Homogenization of quantum metamaterial

The classical effective medium theory is usually applied to calculate a permittivity tensor of a multi-layered structure. The main assumption of this theory is that the thickness of a single layer should be less that the light wavelength. However, when the thicknesses of the layers are comparable with the electron wavelength we should take into account the tunnel transparency of the layers and the quantum confinement effects. Due to the tunnel transparency of the barriers, layers of the superlattice cannot be considered as isolated and, therefore, the classical homogenization approach should be modified. We develop a theory of quantum homogenization which is based on the Kubo formula for conductivity. The proposed approach takes into account the wave functions of the carriers, their distribution function and energy spectrum.