Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling

We study the interaction-induced localization – the so-called self-trapping – of a neutral impurity atom immersed in a homogeneous Bose-Einstein condensate (BEC). Based on a Hartree description of the BEC we show that – unlike repulsive impurities – attractive impurities have a singular ground state in 3d and shrink to a point-like state in 2d as the coupling approaches a critical value β⋆. Moreover, we find that the density of the BEC increases markedly in the vicinity of attractive impurities in 1d and 2d, which strongly enhances inelastic collisions between atoms in the BEC. These collisions result in a loss of BEC atoms and possibly of the localized impurity itself. Impurities immersed in liquid helium have proven to be a valuable tool for probing the structure and dynamics of a Bose-condensed fluid [1]. An example is the use of impurities for the direct visualization of quantized vortices in superfluid He [2]. Recently, the experimental realization of impurities in a BEC [3, 4] and the possibility to produce quantum degenerate atomic mixtures [5, 6] have generated renewed interest in the physics of impurities. In particular, it was pointed out that single atoms can get trapped in the localized distortion of the BEC that is induced by the impurity-BEC interaction [7–11]. More precisely, the impurity becomes self-trapped if its interaction with the BEC is sufficiently strong to compensate for the high kinetic energy of a localized state, an effect akin to the self-trapping of impurities in liquid helium [1]. However, there are two fundamental differences between liquid helium and a BEC. First, for typical experimental parameters the healing length of a BEC is three orders of magnitude larger than for liquid helium [12], and thus the so-called ’bubble’ model [13] cannot be applied. A second important difference is that the impurity-BEC interactions are tunable by an external magnetic field in the vicinity of Feshbach resonances [14,15]. In view of recent experimental progress [16] it thus seems possible to investigate the self-trapping problem in the same physical system over a wide range of interaction strengths – for both attractive and repulsive impurities. In the present paper we study the self-trapping properties of impurities for strong attractive and repulsive impurity-BEC coupling within the framework of a Hartree description of the BEC. This necessitates the use of an essentially non-perturbative approach, which is in contrast to previous analytical studies [7–9] where the effect of the impurities on the BEC was treated as a small perturbation. We first consider the effect of a highly localized δimpurity on the BEC in one dimension. This approach indicates that the density of the BEC is substantially increased in the vicinity of an attractive impurity, which enhances inelastic collisions and may result in the loss of the impurity atom. In addition, we bring forward a scaling argument to show that attractive impurity-BEC interactions can lead to a point-like ground state of the impurity