Fluctuation statistics of mesoscopic Bose-Einstein condensate: reconciling the master equation with the partition function to revisit the Uhlenbeck-Einstein dilemma

TL;DR

This paper develops a unified theoretical framework combining master equation and partition function approaches to accurately analyze fluctuation statistics of mesoscopic Bose-Einstein condensates across all temperatures.

Contribution

It introduces a self-consistent grand canonical analysis that aligns with microscopic results and extends to higher moments using stochastic path integrals, improving fluctuation predictions.

Findings

Analytic expressions match numerical results at all temperatures

Accurate variance results below the critical temperature

Hybrid approach achieves perfect agreement with simulations

Abstract

The atom fluctuations statistics of an ideal, mesoscopic, Bose-Einstein condensate is investigated from several different perspectives. By generalizing the grand canonical analysis (applied to the canonical ensemble problem), we obtain a self-consistent equation for the mean condensate particle number that coincides with the microscopic result calculated from the laser master equation approach. For the case of a harmonic trap, we obtain an analytic expression for the condensate particle number that is very accurate at all temperatures, when compared with numerical canonical ensemble results. Applying a similar generalized grand canonical treatment to the variance, we obtain an accurate result only below the critical temperature. Analytic results are found for all higher moments of the fluctuation distribution by employing the stochastic path integral formalism, with excellent accuracy. We further discuss a hybrid treatment, which weds the master equation/stochastic path integral analysis with the results obtained based on canonical ensemble quasiparticle formalism [V. V. Kocharovsky et al., Phys. Rev. A 61, 053606 (2000)], producing essentially perfect agreement with numerical simulation at all temperatures.