Eur. Phys. J. D (2015) 69: 139
https://doi.org/10.1140/epjd/e2015-50736-1

Regular Article

Confinement effects on an ultra-cold matter wave-packet by a square well impurity near the de-localization threshold: analytic solutions, scaling, and width properties

¹ Facultad de Ciencias, Universidad Nacional Autónoma de México, Av. Universidad 3000, Circuito Exterior S/N Delegación Coyoacán, C.P. 04510 Ciudad Universitaria, D.F. México
² Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Ap. Postal 48-3, Cuernavaca, Morelos 62251, México

^a e-mail: trujillo@fis.unam.mx

Received: 7 October 2014
Received in final form: 16 February 2015
Published online: 27 May 2015

Abstract

The determination of the maximum number of atoms and the density profile of an ultra-cold wave-packet, under confinement conditions by an attractive impurity near the de-localization threshold, have been an open problem in ultra-cold atom physics. In this work, we study the effect of a wave-guide impurity on an ultra-cold matter wave-packet at the threshold of de-localization. The impurity is modeled by a 1-D square well potential with depth V₀ and length 2R₀. Coupling of the square well potential to a contact impurity of strength β at the center is also considered. The time-independent non-linear Schrödinger equation describing a Bose-Einstein condensate at the delocalization threshold is exactly solved. The density profile, maximum non-linear coupling constant, g_max, and maximum number of atoms, N_max, prompt to be localized by the defect potential in the ground and first excited states are also reported. It is shown that g_max and the density profiles become only functions of the reduced impurity size ξ = √V₀R₀ . It is also found that the first excited state at the threshold of de-localization exists only for ξ≥π/(2√2) , always holding a lower number of atoms than the corresponding ground state for the same reduced impurity size. Also, the addition of a repulsive contact impurity leads to a non-linear coupling constant at the de-localization threshold lower than that of the square well potential. In spite of the non-linear character of the Gross-Pitaevskii equation, it is found that a general scaling-law holds for defects with the same ξ, related with the same g_max, having the same reduced density profile in the quasi-free direction. We report the full width at half maximum for the wave-function and density profile, finding a large spread for small reduced confining conditions. Implications of these results for the determination of the wave-packet properties under confinement in atom chip and Bose-Einstein condensates are presented with the aim to motivate further experimental work.