Basic Mean-Field Theory for Bose-Einstein Condensates

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 T _c , 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.