1 M ay 2 00 7 Bright solitary waves and trapped solutions in Bose - Einstein condensates with attractive interactions

We analyse the static solutions of attractive Bose-Einstein condensates under transverse confinement, both with and without axial confinement. By full numerical solution of the Gross-Pitaevskii equation and variational methods we map out the condensate solutions, their energetic properties, and their critical points for instability. With no axial confinement a bright solitary wave solution will tend to decay by dispersion unless the interaction energy is close to the critical value for collapse. In contrast, with axial confinement the only decay mechanism is collapse. The stability of a bright solitary wave solution increases with higher radial confinement. Finally we consider the stability of dynamical states containing up to four solitons and find good agreement with recent experiments. Bright solitary waves and trapped solutions of attractive BECs 2 The presence of attractive interactions in Bose-Einstein condensates (BECs) leads to rich and intriguing nonlinear phenomena. A key example is the formation of bright soliton-like structures [1, 2, 3]. A bright soliton is a one-dimensional (1D) density wave that propagates without spreading due to a balance between attractive interactions and dispersion. In 3D and under transverse confinement, the analog is a bright solitary wave (BSW) solution, which is self-trapped in the axial direction [4, 5, 6, 7]. Due to their self-trapped nature, BSWs hold significant advantages for atom-optical applications, such as atom interferometry. Another important property of attractive BECs in 3D is the collapse instability. In 3D a homogeneous condensate with attractive interactions is always unstable to collapse [8]. However, the presence of trapping can stabilise the condensate against collapse up to a critical number of atoms [9, 10]. Indeed, the collapse instability has been crucial in the experimental formation of BSWs [1, 2, 3]. A highlypopulated repulsively-interacting BEC has its interactions switched to attractive via a Feshbach resonance. This induces the collapse of the condensate, with one [2] or more [1, 3] BSWs emerging from the collapse. For the case of multiple BSWs, a π-phase difference leads to a repulsive solitonic interaction that is important in stabilising their collisions against collapse [7, 11, 12, 13]. Table 1 summarises the three experiments to date that have generated BSWs of attractive BECs, at Rice University [1], ENS in Paris [2], and JILA [3]. They feature cylindrically-symmetric traps with radial harmonic confinement of frequency ωr, and either a confining or expulsive axial harmonic potential. Note that due to the presence of axial confinement these are not true solitonic states and from now on we will generally define BSWs to be solutions under zero axial confinement. The strength of the atomic interactions, characterised by the s-wave scattering length as (as < 0 for attractive interactions), relative to the trapping potential is crucial in determining the onset of collapse, and can be parameterised in terms of the dimensionless parameter [14],