Images of the dark soliton in a depleted condensate

The dark soliton created in a Bose-Einstein condensate becomes grey in the course of time evolution because its notch fills up with depleted atoms. This is the result of quantum mechanical calculations which describe the output of many experimental repetitions of creation of the stationary soliton, and its time evolution terminated by a destructive density measurement. However, such a description is not suitable to predict the outcome of a single realization of the experiment where two extreme scenarios and many combinations thereof are possible: one will see either (1) a displaced dark soliton without any atoms in the notch, but-with a randomly displaced position, or (2) a grey soliton with a fixed position, but a random number of atoms filling its notch. In either case the, average over many I realizations will reproduce the mentioned quantum mechanical result. In this paper we use N-particle wavefunctions, which follow from the number-con serving Bogoliubov theory, to settle this issue.