Superfluidity is the phenomenon of matter flow-referred to as a superflowwithout any resistance from the walls or other obstacles. It is common to refer to superfluidity as a macroscopic quantum phenomenon. The word “quantum” here is only conditionally correct. Furthermore, it is somewhat confusing since it creates an impression that superfluidity cannot exist within the domain of classical physics. It is unquestionably true that the system of classical particles cannot support a superflow, and it is from that classical-particle perspective that superfluidity is associated with the quantum world. However, it turns out that for certain complex-valued classical fields, the superfluid phenomenon is absolutely natural. Hence, superfluidity looks quantum only from either the classical-particle or quantum-field point of view, while being classical from the complementary perspective. In general, this is reflective of the dual nature of quantum fields, which under certain conditions can mimic the behavior of purely classical particles, while under other conditions can also behave as classical fields.
Superfluidity is the phenomenon of matter flow-referred to as a superflowwithout any resistance from the walls or other obstacles. It is common to refer to superfluidity as a macroscopic quantum phenomenon. The word “quantum” here is only conditionally correct. Furthermore, it is somewhat confusing since it creates an impression that superfluidity cannot exist within the domain of classical physics. It is unquestionably true that the system of classical particles cannot support a superflow, and it is from that classical-particle perspective that superfluidity is associated with the quantum world. However, it turns out that for certain complex-valued classical fields, the superfluid phenomenon is absolutely natural. Hence, superfluidity looks quantum only from either the classical-particle or quantum-field point of view, while being classical from the complementary perspective. In general, this is reflective of the dual nature of quantum fields, which under certain conditions can mimic the behavior of purely classical particles, while under other conditions can also behave as classical fields.
What is a superfluid? The word “superfluid” was coined to describe a qualitatively different state of a fluid that can occur at low temperatures, in which the resistance to flow is identically zero, so that flow round a closed path lasts for ever – a persistent current. Superfluidity can occur either for uncharged particles such as helium atoms or for charged particles such as the electrons in a metal. In the latter case the flow constitutes an electric current and we have a superconductor. Since an electric current is accompanied by a magnetic field, it is much easier to demonstrate the presence of a persistent current in a superconductor than in a neutral superfluid. In this chapter I shall describe the properties of superfluids, starting with the simplest and working up to more complicated examples. But in this introductory section I shall depart from the historical order even further by turning to the last page of the detective story so as to catch a glimpse of the conclusion from almost a century of experimental and theoretical research; we shall then have an idea of where we are heading. The conclusion is that superfluidity is more than just flow without resistance. It is an example of a transition to a more ordered state as the temperature is lowered, like the transition to ferromagnetism. At such a transition new macroscopically measurable quantities appear: in the case of a ferromagnet, the spontaneous magnetization. Such new measurable quantities are known as the order parameter of the low-temperature phase. The new quantities in the case of a superfluid are more subtle and more surprising: the amplitude and phase of the de Broglie wave associated with the motion of the superfluid particles. A superfluid can therefore exhibit quantummechanical effects on a macroscopic scale.
Up to this point in our discussion of the superfluidity phenomenon, we have deliberately not mentioned quantum mechanics: this is in contrast with the conventional approach that usually starts with the notion of the condensate wavefunction. In previous chapters, we have shown that classical systems, whether they are dynamical or purely statistical, can be in the superfluid state. Thus, an important connection to follow in the quantum case is the development of the classical-field component that then undergoes the superfluid transition. This can be achieved naturally within the coherent states approach advocated by Langer [1]. As we will see shortly, this connection is rather straightforward and, surprisingly, brings about the counterintuitive (from the classical-particle perspective) question: “Do we ever have an insulating quantum solid state in bosonic systems?” In Chapter 9, we answer this question positively.
Actually, the fundamental feature of Bose condensation is the macroscopic occupation of a single quantum state, whatever that state may be. As long as such a macroscopically-occupied state exists, the system will exhibit superfluid behavior. Superfluid flow thus appears as a "collective" phenomenon, in which the particles move together in order to preserve the macroscopic occupation of a single state. The equilibrium is metastable, because obviously there exists a state of lower energy, that of the fluid at rest; the essential point is that no transitions to the latter state will take place. Metastable superfluid flow is only possible because it involves condensate motion. The superfluid velocity is thus proportional to the gradient of the phase of the condensate wave-function. The superfluid motion is thus a uniform translation, in contrast with ordinary viscous flow, for which the fluid velocity varies from the walls to the center of the tube.
As we all know, single microscopic particles such as electrons or atoms do not behave according to the familiar classical (Newtonian) physics which describes the everyday world, but rather according to the prescriptions of quantum mechanics, the outcome is often bizarre and counterintuitive. When a macroscopic number of such particles (say ∼ 10) is assembled under conditions where the effects of quantum mechanics are important, the results can be even more spectacular; they include macroscopic phenomena such as superfluidity and superconductivity, which are completely impossible to explain in classical terms. Most, though not all of these exotic phenomena, are a result of a phenomenon which I shall refer to generically as “quantum condensation,” in which a macroscopic number of particles or pairs of particles—a nonzero fraction of all particles present—occupy a single quantum state; this phenomenon is traditionally referred to as Bose-Einstein condensation (BEC) when it occurs in a system of bosons such as liquid He, and as Cooper pairing when it occurs, for pairs of particles, in a system of fermions such as the electrons in a superconducting metal. As a result of quantum condensation, the characteristically quantum-mechanical behavior which we are used to seeing at the level of single electrons is as it were amplified by the cooperation of condensed particles and can manifest itself at the macroscopic level. The present essay is intended as a general introduction to the subject of quantum condensation and its consequences, as manifested in a variety of physical systems. I will try to review the fundamental physical ideas, usually without de-
Quantum fluids are fluids, indeed. We have seen in the previous chapter what it means from a classical physics point of view. But quantum fluids are by definition ruled by quantum physics, as well. The goal of this chapter is to introduce some key concepts of quantum physics necessary to understand what makes a fluid a quantum one or rather what quantum property makes an ensemble of particles a quantum fluid. We are here precisely at the edge of our basic intuition of quantum physics. Indeed, one usually associates the notion of quantum object to a microscopic nature and it may thus a priori seem paradoxical to link the word quantum to a macroscopic object, such as the notion of fluid. Thinking it over a bit deeper, quantum physics is certainly present at all scales and we are confronted with quantum mechanics through the many electronic devices which have invaded our everyday life. And it actually turns out that, as we will see later on, most of these quantum devices are directly connected to the notion of quantum fluids, the case of electron fluids being certainly the most common one in such devices. Quantum physics has for sure deeply modified our everyday life. But it has also changed our perception of the world, and as we shall see in this chapter, it is precisely this new vision of the world which allows us to understand the concept of quantum fluids underlying so many physical systems.
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.