Photonic band gaps and defect states in periodic condensates
Photonic band gaps do appear if light is propagating through a periodic dielectric medium. The spectrum then contains characteristic gaps, i.e., frequencies at which the light is not allowed to propagate. This is due to the interaction with the medium and is quite similar to the case of electrons in a crystal.
A Bose-Einstein condensate can be made periodic by placing it in a periodic potential like that of an optical lattice. As the atoms in such a periodic condensate provide a kind of dielectric one can expect that photonic band gaps will appear, too.
Nevertheless, the case of a condensate is something special because of the coherence of the atoms. The latter produces a particular interaction with the light field that leads to the formation of polaritons, i.e., coherent superpositions of excited atoms and photons. Thus, in a condensate we rather deal with polariton band-gaps instead of photonic band gaps.
We have shown that the structure of a polaritonic band gap is quite different from that of an ordinary dielectric. Even in a homogeneous condensate the polariton spectrum contains avoided crossings. In a periodic condensate two avoided crossings are combined to form a band gap.
The following applet illustrates how the interaction between photons and excited atoms leads to avoided crossings and band gaps in the polariton spectrum.
Red lines correspond to photons, blue lines to excited atoms. A superposition is indicated by violet. If the BEC is homogeneous (right scrollbar in the upper position) avoided crossings do occur. Shown are polariton spectra for momentum k (solid lines) and -k (dashed lines). In a periodic BEC these two avoided crossings are combined to form band gaps.
We have also studied the case that the BEC is not perfectly periodic but has a bump at some position. This leads to the formation of defect states which are identified by the appearance of a sharp frequency inside the band gap. Using the Koster-Slater model we have shown that the distance of this frequency to the band edge is in the order of 102 to 104 Hz.