6533b835fe1ef96bd129f2a3

RESEARCH PRODUCT

Finite-temperature correlations in the trapped Bose-Einstein gas

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There is a large literature (cf. eg. [1, 2]) which, under conditions of translational invariance, has used functional integral methods to calculate, ab initio, the equilibrium finite temperature 2-point correlation functions (Green ’s functions) \[\left\langle {\hat \psi (r,\tau ){{\hat \psi }^\dag }(r',\tau ')} \right\rangle \] \(G\left( {r,r'} \right) \equiv \left\langle {\psi \left( {r,\tau } \right){{{\hat{\psi }}}^{\dag }}\left( {r',\tau '} \right)} \right\rangle \) for a Bose gas in each of d=3, d=2, d=1 space dimensions: (…) means thermal average and τ, τ′ are ‘thermal times’ for which 0<τ,<τ′β and β−1=k B T, T the temperature. These functional integral methods [1, 2] solve the many-body problem, to good and controlled approximations, for a weakly coupled Bose gas typically modelled by the quantum repulsive Nonlinear Schrodinger (NLS) equation in d+1 dimensions — essentially as was first used by N.N. Bogoliubov (1947) for d=3 and as is now described by ego. Huang [3]: in (adjoint) are Bose field operators,\(\left[ {\hat \psi (r),{{\hat \psi }^\dag }(r')} \right]\) forℏ=1 and r, r′ are two spatial points in d dimensions: translational invariance means G(r, r′)=G(r−r′) depending only on r−r′≡R. Results show a critical temperature T c such that for T<T c there is a long range constant valued condensate density ρ0 for, and only for, d=3; and for d=3 G is asymptotic to constant {ρ0 for large enough R = |R|. For d=2, 1, G falls off with R ultimately to zero, and eg. for d=1 exact calculations [4] show an exponential decay with R.