Emergent Nonlinear Phenomena in Bose-Einstein Condensates Theory and
This book, written by experts in the fields of atomic physics and nonlinear science, consists of reviews of the current state of the art at the interface of these fields, as is exemplified by the modern theme of Bose-Einstein condensates. Topics covered i
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.1 Introduction The phenomenon of Bose–Einstein condensation, initially predicted by Bose [1] and Einstein [2, 3] in 1924, refers to systems of particles obeying the Bose statistics. In particular, when a gas of bosonic particles is cooled below a critical transition temperature Tc , the particles merge into the Bose–Einstein condensate (BEC), in which a macroscopic number of particles (typically 103 to 106 ) share the same quantum state. Bose–Einstein condensation is in fact a quantum phase transition, which is connected to the manifestation of fundamental physical phenomena, such as superfluidity in liquid helium and superconductivity in metals (see, e.g., [4] for a relevant discussion and references). Dilute weakly-interacting BECs were first realized experimentally in 1995 in atomic gases, and specifically in vapors of rubidium [5] and sodium [6]. In the same year, first signatures of Bose–Einstein condensation in vapors of lithium were also reported [7] and were later more systematically confirmed [8]. The significance and importance of the emergence of BECs has been recognized through the 2001 Nobel prize in Physics [9, 10]. During the last years there has been an explosion of interest in the physics of BECs. Today, over fifty experimental groups around the world can routinely produce BECs, while an enormous amount of theoretical work has ensued. From a theoretical standpoint, and for a wide range of experimentally relevant conditions, the dynamics of a BEC can be described by means of an effective mean-field theory. This approach is much simpler than treating the full many-body Schr¨ odinger equation and can describe quite accurately the static and dynamical properties of BECs. The relevant model is a classical nonlinear evolution equation, the so-called Gross–Pitaevskii (GP) equation [11, 12]. In fact, this is a variant of the famous nonlinear Schr¨ odinger (NLS) equation [13], which is a universal model describing the evolution of complex field envelopes in nonlinear dispersive media. The NLS is widely relevant to other areas of applications ranging from optics to fluid dynamics and plasma physics [14], while it is also interesting from a mathematical viewpoint [13–15]. In the case
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of BECs, the nonlinearity in the GP model is introduced by the interatomic interactions, accounted for through an effective mean-field. Importantly, the study of the GP equation allows the prediction and description of important, and experimentally relevant, nonlinear effects and nonlinear states (such as solitons and vortices), which constitute the subject of this book. This chapter is devoted to the mean-field description of BECs, the GP model and its properties. In particular, we present the derivation of the GP equation (see also Chap. 18) and some of its basic features. We discuss the different cases of repulsive and attractive interatomic interactions and how to control them via Feshbach resonances. We describe the external potentials that can be used to confine BECs, as well as how their form leads to specific ty
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