Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.
Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
This chapter presents an overview of the specific experimental tools developed in the field of ultra-cold gases to achieve and probe superfluidity in vapours of bosonic atoms. First, it describes the main cooling techniques: the magneto-optical trap, which brings an atomic vapour from room temperature down to the sub-mK range, and the evaporative cooling scheme, which bridges the gap to the superfluid regime. It then proceeds to the study of interaction effects. It shows that these play a central role in the understanding of both static and dynamic properties of trapped Bose–Einstein condensates (BECs), despite the dilute character of these gases. As a consequence, gaseous BECs and superfluid helium obey the same laws of hydrodynamics and exhibit similar dynamical properties, although their densities differ by several orders of magnitude. The third section is devoted to the coherence properties of gaseous BECs. It shows that, by contrast with liquid helium, specific features of ultra-cold gases make them suitable for a direct probing of the quantum coherence associated with Bose–Einstein condensation. Finally, the possibility of tailoring trapping potentials and realizing low-dimensional systems is discussed, where one or two directions of motion are frozen.
Bose Einstein Condensate (BEC) is the fifth state of matter in condensed matter physics. It is the most impressive example of quantum behaviour on a macroscopic scale but the most difficult state to observe. When low-density boson gas is cooled to a temperature nearly absolute zero, then BEC is formed. In this condition, the bosons have almost no free energy to move relative to each other. They clump together, come into the lowest quantum state, and behave as a single atom. Then microscopic quantum mechanical phenomena work on a macroscopic scale. This chapter summarizes the history and the developments regarding BEC with a brief discussion of the key scientists in this field. In the next section, the summary of the all chapters is given.
In 1924/25, Satyendranath Bose and Albert Einstein predicted that particles of integer spin (today called bosons) should undergo a statistical phase transition when cooled to temperatures very close to absolute zero (Bose, 1924; Einstein, 1925). This phenomenon, nowadays called Bose-Einstein condensation, has been experimentally observed seventy years later using (nanokelvin temperatures) of bosonic neutral atoms (Anderson et al., 1995; Bradley et al., 1995; Davis et al., 1995c). This spectacular experimental achievement was awarded the Nobel price in 1995 and has triggered a new research direction ultracold gases with more than 200 groups worldwide today. 1 Bose-Einstein condensates (BEC) of atomic represent a fascinating exotic state of matter with properties entirely determined by the laws of mechanics. Although they consisting of several thousands up to millions of atoms, the state can in most cases be described by a single collective wave function with a common phase. This wave function usually has a spatial extent ranging between 1-100 microns, allowing us to observe mechanics essentially by eye using basic magnification optics. The Bose-Einstein condensate can be described as a coherent matter wave in close analogy to the optical field emitted by (or inside) a laser. This analogy has brought about the field of quantum-atom-optics which aims to implement standard elements and experiments known from laser optics using matter waves. As heavy rest mass particles, such as atoms, are very sensitive to gravity or acceleration/rotations, matter-wave interferometers promise orders of magnitude in sensitivity gain compared to their photonic counterparts (Berman, 1996). Over the last decade, experimental tools for the creation and manipulation of matter have reached an impressive degree of sophistication. This allows the construction of tailored Hamiltonians for the system and the quantum simulation of more general physical situations, not connected to atomic physics alone. Optical lattice potentials allow to mimic solid state physics with a high degree of control over essentially all experimental parameters. Particle interactions can be tuned via Feshbach resonances, allowing the study of superconductivity, superfluidity and the formation of
This chapter focuses on integrating the distribution function over all the states of conserved Bosons to obtain a relation between particle density and the chemical potential. It describes the phenomenon of Bose-Einstein condensation in a perfect boson gas. The chapter discusses the experimental observation of Bose-Einstein condensation in perfect gases. There are two routes to obtaining Bose-Einstein condensation in a perfect gas. One is to reduce the mass of the particle, thus reducing critical density to a low value at a reasonable temperature. The other route is to lower the temperature to the point where the critical density is so low that the particles are too far apart to interact with each other.
In 1925 Einstein predicted that at low temperatures particles in a gas could all reside in the same quantum state. This gaseous state, a Bose–Einstein condensate, was produced in the laboratory for the first time in 1995 and investigating such condensates has become one of the most active areas in contemporary physics. The study of Bose–Einstein condensates in dilute gases encompasses a number of different subfields of physics, including atomic, condensed matter, and nuclear physics. The authors of this graduate-level textbook explain this exciting new subject in terms of basic physical principles, without assuming detailed knowledge of any of these subfields. Chapters cover the statistical physics of trapped gases, atomic properties, cooling and trapping atoms, interatomic interactions, structure of trapped condensates, collective modes, rotating condensates, superfluidity, interference phenomena, and trapped Fermi gases. Problem sets are also included in each chapter.