  The one-dimensional extended bosonic Hubbard model has been shown to exhibit
a variety of phases ranging from Mott insulator and superfluid to exotic
supersolids and Haldane insulators depending on the filling and the relative
value of the contact ($U$) and near neighbor ($V$) interaction strengths. In
this paper we use the density matrix renormalization group and the time
evolving block decimation numerical methods to study in detail the dynamics and
excitation spectra of this model in its various phases. In particular, we study
in detail the behavior of the charge and neutral gaps which characterize the
Mott, charge density and Haldane insulating phases. We also show that in
addition to the gapless modes at $k=0$, the supersolid phase exhibits gapless
modes at a finite $k$ which depends on the filling.

Quantum Gases;Quantum Physics