Counting statistics and fluctuations in Bose–Einstein condensation

Jianhui Wang, Jizhou He, Yong-li Ma

Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.

Vito Barbarani

Christoph Weiß, Martin Wilkens

We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.

Oleś, Bartłomiej

Katrina Smith-Mannschott

This thesis aims to investigate the dynamic properties of ultra-cold Bose-Einstein condensates (BEC) trapped in a deep double well potential. Despite its simplicity, this system shows a rich variety of dynamical behaviors: Josephson oscillations, self-trapping behavior and critical dynamics. In contrast to previous studies, we aim to understand not only the first moment of the evolving occupation probability, but all its moments. This is known in mesoscopic physics as the "counting statistics" problem. Using the quantum dimer, we are the first to address the effects of interatomic interactions on counting statistics. We expect that our predictions will be easily tested in recent experimental realizations of the BEC double well trap.

Hu Huang, Shiqun Li