Swift Loss of Coherence of Soliton Trains in Attractive Bose-Einstein Condensates

Experiments on ultracold attractive Bose-Einstein condensates (BECs) have demonstrated that at low dimensions atomic clouds can form localized objects, propagating for long times without significant changes in their shapes and attributed to bright matter-wave solitons, which are coherent objects. We consider the dynamics of bright soliton trains from the perspective of many-boson physics. The fate of matter-wave soliton trains is actually to quickly lose their coherence and become macroscopically fragmented BECs. The death of the coherent matter-wave soliton trains gives birth to fragmented objects, whose quantum properties and experimental signatures differ substantially from what is currently assumed.

Figure 1

Evolution in time of a two-hump soliton train in coordinate space. Shown is the evolution in time of the density $\rho (x,t)$ of the attractive system with $N=2000$ bosons and ${\lambda }_0=-0.002$ computed by the popular Gross-Pitaevskii equation on the mean-field level in (a)–(c), and the corresponding dynamics computed on the many-body level in (d)–(f). Panels (a) and (d) show the dynamics of a 0-phase symmetric train; panels (b) and (e) show the dynamics of a $\pi$-phase antisymmetric train; panels (c) and (f) show the dynamics of a 0-phase asymmetric train. On the many-body level the forces between the two density humps are weaker, and there is essentially only repulsion and no attraction between them at all but short times. All quantities shown are dimensionless.

Figure 2

Death of soliton trains by fragmentation. The upper panels show the occupation numbers of the reduced one-body density matrix of the initially prepared 0-phase symmetric (left) and $\pi$-phase antisymmetric (right) soliton trains of Fig. 1. It is seen that the initially condensed systems become quickly fragmented. The emergence of fragmentation signifies the death of the condensed objects known as bright soliton trains. The lower two panels depict the respective natural orbitals after fragmentation of the initially prepared 0-phase symmetric (left) and $\pi$-phase antisymmetric (right) coherent wave packets. The two natural orbitals are delocalized functions of “gerade” (even) and “ungerade” (odd) symmetries. The delocalization of the natural orbitals in combination with the exchange interaction stabilizes the fragmented attractive Bose-Einstein condensates in comparison with the condensed soliton trains. All quantities shown are dimensionless.

Figure 3

Detecting the death of soliton trains by means of first-order correlation functions. The upper two panels show the first-order correlation function of the initially prepared 0-phase symmetric soliton train of Fig. 1 in coordinate (left) and momentum (right) space. It is completely flat, signifying a coherent and condensed state. Within Gross-Pitaevskii theory the system remains condensed for all times; thus, the upper two panels represent the first-order correlation function in this case for all times. On the many-body level, the correlation functions are completely different and structured. The pattern emerging in coordinate space indicates no coherence between the two humps (left-hand middle and lower panels), the respective pattern in momentum space represents a fragmented state (right-hand middle and lower panels). Thus, measuring the first-order correlation functions in either coordinate or momentum space provides a direct indication of the death of the soliton train. All quantities shown are dimensionless.