Introduction

Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally.

The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.

Introduction

Spintronics, the study of the interplay between the electrical transport and magnetic properties of magnetically ordered solids, has made steady progress over the past few decades. Spintronics involves both phenomena such as giant magnetoresistance, in which transport properties are influenced by magnetic order configurations, and phenomena such as spin-transfer torques in which transport currents can be used to modify magnetic configurations. Pure spin currents, which do not involve charge flow, are routinely detected via the spin-transfer torques they exert on magnetic condensates and the electrical signals they give rise to when spins accumulate near sample boundaries or at electrodes. There are hopes that spin currents have advantages over charge currents that can be exploited to enable faster or lower-power electronic devices. In this chapter, we discuss the notion of spinsuperfluidity in thin film magnetic systems, either ferromagnetic or antiferromagnetic and either metallic or insulating, that have approximate easy-plane magnetic order [1, 2, 3, 4, 5, 6, 7]. In spintronics, spin-superfluidity refers to the capacity for spin currents to be carried without dissipation by a metastable configuration of a magnetic condensate rather than by an electron or magnon quasiparticle current.

Our chapter is organized as follows. In Section 27.2, we introduce the concept of spin superfluidity using the common language of magnetism researchers by applying Landau-Lifshitz equations to easy-plane magnets.

Introduction

Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems.

Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles.

It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).

Introduction

The superfluid transition in liquid 3He can be regarded as a Bose-Einstein condensation of the fermionic Cooper pairs (or a variation thereof) resulting in a dense condensate, at the other end of the scale from the tenuous cold-gas equivalents. While the cold gases provide interesting laboratories for looking at problems of wider relevance, their limited size and densities can be a disadvantage. The superfluid 3He condensate, while less straightforward to achieve, has a number of beneficial properties which make it ideal for many experiments.

The analogies between superfluid dynamics and the behaviour of pure Euler liquids have long been apparent. However, the superfluid case is complicated, at medium temperatures (0 < T < Tc), by the presence of ‘normal’ fluid, i.e., the residual gas of those unpaired particles not contributing to the condensate. Superfluid 3He is simpler in this regard than superfluid 4He in that the condensate (or superfluid fraction) can be identified with the Cooper pairs and the normal fluid fraction with the remaining unpaired 3He atoms. In the case of our experiments, we avoid this latter complication by working at the zero-temperature limit, where the normal fraction is negligible. In this region, the superfluid flow does indeed correspond to that of an Euler fluid and is irrotational. However, we can simulate rotation by introducing vortices, or line defects, along the cores of which the condensation is suppressed. Around these vortices, circulation is allowed, since the local liquid flow remains irrotational.

Introduction

Although classical General Relativity (GR) has had enormously interesting experimental predictions, going from the deflection of light to the expansion of the universe, we are still missing an efficient way to consume the marriage between GR and quantum mechanics.

We can easily identify the two main reasons that make this desired marriage extremely difficult. Quantum mechanics, as well as Quantum Field Theory (QFT), is based on notions such as particle and interaction that are defined relative to an absolute space-time. They can be extended to curved backgrounds, but when the geometry starts to affect concepts such as global time, paradoxes unavoidably appear. Moreover, the main message of GR, namely that geometry itself is a dynamical notion, creates its own set of conceptual problems. Should we think of geometry as on an equal footing as we think of other fields such as Yang Mills that we quantize on a privileged absolute Minkowski space-time? And, if that is the right way to proceed, how should we deal with the problems of renormalizability and unitarity that this naive approach immediately creates?

As it is well known the root of these perturbative problems lies in the dimensions of the coupling (the Newton constant) defining the strength of the gravitational interaction. In Wilsonian terminology, the gravitational interaction is defined by an irrelevant operator that flows nicely into the infrared (IR) but goes out of control in the ultraviolet (UV) whenever we go beyond the Planck scale, a length scale that acquires the Wilsonian meaning of a natural cutoff.

Introduction

Bose-Einstein condensation is a phenomenon in which a macroscopic number of particles occupy the same single-particle state as a consequence of symmetrization of a many-body wave function and thereby quantum effects are amplified to the macroscopic level. The amplified single-particle state serves as a complex order parameter whose phase behaves as an emergent thermodynamic quantity, with its temporal and spatial variations giving the chemical potential and the superfluid velocity, respectively. If the interparticle interaction is repulsive, a Bose-Einstein condensate (BEC) acquires stability against excitations out of the condensate because they would cost the Fock exchange energy. This rigidity of the order parameter or the condensate wave function endows a BEC with several remarkable transport properties which are collectively known as superfluidity. In particular, the superfluid fraction can be 100% at zero temperature even though the condensate fraction is depleted due to interparticle interactions. Interparticle interactions also make excited particles phase-locked to the condensate, leading to Bogoliubov quasiparticles, which are the manifestation of phase-coherent particle– hole excitations.

