Eur. Phys. J. B (2013) 86: 343
https://doi.org/10.1140/epjb/e2013-40551-y

Regular Article

Interaction effects on atomic laboratory trapped Bose-Einstein condensates

¹ Istituto Nazionale di Alta Matematica Francesco Severi, Gruppo Nazionale di Fisica Matematica, Città Universitaria, P.le A. Moro 5, 00185 Roma, Italy
² Dipartimento SBAI, Sezione di Matematica, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy

^a e-mail: fabio.briscese@sbai.uniroma1.it

Received: 28 March 2013
Received in final form: 6 June 2013
Published online: 1 August 2013

Abstract

We discuss the effect of inter-atoms interactions on the condensation temperature T_c of an atomic laboratory trapped Bose-Einstein condensate. We show that, in the mean-field Hartree-Fock and semiclassical approximations, interactions produce a shift ΔT_c/T_c⁰ ≈ b₁(a/λ_Tc) + b₂(a/λ_Tc)² + ψ[a / λ_Tc] with a the s-wave scattering length, λ_T the thermal wavelength and ψ[a / λ_Tc] a non-analytic function such that ψ[0] = ψ′[0] = ψ′′[0] = 0 and |ψ′′′[0]| = ∞. Therefore, with no more assumptions than Hartree-Fock and semiclassical approximations, interaction effecs are perturbative to second order in a / λ_Tc and the expected non-perturbativity of physical quantities at critical temperature appears only to third order. We compare this finding with different results by other authors, which are based on more than the Hartree-Fock and semiclassical approximations. Moreover, we obtain an analytical estimation for b₂ ≃ 18.8 which improves a previous numerical result. We also discuss how the discrepancy between b₂ and the empirical value of b₂ = 46 ± 5 may be explained with no need to resort to beyond-mean field effects.