A remark on the isotropic model

The applicability of the so-called isotropic complete photonic-band-gap (CPBG) models [S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990)] to capture essential features of the spontaneous emission (SE) of a fluorescent atom or molecule near a band-gap-edge of a CPBG structure is discussed. PACS number: 32.80.-t Photon interactions with atoms PACS number: 42.70.Qs Photonic bandgap materials www.amolf.nl/research/photonic materials theory/moroz/moroz.html [email protected] present address In their influential article [1], John and Wang considered the quantum electrodynamics (QED) of an atom, minimally coupled to the radiation filed, in the presence of a complete photonic-band-gap (CPBG). In the latter case, there is a frequency interval in which, independently of the photon direction and polarization, no photon modes can propagate. Such a QED vacuum differs significantly from the conventional QED vacuum. In the vicinity of a CPBG edge ωc a number of exotic phenomena were predicted, among others, radical changes in the spontaneous emission (SE) and an anomalous Lamb shift. (It is worthwhile to notice that some of the exotic phenomena have been discussed much earlier in less known papers by Bykov [2].) Atom properties were shown to depend strongly on the exponent η of the density of states (DOS) asymptotic ρ(ω) ≈ const |ω − ωc| η (1) near the CPBG edge. Calculations involving photonic crystals in more than one dimension (1D) are notoriously difficult. Therefore, in order to determine η, John and Wang made use of approximations, subsequently employed in a number of recent discussions of the SE near a CPBG edge [3], and often called isotropic and anisotropic CPBG models. In the first model, the DOS near the band edge ωc is obtained from an approximated dispersion relation of the CPBG material ωk ≈ ωc+A(k−k0) , where A ≈ ωc/k 2 0 and k0 is a vector at the Brillouin zone boundary. The second model is then a slight generalization of the first one. However, already in the Wigner-Weisskopf approximation the SE of an isolated fluorescent atom (or molecule) in a fixed position r within the unit cell is determined by the local DOS (LDOS) and not the DOS [4]. Considerations based on the DOS can only be valid in the two hypothetical cases which are difficult to achieve: (i) when the atom is allowed to freely propagate within a CPBG structure, as in experiments with cavity QED, and (ii) when atoms are distributed homogeneously within the entire unit cell. Only then the averaged and not the local properties of the QED vacuum within the unit cell are probed. The fluorescent atoms are usually not distributed uniformly. If the atoms can be considered as independent and are only radiatively coupled to the crystal, the measured SE is determined by the weighted average (with the atomic position probability distribution) of the LDOS. In contrast to the DOS, the LDOS behavior near a CPBG edge shows more complex