  We consider the problem of finding the bound-state spectrum of an impurity
immersed in a weakly interacting two-dimensional Bose-Einstein condensate
supporting a single vortex. We obtain approximate expressions for the energy
levels and show that, due to the finite size of the condensate, the impurity
can access only a finite number of physical bound states. By virtue of the
topological quantization of the vorticity and of the emergence of the Tkachenko
lattice, this system is promising as a robust and scalable platform for the
realization of qubits. Moreover, it provides a potentially new paradigm for
polaron physics in Bose-Einstein condensates and a glimpse towards the study of
quantum turbulence in low-dimensionality systems.
