Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium. Introduction Liquid helium ( 4 He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena. Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4 He atoms, there is no limitation on the number of particles that can occupy a single quantum state.
Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium. Introduction Liquid helium ( 4 He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena. Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4 He atoms, there is no limitation on the number of particles that can occupy a single quantum state.
As was shown by Einstein in 1924, Bose-Einstein statistics gives rise to a phase transition in which a macroscopic part of a gaseous state condenses into a phase with ground state properties, offering a unique view on the behavior of matter at zero temperature. From the very start the nature of Bose-Einstein condensation (BEC) has provoked statements concerning its relation with respect to superconductivity in electron gases (Einstein 1925) and the superfluidity of liquid helium (London 1938), but this connection is not easily grasped experimentally. Remarkably, although the concept of BEC appears routinely in very different contexts, ranging from excitons in semiconductors to nuclear matter, it had never been observed in its pure form, i.e., in a gaseous phase, until in 1995 BEC in ultracold Rb vapor was reported by Cornell and Wieman. Bose-Einstein condensation is probably the most exciting phenomenon currently under experimental investigation in ultracold atomic gases. In the tutorial the physics will be addressed which is at the roots of the current interest of BEC in atomic gases.
The basic assumption of this paper is that in the presence of Bose-Einstein condensation (BEC) in liquid 4He, the wave functions of occupied many particle states are the superposition of two components; a phase coherent component, proportional to the ground state wave function and a phase incoherent component. It is shown that this assumption satisfies necessary conditions imposed by the presence of BEC and that the wave functions of the ideal Bose gas and wave functions of the Bijl-Feynman type are of this form. It is shown that this single assumption provides simple microscopic explanations of essentially all the exotic properties of helium II; why BEC implies two fluid behavior, how the superfluid and condensate fractions are linked, how Landau theory is linked to the presence of BEC, why the superfluid exhibits flow without viscosity and macroscopic quantum effects while the normal fluid does not, how the anomalous expansion and loss of spatial order, observed in helium II as it is cooled, is linked to BEC, and how the presence of sharp peaks observed in the dynamic structure factor are linked with BEC. The theory provides predictions that are in quantitative agreement with a wide range of presently unexplained experimental data on helium II.
Motivated by recent experiments on alkali gases in atom traps, a largely pedagogical account is given of the implications of the existence of a single-particle Bose-Einstein condensate for the phenomenon of superfluidity. The first conclusion is that, for alkalis in traps at the lowest temperatures, Bose-Einstein condensation coexists with superfluidity. Both experimental evidence and basic microscopic theory are reviewed in this context, with superfluidity in finite systems, quantized vortices and the threshold for breakdown of superfluidity all being referred to. The contrast between liquid 4He below the λ point and the alkali atom gases is then emphasized. Both exhibit superfluidity, but the manifestations of a Bose-Einstein condensate are quite different. For the atomic vapours, near to 100% of the atoms are condensed at the lowest temperatures, whereas from (i) neutron scattering and (ii) computer simulations in liquid 4He present evidence is that 7% is the condensate fraction. This leads finally into a brief discussion of superfluidity without a condensate, with specific reference to the two-dimensional Bose Coulomb gas. The main conclusion is that Bose-Einstein condensation and superfluidity are distinct consequences of deeper topological properties of the many-body wavefunction.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
Surprisingly, it was 30 years after the first liquefaction of (4)He in 1908 that the discovery that liquid (4)He is not just a 'cold' liquid was made. Below T = 2.18 K, it is a 'quantum' liquid which exhibits spectacular macroscopic quantum behaviour that can be seen with the naked eye. Since the observation of superfluidity in liquid (4)He is one of the greatest discoveries in modern physics, we present a day-to-day chronology of the tangled events which preceded the seminal discovery of zero viscosity in 1938 by Kapitza in Moscow and by Allen and Misener in Cambridge. On the theory side, London argued in 1938 that the microscopic basis for this new superfluid phase was the forgotten phenomenon of Bose-Einstein condensation (BEC) first suggested by Einstein in 1925. In 1941, Landau developed a very successful theory of superfluid (4)He, but it was not anchored in a microscopic theory of interacting atoms. It took another 20 years for theorists to unify the two seemingly different theories of Landau and London. Experiments on trapped superfluid atomic gases since 1995 have shone new light on superfluid (4)He. In the mid-1930s, London had emphasized that superconductivity in metals and superfluidity in liquid (4)He were similar. Experiments on trapped two-component Fermi gases in the last five years have shown that a Bose condensate is indeed the basis of both of these superfluid phases. This confirms the now famous Bardeen-Cooper-Schrieffer-BEC crossover scenario developed for superfluidity by Leggett and Nozières in the early 1980s but largely ignored until a few years ago. The study of superfluid (4)He will increasingly overlap with strongly interacting dilute quantum gases, perhaps opening up a new era of research on this most amazing liquid.
Bose–Einstein condensates in dilute atomic gases, which were first realized experimentally in 1995 for rubidium, sodium, and lithium, provide unique opportunities for exploring quantum phenomena on a macroscopic scale. These systems differ from ordinary gases, liquids, and solids in a number of respects, as we shall now illustrate by giving typical values of some physical quantities. The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 10 13 –10 15 cm −3 . By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 10 19 cm −3 . In liquids and solids the density of atoms is of order 1022 cm −3 , while the density of nucleons in atomic nuclei is about 10 38 cm −3 . To observe quantum phenomena in such low-density systems, the temperature must be of order 10 −5 K or less. This may be contrasted with the temperatures at which quantum phenomena occur in solids and liquids. In solids, quantum effects become strong for electrons in metals below the Fermi temperature, which is typically 10 4 –10 5 K, and for phonons below the De bye temperature, which is typically of order 10 2 K. For the helium liquids, the temperatures required for observing quantum phenomena are of order 1 K. Due to the much higher particle density in atomic nuclei, the corresponding degeneracy temperature is about 10 11 K. The path that led in 1995 to the first realization of Bose–Einstein condensation in dilute gases exploited the powerful methods developed over the past quarter of a century for cooling alkali metal atoms by using lasers.