The spatial Cauchy problem for a dissipative infinite quantum waveguide supporting a single propagating mode

We consider an infinite waveguide supporting a single propagating mode for which the excitation (say the electric or the magnetic field) is assigned in a given section assumed as the origin of the coordinate z along its axis. In steady state the state evolution along z is the solution of the spatial Cauchy problem along such coordinate. As soon as the radiation involves a low number of photons, or even reduces to a single one, the classical treatment must be replaced by a quantum one. Moreover the effect of dissipation must be taken into account by either a microscopic treatment of the properties of the dielectric material and of the metallic boundaries, or, as long as only the mathematical form of the equations be of interest, by using a spatial version of Lindblad equations. We choose here the second alternative. From the obtained equations it is possible to extend the treatment to multimode finite terminated waveguides.