Interacting Bright Solitons in Trapped BECs: Mean-Field Theory and Way Beyond

Bright multi-soliton trains have been experimentally observed in harmonic traps,\footnote{K. E. Strecker \textit{et. al.,} Nature \textbf{417},150 (2002).} in experiments made especially difficult by the fact that a too-bright soliton (or ground state of an attractive BEC) will self destruct in 3D. These are fascinating systems as, as recently pointed out,\footnote{A. I. Streltsov \textit{et. al}., Phys. Rev. Lett. \textbf{106}, 240401 (2011).} usual mean-field descriptions via the NLSE (or GP) equation, will often fail, as many-body effects may play a crucial role. Here we discuss 4 regimes of the statics and dynamics of double-bright-soliton systems by giving an overview of the full energy correlation diagram in a two (non-linear) mode description. 1) Strongly trapped soliton pairs are found to be quite well described by GP, and also via two mode, dynamics, up-holding prior theoretical analysis of existing experiment.\footnote{Strecker, op. sit.} Separated soliton pairs may be: 2) Schr\"{o}dinger cats; 3) fully fragmented; or, 4) decohering\footnote{Streltsov, op. sit.} wave packets if followed via two-mode dynamics, using coherent mean-field (GP) initial conditions, and end up producing states of controllable partial entanglement.