Density wave and supersolid phases of correlated bosons in an optical lattice

Motivated by the recent experiment on the Bose-Einstein condensation of $^{52}$Cr atoms with long-range dipolar interactions (Werner J. et al., Phys. Rev. Lett., 94 (2005) 183201), we consider a system of bosons with repulsive nearest and next-nearest neighbor interactions in an optical lattice. The ground state phase diagram, calculated using the Gutzwiller ansatz, shows, apart from the superfluid (SF) and the Mott insulator (MI), two modulated phases, \textit{i.e.}, the charge density wave (CDW) and the supersolid (SS). Excitation spectra are also calculated which show a gap in the insulators, gapless, phonon mode in the superfluid and the supersolid, and a mode softening of superfluid excitations in the vicinity of the modulated phases. We discuss the possibility of observing these phases in cold dipolar atoms and propose experiments to detect them.