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Quantum electrodynamic theory (QED) in the vicinity of macroscopic structures has achieved new importance due to its applicability, particularly in nanotechnology. There are many powerful methods for studying QED near media with diverse properties and geometries. However, applying them to a particular problem generally necessitates extensive numerical calculations. This is not the case for simple systems of high symmetry, in which the electromagnetic field can be quantised by explicit mode expansion, allowing exact analytic calculations. In the present thesis, we calculate the energy-level shift of a ground state atom near a non-dispersive and non-dissipative dielectric slab. The shift is due to the interaction of the atom with electromagnetic field fluctuations, which in turn are affected by the presence of the slab. Thus, a quantisation of the electromagnetic field in the presence of a layered system is required. We derive the field modes, which comprise of a continuous set of travelling modes (with incident, reflected and transmitted parts) and trapped modes, subject to repeated total internal reflection and emerge as an evanescent field outside the slab, they only exist at certain discrete frequencies. The shift is obtained by means of second-order perturbation theory. It splits up naturally into two contributions, due to the different nature of the modes, and a problem arises when we have to add them all. We have come up with a convenient method of summing over all modes, and its validity has been demonstrated by proving the completeness. The calculation of the shift follows as an application of our method. The result is analysed asymptotically for various regions, reducing to simple formulas that can be utilised in recent experiments, in which the thickness of the substrate matters. |
