Polarization-Mixing in Optical Lattices with Uniaxial Anisotropy

We study optical excitations of an ultracold gas in an optical lattice in the Mott insulator case and with anisotropic properties. The anisotropy is induced by an oriented transition dipole generated by optical pumping and external static fields, and is of the type of uniaxial crystals. Within a cavity, the electronic excitations and the cavity photons of both TE and TM polarizations are coherently mixed to form two polariton branches. Also a photonic branch appears which decouples from the electronic transitions. The photon polarization mixing can be observed by linear optical spectra. For example, for an incident field of TE polarization we obtain TE and TM transmitted and reflected fields. Also we calculate the phase shift in the TE and TM transmitted and reflected fields relative to the incident field phase, which lead to polarization rotations.