Nonlinear Band Spectrum and Elementary Excitations of a Bose Gas Within a Multi-Rods Structure

We calculate the ground state (gs) energy spectrum as well as the elementary excitations of a
weakly-interacting Bose gas confined by a one-dimensional periodic multi-rods structure created
by an external Kronig-Penney potential. We use mean-field theory approximation by solving
analytically the Gross-Pitaevskii equation¹ in order to get the (nonlinear) energy spectrum of the gas.
In addition, we solve the Bogoliubov-de-Gennes (BdG) equations² to calculate the elementary
excitations of the system. In the limit of zero external potential, we recover the well-known Lieb-Liniger
results³ for a weakly-interacting Bose gas. The nonlinear band spectrum shows a "swallow-tail"
shape in the edges of the first Brillouin zone for uneven bands and at zero momentum for even bands.
We also obtain the eigen-energies of the BdG equations, which form a band structure just like the nonlinear band
spectrum. The elementary excitations display a phononic behavior at low momenta from where we
calculate the sound velocity in terms of the slope of their energy at zero momentum.

1. Omar A. Rodríguez-López and M. A. Solís, work in process.
2. Mora, C. and Castin, Y., Phys. Rev. A 67, 53615 (2003)
3. Lieb E. H. and Liniger W., Phys. Rev. 130, 1605 (1963)