When constituent particles have spin degrees of freedom, a BEC features magnetism and spin nematicity which conspire with superfluidity to produce a rich phase diagram. In the absence of an external magnetic field, the mean-field groundstate phase diagram includes two [1, 2], five [3, 4, 5], and eleven phases [6, 7] for spin-1, 2, and 3 BECs, respectively. BECs with spin degrees of freedom are called spinor condensates. Moreover, the magnetic dipole–dipole interaction (MDDI) between atoms lends distinct twists to spinor condensates because it is long-ranged and anisotropic in sharp contrast with other interactions, which are short-ranged and isotropic. In this chapter, we will present a brief overview on the spinordipolar aspects of BEC. An early experimental work of spinor BECs is reviewed in Ref. [8]. Later developments with an emphasis on theoretical aspects are described in Ref. [9]. Many-body physics of polarized dipolar gases is reviewed in Ref. [10].

Introduction

Over the past years, ultracold atoms in optical lattices have emerged as versatile new systems to explore the physics of quantum many-body systems. On the one hand, they can be helpful in gaining a better understanding of known phases of matter and their dynamical behavior; on the other hand, they allow one to realize completely novel quantum systems that have not been studied before in nature [1, 2, 3]. Commonly, the approach of exploring quantum many-body systems in such a way is referred to as “quantum simulations.” Examples of some of the first strongly interacting many-body phases that have been realized both in lattices and in the continuum include the quantum phase transition from a superfluid to a Mott insulator [4, 5, 6], fermionic Mott insulators [7, 8], the achievement of a Tonks- Girardeau gas [9, 10], and the realization of the Bose-Einsten condensate (BEC)– Bardeen-Cooper-Schrieffer (BCS) crossover (see also Chapter 12) in Fermi gas mixture [11] using Feshbach resonances [12].

In almost all of these experiments, detection was limited to time-of-flight imaging or more refined derived techniques that mainly characterized the momentum distribution of the quantum gas [2]. For several years, researchers in the field have therefore aspired to employ in situ single-particle detection methods for the analysis of ultracold quantum gases. Only recently has it become possible to implement such imaging techniques, marking a milestone for the characterization and manipulation of ultracold quantum gases [13, 14, 15, 16, 17]. In our discussion, we will focus on one of the most successful of these techniques based on high-resolution fluorescence imaging. Despite being a rather new technique, such quantum gas microscopy has already proven to be an enabling technology for probing and manipulating quantum many-body systems. For the first time, controllable and strongly interacting many-body systems, as realized with ultracold atoms, could be observed on a local scale [17, 16].

Introduction

Strange but striking phenomena, which are accessed by advanced experimental techniques, become a fuel to stimulate both experimental and theoretical research. Experimentalists concoct new tools for sophisticated measurements, and theorists establish models in order to explain the surprising observation, ultimately expanding our knowledge boundary. A classic example of the seed to the knowledge expansion is the feature of abnormally high heat conductivity in liquid helium reported by Kapitza and Allen's group, who used cryogenic liquefaction techniques in 1938 [1, 2]. It is a precursor to a “resistance-less flow” a new phase of matter, coined as superfluidity in the He-II phase. Immediately after this discovery, London conceived a brilliant insight between superfluidity and Bose-Einstein condensation (BEC) of noninteracting ideal Bose gases [3], which has led to establish the concept of coherence as off-diagonal long-range order emerging in the exotic states of matter. Since then, it is one of the core themes in equilibrium Bose systems to elucidate the intimate link of superfluidity and BEC in natural and artificial materials, where dimensionality and interaction play a crucial role in determining the system phase.

Let us consider the noninteracting ideal Bose gases whose particle number N is fixed in a three-dimensional box with a volume V. According to the Bose-Einstein statistics, the average occupation number Ni in the state i with energy is given by with the chemical potential and a temperature parameter (Boltzmann constant kB and temperature T). For the positive real number of is restricted to be smaller than, and the ground-state particle number N0 diverges as approaches the lowest energy. Its thermodynamic phase transition refers to BEC, in which the macroscopic occupation in the ground state is represented by the classical field operator, where is the particle density and is the phase.

