In January 1938, the Russian physicist Pyotr Kapitza reported that, at temperatures below 2.2 degrees above absolute zero, liquid helium seemed to flow with no loss of energy due to friction. The term “superfluid” was suggested to describe this new behavior—which was eventually understood as a dramatic illustration of the quantum-mechanical wave nature of matter. The study of such macroscopic quantum effects now lies at the heart of modern physics. The complete analogue of the Landau two-fluid hydrodynamic equations, including various transport coefficients associated with the normal fluid, has recently been derived for the trapped atomic Bose gases. In the next few years, experiments should be able to probe this two-fluid domain in Bose gases, a region first explored with such spectacular success in superfluid helium by Kapitza and Allen and Misener. The complexity of analyzing any liquid at a microscopic level has resulted in many aspects of the theory of superfluid.
In January 1938, the Russian physicist Pyotr Kapitza reported that, at temperatures below 2.2 degrees above absolute zero, liquid helium seemed to flow with no loss of energy due to friction. The term “superfluid” was suggested to describe this new behavior—which was eventually understood as a dramatic illustration of the quantum-mechanical wave nature of matter. The study of such macroscopic quantum effects now lies at the heart of modern physics. The complete analogue of the Landau two-fluid hydrodynamic equations, including various transport coefficients associated with the normal fluid, has recently been derived for the trapped atomic Bose gases. In the next few years, experiments should be able to probe this two-fluid domain in Bose gases, a region first explored with such spectacular success in superfluid helium by Kapitza and Allen and Misener. The complexity of analyzing any liquid at a microscopic level has resulted in many aspects of the theory of superfluid.
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.
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
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.
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.
In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures. This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6). History of two-fluid equations The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.
Water flowing down a pipe experiences viscous drag that causes it to lose energy and slow down. A superfluid, such as liquid helium (3He and 4He) or electrons in a superconductor, can flow without friction when in contact with a wall. This fascinating phenomenon, or superfluidity, was discovered by Kapitza [1], Allen and Misener [2] in 1937. The formation of a superfluid is related to the formation of a Bose--Einstein condensate (BEC). 4He is a boson as its total spin is zero, an integer. Condensation in fermionic 3He liquid occurs through composite bosons of paired atoms in a process similar to electron pairing in a superconductor. To explain the peculiar properties of superfluid helium, Landau considered a superfluid to be comprised of two parts: a ‘pure’ superfluid component with zero entropy and viscosity, and a normal component that behaves like ordinary fluid. An interesting manifestation of this two-fluid model is the existence of two types of oscillations, the ‘first and second sounds’ as named by Landau. The ‘first sound’ is a wave that propagates at constant entropy per particle, just like sounds in ordinary fluids. It propagates with pressure and density oscillations. ‘Second sound’, unlike what its name suggests, is not audible as it propagates at nearly constant pressure (isobaric oscillation). Here, entropy density takes the role of pressure/density in normal sound waves. Historically, the observation of the second sound in superfluid helium, and the measurement of its speed, played a crucial role in understanding the nature of superfluid helium. The speed of the second sound can be used to determine the superfluid fraction, i.e. the ratio of the superfluid density to the total fluid density [3,4]. In the past decades, the field of ultracold atomic gases has developed into a fertile ground for studying superfluid physics, as the inter-particle interactions between two atoms can be tuned using the so-called Fano–Feshbach resonances. Systematic studies on Fermi gases can be preformed in the limit of strong interactions, whose properties are difficult to predict but important to the understanding of strongly correlated quantum systems of many particles, such as high-temperature superconductors, quark–gluon plasmas and neutron stars. Quantitative determination of the superfluid fraction in strongly interacting Fermi gases has been an elusive goal of scientists for many years, as the methods that allow the determination of the superfluid fraction in liquid helium cannot be directly applied to ultracold gases. For example, to excite the second sound in liquid helium, a local heat pulse can be created by passing a current through a heating wire submersed in the liquid. The heat pulse can then be detected at the other end of the liquid using a temperature sensitive sensor as illustrated in Fig. 1(a). Such methods cannot be implemented with ultracold atomic gas ensembles that typically contain only about hundreds of thousand atoms at μK (1 millionth of a kelvin) temperatures. Further adding to the difficulties, the propagation of the second sound in liquid helium usually involves no density variation/oscillation. This means that the second sound could not be detected easily with the absorption imaging technique typically used for probing cold atoms, assuming that the strongly interacting Fermi gases behave very much like strongly interacting bosonic liquid helium (4He). (a) Excitation and detection of the second sound in superfluid helium; (b) local heating through the power modulation of an excitation laser beam; (c) the detection of the second sound in a strongly interacting Fermi gas with direct absorption imaging. Recently, the group of Professor Rudolf Grimm from the University of Innsbruck (Austria), in collaboration with a theoretical team from the Trento University (Italy) led by Professors Sandro Stringari and Lev Pitaevskii, successfully measured the speed of second sound in a resonantly interacting Fermi gas using a simple and elegant method. Their results were published in [5]. To observe the second sound, the authors created a local heat pulse at the center of an elongated trapped gas by creating a short burst of modulation to the power of a light beam intercepting this gas (see Fig. 1b). They then measured the speed of the second sound by observing the movement of a thermal-induced density dip along the elongated gas (Fig. 1c). Taking advantage of the inhomogeneous density distributions of the superfluid atoms in the trap, they ingeniously determined the temperature dependence of the superfluid fraction without needing to vary the temperature of the samples. The observation of the second sound in the strongly interacting Fermi gases opens up a new horizon for studying the superfluid properties with tunable interactions. Knowing the speed of second sound can provide a lot of valuable information, allowing physicists working on many-body theories to falsify their models and theories. It demonstrates the importance of using experimentally measured quantities, such as the equation of state employed here, to calibrate and to supplement theoretical investigations. Significant new advances will follow in the understanding of superfluidity of many strongly interacting particles, where faithful theoretical modeling and calculations are impossible at the microscopic level.