The saddle-point method for condensed bose gases

The application of the conventional saddle-point approximation to condensed Bose gases is thwarted by the approach of the saddle-point to the ground-s tate singularity of the grand canonical partition function. We develop and test a variant of the saddle-point method which takes proper care of this c omplication, and provides accurate, flexible, and computationally efficient access to both canonical and microcanonical statistics. Remarkably, the er ror committed when naively employing the conventional approximation in the condensate regime turns out to be universal, that is, independent of the sy stem's single-particle spectrum. The new scheme is able to cover all temper atures, including the critical temperature interval that marks the onset of Bose-Einstein condensation, and reveals in analytical detail how this onse t leads to sharp features in gases with a fixed number of particles. In par ticular, within the canonical ensemble the crossover from the high-temperat ure asymptotics to the condensate regime occurs in an error-function-like m anner; this error function reduces to a step function when the particle num ber becomes large. Our saddle-point formulas for occupation numbers and the ir fluctuations, verified by numerical calculations, clearly bring out the special role played by the ground state. (C) 1999 Academic Press.