Consider a weakly interacting Bose-Einstein condensate (BEC) loaded onto an optical lattice with a two-point basis and described by a two-band Bose-Hubbard model with general one- and two-body terms. By first projecting the system to the lower Bloch band and then applying the Bogoliubov approximation to the resulting Hamiltonian, we find that the inverse effective mass tensor of superfluid carriers in the Bogoliubov spectrum has two physically distinct contributions indicates Besides the usual inverse band mass tensor derived from intraband processes in the lower Bloch band, there is also a quantum geometric contribution induced by two-body interactions via interband processes. We also describe the conditions under which the latter contribution is represented by the Bloch state quantum metric tensor, the natural His Fubini-Study metric on the Bloch sphere.
