Order parameter at the boundary of a trapped Bose gas.

Through a suitable expansion of the Gross-Pitaevskii equation near the classical turning point, we obtain an explicit solution for the order parameter at the boundary of a trapped Bose gas interacting with repulsive forces. The kinetic energy of the system, in terms of the classical radius R and of the harmonic oscillator length a HO , follows the law Ekin/N ∝ R [log(R/a HO ) + const.], approaching, for large R, the results obtained by solving numerically the Gross-Pitaevskii equation. The occurrence of a Josephson-type current in the presence of a double trap potential is finally discussed. PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj Typeset using REVTEX 1 The recent experimental realization of Bose-Einstein condensation in atomic gases confined in magnetic traps [1–3] is stimulating a novel interest in the study of inhomogeneous Bose condensed systems where the order parameter exhibits an important spatial dependence on a macroscopic scale [4]. The purpose of the present work is to investigate the behavior of the wave function of the condensate near the classical turning point, that is, at the boundary of the trapped gas. This region is particularly important for the determination of the kinetic energy associated with the atoms of the condensate [5,6]. It is also crucial for the description of Josephson-type effects taking place in the presence of a barrier in the confining potential. The order parameter ψ(r) associated with the ground state of a dilute Bose gas obeys the Gross-Pitaevskii equation: