Spontaneous emission near the band edge of a three-dimensional photonic crystal: afractional calculus approach

Abstract

We suggest a better mathematical method, fractional calculus, for studying the behavior ofthe atom–field interaction in photonic crystals. By studying the spontaneous emission of anatom in a photonic crystal with a one-band isotropic model, we found that the long-timeinducing memory of the spontaneous emission is a fractional phenomenon. This behaviorcould be well described by fractional calculus. The results show no steady photon–atombound state for the atomic resonant transition frequency lying in the proximity of theallowed band edge which was encountered in a previous study (Woldeyohannes and John2003 J. Opt. B: Quantum Semiclass. Opt. 5 R43). The correctness of this result is validatedby the ‘cut-off smoothing’ density of photon states (DOS) with fractional calculus. Byobtaining a rigorous solution without the multiple-valued problem for the system,we show that the method of fractional calculus has a logically concise property.