Localization-delocalization transition of dipolar bosons in a four-well potential

We study interacting dipolar atomic bosons in a four-well potential within a ring geometry and outline how a four-site Bose-Hubbard (BH) model including next-nearest-neighbor interaction terms can be derived for the above four-well system. We analyze the ground state of dipolar bosons by varying the strength of the interaction between particles in next-nearest-neighbor wells. We perform this analysis both numerically and analytically by reformulating the dipolar-boson model within the continuous variable picture applied in [Phys. Rev. A {\bf 84}, 061601(R) (2011)]. By using this approach we obtain an effective description of the transition mechanism and show that when the next-nearest-neighbor interaction crosses a precise value of the on-site interaction, the ground state exhibits a change from the uniform state (delocalization regime) to a macroscopic two-pulse state, with strongly localized bosons (localization regime). These predictions are confirmed by the results obtained by diagonalizing numerically the four-site BH Hamiltonian.