Bogoliubov theory of Bose-condensates of spinor exciton-polaritons

In the present work, we present the formalism allowing to calculate the energy spectrum of polariton spinor Bose condensates. We investigate the dispersion of the elementary excitations of the weakly depleted polariton condensate within the framework of the Bogoliubov approach. We show that for linearly polarized condensates the anisotropy of the polariton-polariton interactions leads to important differences in polariton dispersion in linear co- and cross-linear polarizations. Surprisingly, we find that contrary to the standard Bogoliubov model (which does not take into account the spin of the condensate) the dispersion is not necessary linear at k = 0, but can be parabolic