Finite-temperature correlations in the trapped Bose-Einstein gas

New analytical results are reported for the two-point correlation function at nonzero temperatures in d=3, 2 and 1 dimensions. Only for d=3 are correlations below T-c both long range and coherent at long range consistent with the existence of a coherent state of the condensate in the trap. Asymptotically these correlations are found to decay as (rho (S) .|r(1)-r(2)|)(-1) towards a nonzero value. The condensate density rho (S), with S = (r(1) + r(2))/2, is a slowly varying function and introduces a small amplitude variation in the decay of correlations when considered as a function of |r(1)-r(2)|. The prediction of a |r(1)-r(2)|(-1)-dependence at least has very recently been confirmed in a beautiful experiment ( Bloch I., Hansch T. W. and Esslinger T., Nature, 403 (2000) 166). The two-dimensional condensate is marginally stable in the sense that for 0<T<T-c correlations decay by a power law.