  Dispersion properties of metals and propagation of quantum bulk plasmon in
the high photon energy regime are studied. The nonlocal dielectric permittivity
of a metal is determined by the quantum plasma effects and is calculated by
applying the Wigner equation in the kinetic theory and taking into account the
electron lattice collisions. The properties of epsilon near zero material are
investigated in a thin gold film. The spectrum and the damping rate of the
quantum bulk plasmon are obtained for a wide range of energies, and the
electron wave function is analytically calculated in both classical and quantum
limits. It is shown that the quantum bulk plasmons exist with a propagation
length of 1 to 10nm, which strongly depends on the electron energy. The
propagation length is found to be much larger than the propagation length in
the classical regime which is comparable to the atomic radius and the average
inter particle distance. It is found that the spatial localization of the
electron wave function is extended due to the quantum effects. Also, it is
shown that the damping of electromagnetic waves decreases when the energy of
the photon decreases which is opposite to the conclusions obtained from the
classical Drude model. The importance of the quantum effects in the development
of next-generation metamaterials is also discussed.
