM n(s)). Adams and co-workers [56] measured the kinetic energy K of bulk solid helium as a function of temperature through Tc. They found no change in K with temper- ature and, from K=(3/2m)A1α2 and A1 = 1 − n0, they inferred no condensate n0 (or drop in α2) below Tc within precision. JLowTempPhys
Fig. 13 The condensate fraction n0 versus Q at 150 mK and65mKinhcpsolidhelium at V = 19.8cm3/mol obtained from fits to data shown in Fig. 12. The mean n0 is n0 = 0.0 ± 0.3 % (from Ref. [5]) (Color figure online)
In a second attempt [5], we measured the DSF, J(Q,y), of a bulk solid which had a large surface area, since at that time solids with large surface to volume ratios ap- peared to have large superfluid fractions. The J(Q,y)observed at three temperatures is shown in Fig. 12. The observed J(Q,y) at T = 500 mK is fitted with the model J(Q,s)= n(s)R(Q,s) with a condensate fraction fixed at n0 = 0, since no BEC is expected at 500 mK. This same function, all parameters unchanged, is plotted with the data at T = 150 mK and T = 65 mK. The function determined at T = 500 mK fits the data at T = 150 mK and T = 65 mK equally well. This shows that J(Q,y)is unchanged, within precision as we cool the solid through Tc 200 mK. Particularly, there is no additional intensity at y ≈ 0atT = 65 mK suggestive of a condensate in 2D or 3D. To put estimates on n0, we fitted the observed J(Q,y)at T = 150 mK and 65 mK ∗ holding n (k) at its 500 mK shape and allowing n0 to be a free fitting parameter, The values of n0 obtained at several Q values are shown in Fig. 13. The mean value in each case is n0 = 0 ± 0.3 %. For a bulk polycrystalline solid we take n0 = 0.3%as the upper limit of n0.
7 Future Opportunities
Superfluidity of liquid 4He confined in porous media has been extensively investi- gated [42]. The porous media range from highly porous aerogels [42, 67], to Vy- cor [40, 41, 68] which has 70Ð80 Å diameter pores, to smaller pore diameter media such as gelsils [63] (e.g. 25 Å diameter) and FSM [62] (e.g. 28 Å diameter). Confine- ment introduces finite size, surfaces and disorder which lower the superfluid transi- tion temperature Tc in the porous media below the bulk value, Tλ. The smaller the pore diameter, the lower is Tc. In contrast, the onset temperature of BEC (denoted TBEC) does not appear [27, 43, 65, 69] to be lowered as much by confinement as Tc. There appears to be a temperature range, Tc Fig. 14 Phase diagram of liquid 4He in MCM-41 (47 Å), gelsil (25 Å), and FSM (28 Å). The superfluid (SF) phase in FSM and in gelsil is at the lower left, the blue region.Thegreen line is the Tc of the SF phase in FSM observed by Taniguchi et al. [62]andred line the Tc in gelsil observed by Yamamoto et al. [63]. The yellow region shows the Localized Bose-Einstein condensation (LBEC) region in gelsil. In the LBEC region, well defined phonon-roton modes are observed [43, 64, 65], up to but not beyond the pressures and temperatures indicated by the blue dashed line in gelsil and MCM-41 which is taken as the upper limit of the LBEC region. The red solid line is the freezing onset of 4He in gelsil measured by Shirahama et al. [66]. The black solid and dashed lines show the freezing onset of 4He in MCM-41 observed [65] from the static structure factor, S(Q).Thedashed red lines show the phase boundaries in bulk 4He (Color figure online) “localized” BEC, localized by disorder. The BEC exists in patches or islands of BEC separated by a sea of normal liquid. As a result the phase in each island is indepen- dent and there is no continuous, connected phase across the sample and as a result no observable superflow across the sample when there is only localized BEC. In Fig. 14 we show the phase diagram of helium in 25 Å pore diameter gelsil, 28 Å FSM and 47 Å MCM-41 with a “localized” BEC region lying between the superfluid and fully normal liquid phases. The existence of BEC can be inferred from the existence of well-defined phonon- roton modes in the confined liquid [27, 43, 69]. The temperature at which well- defined modes at higher wave vector disappear is identified with TBEC. In bulk liq- uid 4He, well defined phonon-roton (PR) modes are observed only in the superfluid phase [70, 71] where there is BEC. In bulk liquid where there is no disorder, BEC and superflow coincide. There are sound theoretical reasons for associating well defined PR modes in the roton region with BEC. The PR modes clearly disappear above a well-defined temperature in porous media. The existence of BEC has also been inferred from the specific heat [72] but the specific heat at higher temperature is con- trolled by the lower energy layer modes [43]. The existence of layer modes may not require the existence of BEC. There is an opportunity to observe BEC directly in liquid 4He in porous media directly using the methods set out above. The condensate fraction in porous media is expected to be somewhat less but comparable to the bulk value at an equivalent pressure. Measurements of n0(T ) and TBEC at SVP in Vycor, gelsil and MCM-41 to verify values of TBEC inferred from the temperature at which P-R modes disappear JLowTempPhys would be most interesting. Measurements as a function of pressure to verify the TBEC (data points + dashed line in Fig. 14) inferred from temperature at which P-R modes disappear are especially interesting. 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