Period-Doubled Bloch States in a Bose–Einstein Condensate

We study systematically the period-doubled Bloch states for a weakly interacting Bose–Einstein condensate in a one-dimensional optical lattice. This kind of state is of form \psi_k=e^ikx\phi_k(x), where \phi_k(x) is of a period twice the optical lattice constant. Our numerical results show how these nonlinear period-doubled states grow out of linear period-doubled states at a quarter away from the Brillouin zone center as the repulsive interatomic interaction increases. This is corroborated by our analytical results. We find that all nonlinear period-doubled Bloch states have both Landau instability and dynamical instability.