Formation and dynamics of many-boson fragmented states in attractive one-dimensional ultra-cold gases

Dynamics of attractive ultra-cold bosonic clouds in one dimension are studied by solving the many-particle time-dependent Schr\"odinger equation. The initially coherent wave-packet can dynamically dissociate into two parts when its energy exceeds a threshold value. Noticeably, the time-dependent Gross-Pitaevskii theory applied to the same initial state does not show up the splitting. We call the split object {\it fragmenton}. It possesses remarkable properties: (1) it is two-fold fragmented, i.e., not coherent; (2) it is dynamically stable, i.e., it propagates almost without dispersion; (3) it is delocalized, i.e., the two dissociated parts still communicate with one another. A simple static model predicts the existence of fragmented states which are responsible for formation and dynamics of fragmentons.