The single-mode Jaynes-Cummings model has been of paramount importance in the development of quantum optics. Recently, also the strong coupling to more than a single mode of an electromagnetic resonator has drawn considerable interest. We investigate how this superstrong coupling regime can be harnessed to coherently control quantum systems. In particular, we show that elliptical cavities and Maxwell-Fish-Eye lenses can be used to implement a pulsed excitation-exchange between two quantum emitters. This periodic exchange is mediated by single photon pulses and can be extended to a photon-exchange between two atomic ensembles, for which the coupling strength is enhanced collectively. Our study illustrates how ideas from classical optics can be used in the realm of multimode strong coupling for applications in quantum technology.