Fluctuations in Ideal and Interacting Bose-Einstein Condensates: From the laser phase transition analogy to squeezed states and Bogoliubov quasiparticles

Abstract

We review the phenomenon of equilibrium fluctuations in the number of condensed atoms in a trap containing N atoms total. We start with a history of the Bose-Einstein distribution, the Einstein-Uhlenbeck debate concerning the rounding of the mean number of condensed atoms near a critical temperature, and a discussion of the relations between statistics of BEC fluctuations in the grand canonical, canonical, and microcanonical ensembles. Next we discuss different approaches capable of providing approximate analytical results and physical insight into the problem of fluctuations. In particular, we describe the master equation (similar to the quantum theory of the laser) and canonical-ensemble quasiparticle approaches which give the most accurate and physically transparent picture of the BEC fluctuations. In the last part we describe condensate fluctuations in the interacting Bose gas. In particular, we show that the canonical-ensemble quasiparticle approach works very well for the interacting gases and yields analytical formulas for the characteristic function and all moments of the condensate fluctuations. In most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit. We clarify a crossover between the ideal and weakly-interacting-gas statistics which is governed by a pair-correlation squeezing mechanism.