Loop current states and their stability in small fractal lattices of Bose-Einstein condensates

We consider a model of interacting Bose-Einstein condensates on small Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gaskets. We study eigenstates which are characterized by cyclic supercurrents for each triangular plaquette (loop states). For noninteracting systems we find at least three classes of loop eigenmodes: standard, chaotic, and periodic. Standard modes are those inherited from the basic three-site ring of condensates with phase differences locked to $2\ensuremath{\pi}/3$. Standard modes become unstable in the interacting system but only when the interaction exceeds a certain critical value ${u}_{c}$. Chaotic modes are characterized by very different circular currents per plaquette, so the usual symmetry of loop currents is broken. Circular supercurrents associated with chaotic modes become chaotic for any finite interaction, signaling the loss of coherence between the condensates. Periodic modes are described by alternating populations and two different phase differences. The modes are self-similar and are present in all generations of Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gasket. When the interaction is included, the circular current of such a mode becomes periodic in time with the amplitude growing linearly with the interaction. Above a critical interaction the amplitude saturates, signaling a transition to a macroscopic self-trapping state originally known from a usual Bose-Josephson junction. We perform a systematic analysis of this rich physics.