Breakdown of the mean field for dark solitons of dipolar bosons in a one-dimensional harmonic trap

We directly compare the mean-field and the many-body approaches, in a one-dimensional Bose system in a harmonic trap. Both contact and dipolar interactions are considered. We obtain eigenstates of the many-body system by exact diagonalization with a truncated basis. We propose a multiatom version of the phase-imprinting method to generate dark solitons in the system. We begin with a general analysis of system dynamics and observe the emergence of a dark soliton and a shock wave in the form of sloshing motion of the condensate upon phase imprinting. The center of mass and soliton go out of phase because the soliton does not oscillate with the trap frequency. A detailed investigation of frequencies reveals significant differences in the results obtained in the mean-field versus the many-body picture.