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Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach

Spontaneous emission (SE) from a two-level atom in a photonic crystal (PC) with anisotropic one-band model is investigated using the fractional calculus. Analytically solving the kinetic equation in terms of the fractional exponential function, the dynamical discrepancy of SE between the anisotropic and isotropic systems is discussed on the basis of different photon density of states (DOS) and the existence of incoherent diffusion field that becomes even more clearly as the atomic transition frequency lies close to the band edge. With the same atom-field coupling strength and detuning in the forbidden gap, the photon-atom bound states in the isotropic system turn into the unbound ones in the anisotropic system that is consistent with the experimental observation in $Phys.$ $Rev.$ $Lett.$ \textbf{96}, 243902 (2006). Dynamics along different wavevectors with various curvatures of dispersion is also addressed with the changes of the photon DOS and the appearance of the diffusion fields.