Macroscopic quantum bound states and Josephson effect in Bose-Einstein condensates in optical lattices

Localized ground states of the periodic Gross-Pitaevskii equation (GPE) are considered in the framework of a quantum linear Schrodinger equation with an effective potential determined in a self-consistent manner. We show that, depending on whether the interaction among the atoms is attractive or repulsive, bound states of the linear self-consistent problem are formed in the forbidden zones of the linear spectrum below or above the energy bands, respectively. These eigenstates are found to be exact soliton solutions of the GPE. More complicated solutions of the GPE can be constructed by taking linear combinations of eigenstates as effective potentials in the self-consistent approach. In particular, we discuss bound-state wavefunctions that describe Bose-Einstein condensate (BEC) Josephson junctions. These bound states consist of two localized weakly interacting condensates that exhibit, in the interaction region, matter oscillations that are governed by the phase difference between the wavefunctions of the two condensates.