Introduction

Bose-Einstein condensation (BEC) in atomic gases was first observed in 1995, and has changed the face of atomic physics. It is for atoms or matter waves what the laser is for photons: a macroscopically occupied quantum state. It was regarded as an elusive goal until it was discovered in 1995. Although BEC was immediately viewed as a major accomplishment, its impact has far exceeded expectations. Now, twenty years on, there is no question that the field remains exciting.

It is impossible to give a review over the developments during those twenty years. Instead, in this chapter, I want to illustrate how often predictions or expectations changed in the pursuit of Bose-Einstein condensation. There were many surprises, and some advances and breakthroughs happened although they were predicted to be impossible. A lesson we can learn from this is that we should always carefully read the fine print when something is proven or assumed impossible, and try to figure out if the assumptions can be circumvented!

Since the early history of Bose-Einstein condensation provides several examples for such “impossibility theorems,” I digress into those earlier developments in the first part of this chapter.

Early Theoretical Questions

Validity of the Prediction of Bose-Einstein Condensation

Einstein predicted Bose-Einstein condensation in 1924 in the second of two papers where he generalized Bose's treatment from photons to massive particles [1]. Using statistical arguments introduced by Bose, he found that below a critical temperature, bosonic particles condense in the lowest energy state of the system. In contrast, at very low temperature photons can simply disappear.

However, for about a decade, this prediction was not taken at face value. Einstein himself wrote to Ehrenfest, “From a certain temperature on, the molecules ‘condense’ without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there some truth in it?” [2].

In 1927, George E. Uhlenbeck concluded that the predicted BEC phase transition was an artifact of the replacement of the summation over states by an integral and wrote that no “splitting into two phases” would occur [3].

Introduction

The achievement of atomic Bose-Einstein condensation (BEC) in 1995 initiated a new adventure to explore quantum many-body phenomena in the gas phase, a journey that has yielded much excitement and is still progressing fast. Because of their diluteness, ultracold atomic gases can be precisely modeled and characterized with high accuracy. In addition, new opportunities have emerged in the past decade to prepare cold atoms in the strong coupling regime. By loading the samples into optical lattices [1] or tuning them near a Feshbach resonance [2], for instance, cold atoms offer a unique platform to investigate strongly correlated quantum phenomena. Examples include the superfluid-Mott insulator transition [3] (see also Chapter 13), BEC-BCS (Bardeen-Cooper-Schrieffer) crossover [4, 5, 6, 7] (see also related overview in Chapter 12), Efimov trimer states [8], and Berezinskii- Kosterlitz-Thouless transition in two dimensions [9] (see also Chapter 10). One key motivation in these explorations is to uncover new, universal laws that underpin the behavior of strong-coupled few- and many-body quantum systems.

In this chapter, we discuss “scale invariance” as a generic attribute in various cold atom systems, including all of the aforementioned examples. While scale invariance is conventionally discussed in the context of critical phenomena, the cold atom systems that interest us belong to a new class, which we call “intrinsic scale invariance.”

In the following, we first overview the main features of different types of scale invariance and discuss related examples in cold atom systems.

Introduction

Although it can seem paradoxical, the quantum magnetic states found in insulators offer an interesting route to study Bose-Einstein condensation (BEC) and more generally phenomena related to itinerant, interacting bosonic systems.

In many materials, the repulsion between the electrons causes the ground state to be a Mott insulator with one electron localized per site. In such a case, the charge degrees of freedom are completely locked by large charge gaps (usually of the order of tens of electron volts). However, one can study the fascinating physics of the spin degrees of freedom. As was shown a long time ago, the localized spins are coupled by a so-called superexchange which may be ferro- or antiferromagnetic. The properties of the exchange coupling between the spins depend mainly on the underlying atomic structure of the material. The resulting physics is extremely rich and such materials can have a host of phases ranging from ordered magnetic phases to so-called spin-liquid phases in which spin–spin correlations decay extremely fast with distance. The field of research concerning these materials, dubbed quantum magnetism, is thus an extremely active field in its own right [1].

Additionally, one can exploit the formal mapping that exists between spin operators and bosonic ones, as will be described in the next section, to connect the magnetic materials to itinerant bosonic systems with interactions between the bosons. This opens the way to use such quantum spin systems as quantum simulators. A quantum simulator means that the experimental system can be employed in order to “solve” in a controlled way a problem/Hamiltonian known from a different less controlled physical realization, e.g., inspired by fundamental many-body physics, quantum information or computational applications.

