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Shape dependence and anisotropic finite-size scaling of the phase coherence of three-dimensional Bose-Einstein condensed gases

We investigate the equilibrium phase-coherence properties of Bose-condensed particle systems, focusing on their shape dependence and finite-size scaling (FSS). We consider three-dimensional (3D) homogeneous systems confined to anisotropic L x L x L_a boxes, below the BEC transition temperature $T_c$. We show that the phase correlations develop peculiar anisotropic FSS for any $T<T_c$, in the large-$L$ limit keeping the ratio λ= L_a/L^2 fixed. This phenomenon is effectively described by the 3D spin-wave (SW) theory. Its universality is confirmed by quantum Monte Carlo simulations of the 3D Bose-Hubbard model in the BEC phase. The phase-coherence properties of very elongated BEC systems, λ>>1, are characterized by a coherence length ξ_a \sim A_t ρ_s/T where A_t is the transverse area and ρ_s is the superfluid density.