Seminarium KMMF "Teoria Dwoistości"

Around 1924, Albert Einstein predicted the phenomenon of Bose-Einstein condensation - the abrupt accumulation of atoms in a single orbital resulting from their indistinguishability. Erwin Schrodinger indicated flaws in the derivation of this effect. Although the average number of atoms forming a condensate was correctly predicted by Einstein, the fluctuations turned out to be unphysically large. This led to a debate over decades about the fluctuations of the condensed atom number and their dependence on details of statistical descriptions.In my talk, I will discuss theoretical results in the context of the first-ever measurement of the fluctuations of the condensed atom number achieved by our collaborators from the Aarhus group led by Prof. Jan Arlt in 2016. I will start by showing the basic statistical descriptions: the micro-canonical, canonical and grand canonical ensembles and sketching the derivation of the Bose-Einstein condensation.I will argue that the most relevant description of the current experiments is within the microcanonical ensemble - the ensemble which is the easiest conceptually but also the most difficult to use in practice. This ensemble applied to a one-dimensional case leads to the famous partition problem solved by Ramanujan and Hardy. In three dimensions, it is handled using the so-called fourth statistical ensemble or involved numerics. I will show our techniques, and discuss their limitations. The talk will conclude with a list of the most important recent observations and the unresolved problems in the field of ultracold atoms.