Effects of a rotating periodic lattice on coherent quantum states in a ring topology

We study the landscape of solutions of the coherent quantum states in a ring shaped lattice potential in the context of ultracold atoms with an effective nonlinearity induced by interatomic interactions. The exact analytical solutions in the absence of lattice are used as a starting point and the transformation of those solutions is mapped as the lattice is introduced and strengthened. This approach allows a simple classification of all the solutions into states with periods commensurate/incommensruate with the lattice period and those with/without nodes. Their origins are traced to the primary dispersion curve and the swallowtail branches of the lattice-free spectrum. The commensurate states tend to remain delocalized with increasing lattice depth, whereas the incommensurate ones may be localized. The symmetry and stability properties of the solutions are examined and correlated with branch energies. We identify difference of significance between positive and negative nonlinearity. The crucial importance of rotation is highlighted by its utility in continuously transforming solutions, and accessing in a finite ring with a few sites the full spectrum of nonlinear Bloch waves on an infinite lattice .