Abstract
A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.
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