Excitation spectrum of spin-1 bosonic atoms in an optical lattice with high filling factors

The Green's function and the higher-order correlation functions of spin-1 cold atoms in an optical lattice are defined. Because we consider the problem of spin-1 Bose condensed atoms in an optical lattice with high filling factors, i.e., the number density of Bose condensed atoms no is large, the fluctuation of them can be neglected and we take mean-field approximation for the higher-order terms. The excitation spectra for both the polar case and the ferromagnetic case are obtained and analyzed.