Quantum electrodynamics near a dielectric half-space

Radiative corrections in systems near imperfectly reflecting boundaries are investigated. As an example, the self-energy of an unbound electron close to a single surface is calculated at one-loop level. The surface is modeled by a nondispersive dielectric half-space of a constant refractive index $n$. In contrast to previous, perfectly reflecting models, the evanescent modes in the optically thinner medium are taken into account and are found to play a physically very important role. The Feynman propagator of the photon field is determined and given in two alternative representations, which include the evanescent modes either as a separate contribution or through analytic continuation and deformation of the integration path for the normal component of the complex wave vector $\mathbf{k}$. The evaluation of the self-energy diagram encounters a number of problems that are specific to the boundary dependence and to the imperfect reflection at the boundary. These problems and methods for their resolution are discussed in depth.

Figure 1

A sketch of the simplest system in cavity QED, consisting of a quantum object, e.g. an atom or an electron, that is located a distance $a$ away from a single reflecting wall. If the wall is perfectly reflecting then the electromagnetic field is excluded from the region $z>0$. A more realistic but still simple model is to describe the wall by a constant, frequency-independent refractive index $n$.

Figure 2

The integration path $\mathcal{C}$ in the complex $k_z$ plane.