ABSTRACT
U ltracold atomic gases have emerged as an ideal testground for fundamental many-body physics. The “quantum revolution” trig-
gered by the work of John Bell and the subsequent advances in the experimental control of few-body quantum systems has nowadays entered a new phase associated with the rise of quantum information science and the quest for control over macroscopic quantum systems. The exceptional level of control enabled by cold atom systems has placed them at the frontier of modern quantum physics as a promising route to realizing quantum simulators, engineering topological quantum matter, and emulating interacting gauge fields. By allowing the realization of tunable interactions and synthetic spin-orbit couplings, cold atom systems offer the possibility of engineering topological phases that are not supported by solid state systems, including experimental realizations of ideal models. In this chapter we briefly summarize the recent
developments in this field and point out some of the basic physical ideas behind these advances. For a more detailed survey of the field the reader is referred to specialized books and review articles, for example Refs. [53, 184, 290, 291].
9.1 BRIEF HISTORICAL PERSPECTIVE The experimental observation of Bose-Einstein condensation (BEC) in 1995 [19, 65, 118] marked a breakthrough moment in which the atomic, molecular, and optical (AMO) physics has reached the frontiers of condensed matter physics. This significant achievement was made possible by the earlier development of efficient methods to cool and trap atoms using laser light [90, 100, 345]. With the observation of BEC, the study of many-body systems took central stage in AMO physics. Nonetheless, the condensed matter community remained rather reserved, mainly because the BEC phenomenon can be well understood within an effective Ginzburg-Landau picture [177] in which the coherent many-body state of the quantum system is described by a macroscopic wave function Ψ(r, t), a very familiar concept in superconductivity and superfluidity. A complete and quantitative description of the static and time-dependent properties of a BEC can be obtained by solving the so-called Gross-Pitaevskii equation [195, 347], a nonlinear Schro¨dinger-type equation for the macroscopic wave function,
