Quantum kinetic theory. IV. Intensity and amplitude fluctuations of a Bose-Einstein condensate at finite temperature including trap loss

We use the quantum kinetic theory to calculate the steady state and fluctuations of a trapped Bose-Einstein condensate at a finite temperature. The system is divided in a condensate and a noncondensate part. A quantum-mechanical description based on the number-conserving Bogoliubov method is used for describing the condensate part. The noncondensed particles are treated as a classical gas in thermal equilibrium with temperature $T$ and chemical potential $\ensuremath{\mu}$. We find a master equation for the reduced density operator of the Bose-Einstein condensate, calculate the steady state of the system, and investigate the effect of one-, two-, and three-particle losses on the condensate. Using linearized Ito equations, we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. A Lorentzian line shape is found for the intensity correlation function that is characterized by a time constant ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$ derived in the paper. For the amplitude correlation function, we find ballistic behavior for time differences smaller than ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$, and diffusive behavior for larger time differences.