At the time of the first workshop in this series in 1993, the only experimentally realized Bose condensate (at least in the simple sense conjectured by Einstein) was liquid 4He. In the intervening twenty-plus years, much has happened in the world of Bose-Einstein condensation (BEC). Probably the most exciting development has been the attainment of condensation in ultracold bosonic atomic gases such as 87Rb and 23Na in 1995, followed a few years later by the achievement of degeneracy and eventually Bardeen-Cooper-Schrieffer (BCS) pairing in their fermionic counterparts, and the experimental realization of the theoretically long-anticipated “BEC-BCS crossover” by using the magnetic field degree of freedom to tune the system through a Feshbach resonance. One particularly fascinating aspect of the latter has been the realization of a “unitary gas” at the resonance itself – a system which prima facie has no characteristic length scale other than the interparticle separation, and is therefore a major challenge to theorists. Other systems in which BEC has been realized, sometimes transiently, include exciton-polariton complexes in semiconducting microcavities and, at least in a formal sense, the magnons in a magnetic insulator, as well as ultracold gases with a nontrivial and sometimes large “spin” degree of freedom.

As compared with our “traditional” Bose condensate, liquid 4He, these new systems typically have many more (and more rapidly adjustable) control parameters, and have therefore permitted qualitatively new types of experiment. One particularly fascinating development has been the use of optical techniques to generate “synthetic gauge fields” and thus mimic some of the topologically nontrivial systems which have recently been of such intense interest in a condensed-matter setting. At the same time, there remain long-standing issues from helium physics, such as the nature and consequences of “spontaneously broken U(1) symmetry,” the “Kibble-Zurek” mechanism, and more generally the relaxation of strongly nonequilibrium states to equilibrium; in some cases, the new systems have been used to address these more quantitatively than was possible with 4He. The chapters in this volume address all of these questions and more, and should be of intense interest to both the experimental and the theoretical sides of the BEC community.

Introduction

Magnons are quasiparticles corresponding to quantised spin waves that describe the collective motion of spins [1]. In recent years, it has become possible to realise a BEC of magnons in two remarkably different systems: in superfluid phases of an isotope of helium – Helium-3 at ultralow temperatures [2, 3] and in ferrimagnetic insulators [4, 5, 6, 7]. In the normal, noncondensed state, these magnetic materials exhibit a magnetically ordered state with a gas of magnons whose phases are not correlated. However, once condensation occurs, spins develop phase coherence resulting in a common global frequency and phase of precession. At room temperature, ferrimagnetic insulators, such as yttrium-iron garnet (YIG) films, together with a combination of an in-plane magnetic field and microwave radiation, are required for the magnon condensation [4, 5, 6, 7]. Therefore, in analogy with exciton– polariton condensates, this provides yet another example of a BEC of quasiparticle excitations in a solid-state system.

Experimental Realisation of a Bose-Einstein Condensateof Magnons in an Epitaxial YIG-film

A room-temperature Bose-Einstein condensate of magnons was created in an epitaxial YIG-film using an experimental setup shown in Fig. 25.1a, which, in general, is similar to that used in our previous studies [4, 5]. Given that magnons are quasiparticle excitations, their number is not conserved and, therefore, the chemical potential is zero.

Introduction

The field of Bose-Einstein condensation (BEC) has undergone an explosive expansion during the past twenty years. Newcomers to this field are now often introduced to this as a universal phenomenon, which nonetheless exhibits diverse (and sometimes strikingly different) manifestations. Despite such differences, the common underlying theme creates a unique identity across many different energy and length scales.

The study of BEC as a universal phenomenon was highlighted in a focused conference in 1993, leading to the publication of the well-known “green book” [1], which surveyed the breadth of the field of condensate physics at that time. The success of the conference led to a second meeting in 1995, at which Eric Cornell and Carl Wieman announced the achievement of Bose-Einstein condensation of ultracold 87Rb atoms in a harmonic trap. That work began an explosion of new research in the field of cold atoms, which has continued to this day, and this very success inevitably led many of those studying cold atoms to pay less attention to other types of condensates. Wolfgang Ketterle gives a historical overview of this exciting period of time in Chapter 3. Recently, various scientific meetings have worked to re-establish the physical connections across different BEC systems, and in 2013 a workshop was held with the focused goal of improving communications across disciplines. This present book is an outgrowth of that meeting.

The Situation Before the Revolution

Because of the great success of the cold atom BEC and other BEC systems in the past twenty years, it may be hard for young scientists to understand the climate of BEC research in the early 1990s. At that time, there was only one known example of BEC, namely liquid helium-4, and there was a small but vocal minority of scientists who questioned whether BEC was established even in that system.