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We investigate the many-body state and the static and the dynamic behaviourof the pair-correlation function of a Bose-Einstein condensate with a finiteatom number, which is confined in a quasi-one-dimensional toroidal/annularpotential, both for repulsive, and for attractive interactions. We link thedynamic pair-correlation function that we evaluate with the problem of quantumtime crystals. For weak repulsive interatomic interactions and a finite numberof atoms the pair-correlation function shows a periodic temporal behaviour,which disappears in the limit of a large atom number, in agreement with generalarguments. Finally we provide some insight into older results of attractiveinteractions, where the time-crystalline behaviour exists only in the limit ofa large atom number.

Many-body state and dynamic behaviour of the pair-correlation function of a small Bose-Einstein condensate confined in a ring potential