Fractional relaxations in photonic crystals

A quantum dot interacting with the radiation field of a photonic crystal is considered. An analytical description of the dynamics and the coherence between the two states of the quantum dot is provided. Besides the well-known trapping, a fractional nature of the dynamics appears via relaxations of the Mittag-Leffler type. Furthermore, coherence exhibits a transition from the decay t(-3/2) to t(-1/2) if the transition frequency of the quantum dot is exactly in the middle of the band gap. Similarly, the population of the excited level undergoes a transition from the relaxation 1/t(3) to 1/t. These resonances and transitions belong also to the context of matter-wave emissions in optical lattices.