Modal expansions in dispersive material systems with application to quantum optics and topological photonics

It is proven that in the lossless case the electrodynamics of a generic inhomogeneous possibly bianisotropic and nonreciprocal system may be described by an augmented state-vector whose time evolution is determined by a Hermitian operator. As a consequence, it is shown that a generic electromagnetic field distribution can be expanded into a complete set of normal modes that satisfy generalized orthogonality relations. Importantly, the modal expansions in dispersive systems are not unique because the electromagnetic degrees of freedom span only part of the entire Hilbert space. The developed theory is used to obtain a modal expansion of the system Green's function. Furthermore, it is highlighted that the Hermitian-type formulation of the dispersive Maxwell's equations enables one to extend the powerful ideas of topological photonics to a wide range of electromagnetic systems and to characterize electromagnetic topological phases. In addition, we illustrate how the developed formalism can be applied to quantum optics. We present a simple procedure to quantize the electromagnetic field in a generic bianisotropic and nonreciprocal cavity and derive the quantum correlations of the electromagnetic fields.