The past two decades have seen great progress in our understanding of the properties of condensed helium 3. Research on He3 in our lab (see figure 1 for its modern form) began in 1958. At that time the properties of the more abundant isotope, helium 4, were rather well known: A phase transition at about 2 K transforms it from a normal liquid to a superfluid, a liquid that under some conditions will flow without viscous drag. The explanation for this behavior is rooted in concepts from the condensation of an ideal Bose–Einstein gas. But He3 obeys Fermi–Dirac statistics; would it show any likeness to a Fermi gas or liquid? Would it also exhibit a transition to superfluidity?
The past two decades have seen great progress in our understanding of the properties of condensed helium 3. Research on He3 in our lab (see figure 1 for its modern form) began in 1958. At that time the properties of the more abundant isotope, helium 4, were rather well known: A phase transition at about 2 K transforms it from a normal liquid to a superfluid, a liquid that under some conditions will flow without viscous drag. The explanation for this behavior is rooted in concepts from the condensation of an ideal Bose–Einstein gas. But He3 obeys Fermi–Dirac statistics; would it show any likeness to a Fermi gas or liquid? Would it also exhibit a transition to superfluidity?
It is our great good fortune that the only two substances that remain liquid down to the absolute zero obey different statistics ( 3 He: Fermi-Dirac, 4 He: Bose-Einstein) and thus allow us to study the differing effects of quantum statistics on condensed matter. An extraordinarily diverse and luxuriant field of liquid helium work has consequently developed, particularly in the last 25 years, ranging from refrigeration applications to the behaviour of pulsars, and it is surprising to realize that the ancient controversy over whether the Bose–Einstein condensation phenomenon (BEC) is necessary and sufficient for superfluidity is still not settled.
In the same series of experiments which led to the discovery of superconductivity, Kamerlingh Onnes was the first who succeeded in cooling helium below its boiling point of 4.2 K. In 1908, he was cooling this liquefied helium further in an attempt to produce solid helium. But since helium only solidifies at very low temperatures (below 1 K) and high pressures (above 25 bar), Kamerlingh Onnes did not succeed, and he turned his attention to other experiments. It was, however, soon recognized that at 2.2 K (a temperature reached by Kamerlingh Onnes), the liquid helium underwent a transition into a new phase, which was called helium II. After this observation, it took almost 30 years to discover that this new phase is actually characterized by a complete absence of viscosity (a discovery made by Kapitza and Allen in 1937). The phase was therefore called superfluid . In a superfluid, there can be persistent frictionless flows – superflows – very much like supercurrents in superconductors. This helium II is, up to now, the only superfluid which is available for experiments. The most abundant isotope of helium, 4 He, is a spinless boson. Given this fact, the origin of superfluidity was almost immediately attributed to Bose condensation, which was a known phenomenon by that time.
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.
We have seen in Chapter 3 , that as temperature is increased quantum statistics, both Bose-Einstein or Fermi-Dirac, tend to classical. High temperatures blur out quantum effects in a system, which can be safely described in classical terms. In this chapter, we will consider some systems which manifest quantum properties at macroscopic level, but at very low temperatures. We recall that helium is the only substance which remains liquid at low temperature, down to absolute zero (see Chapter 6 ), a phenomenon related to the quantum behavior of its constituent atoms. Liquid helium was instrumental to the discovery that some metals are able to sustain an electrical current for long periods of time without any driving force. This phenomenon is named superconductivity, and is also a macroscopic manifestation of the quantum behavior of the electrons in metals. Later on it was discovered that liquid helium itself shows a similar property, named superfluidity, consisting in its ability to flow through a capillary without viscosity. Hence, a persistent flow of liquid can be maintained without any driving force. While the quantum agents of superconductivity are electrons, which are spin- 1/2 fermions, those of superfluidity are 4 He atoms, which are spin- 0 bosons. There are other systems able to display similar behavior as liquid 3 He, atomic nuclei and nuclear matter. In fact, superfluidity and superconductivity are new phases of matter related to the bosonic character of 4 He atoms or of pairs of fermions (either electrons, 3 He atoms or nucleons), the same as Bose-Einstein condensation in ultracold atomic gases. The temperatures where these new phases manifest range from 165 K for superconductivity down to a few nano-Kelvin for Bose-Einstein condensation.
This book is about a certain class of superfluids, namely the high temperature superconductors. Perhaps the simplest of the superfluids is liquid He4. Since the very early days of quantum mechanics it has been known that the helium atom He4 is a boson; it is built of an even number of particles (two electrons, two neutrons and two protons), so that a wave function ^(ri,F2) must be symmetrical in FI — r? where FI, r-z are the electron's coordinates. This has the result that half the rotational states of the molecule H62 are absent and that the scattering of alpha particles by helium for low energies deviates from the Rutherford scattering law. It also has the remarkable result that this is not so for the light isotope He3, which shows that the number of neutrons in the nucleus affects properties depending otherwise only on the electrons and nuclear charge.
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.