PHYSICAL REVIEW B VOLUME 43, NUMBER 16 Quantum optics of localized light in a photonic band gap 1 JUNE 1991 Sajeev John and Jian Wang Department of Physics, Uniuersity of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7 (Received 21 September 1990) We describe the quantum electrodynamics of photons interacting with hydrogenic atoms and molecules in a class of strongly scattering dielectric materials. These dielectrics consist of an or- dered or nearly ordered array of spherical scatterers with real positive refractive index and exhibit a complete photonic band gap or pseudogap for all directions of electromagnetic propagation. For hydrogenic atoms with a transition frequency in the forbidden optical gap, we demonstrate both the existence and stability of a photon-atom bound state. For a band gap to center frequency ratio Aco/too-5%, the photon localization length g~„) 10L, where L is the lattice constant of dielectric array. This strong self-dressing of the atom by its own localized radiation field leads to anomalous Lamb shifts and a splitting of the excited atomic level into a doublet when the transition frequency lies near a photonic band edge. We estimate the magnitude of this splitting to be 10 at the vacu- um transition energies. The stability of this photon-bound state with respect to electromagnetic as well as vibrational decay mechanisms is examined. For an isolated molecule embedded in the solid fraction of the dielectric structure, the dominant mechanism for absorption and spontaneous emis- sion is via optically driven electron-phonon interactions and the resulting phonon-absorption and -emission sidebands. Raman or Brillouin scattering of a localized photon into a propagating mode, or vice versa, directly by photon-phonon interaction is forbidden. For atoms not in contact with the solid fraction of the dielectric host, the electromagnetic two-photon spontaneous emission rate is on the scale of several days. For two identical atoms separated by a distance R within the photonic band gap, energy transfer from an excited atom to an unexcited atom occurs by a phase-shifted reso- nance dipole-dipole interaction which vanishes exponentially for R ) g~„. This leads to the forma- tion of a narrow photonic impurity band within the gap when a finite density of atoms is present. This impurity band exhibits a difterent kind of nonlinear optical properties. When two neighboring atoms are both excited, single-photon spontaneous emission at -2%co occurs by a third-order elec- tromagnetic process with rate I -a'coo(ao/R)', where ao is the atomic Bohr radius and e is the fine-structure constant. I. INTRODUCTION Recent experimental investigations' of electromag- netic propagation in strongly scattering dielectric materi- als have demonstrated the observability of the strong 1o- calization of photons in analogy with the Anderson localization of electrons in disordered solids. These dielectric materials constitute the optical analog of elec- tronic semiconductors. The spherical dielectric scatter- ers have a size comparable to the wavelength of light, and the associated Mie resonances play an analogous role to the electronic energy levels in an atom. For a 1arge col- lection of spheres, coherent multiple scattering gives rise to the phenomenon of weak localization. ' When strong spatial correlations exist between individual scatterers, the phase space available to the multiply scat- tered wave can be severely reduced. For electrons in a periodic solid, this gives rise to electronic band structure. Likewise for optical waves, correlations among optically connected spheres lead to a suppression of the photon density of states over a narrow frequency range. This suppression is most dramatic at sphere densities for which there is a synergetic interplay between single- scattering Mie resonances and macroscopic Brag g scattering. The resulting photonic band gap has been ex- perimentally observed by Yablonovitch and emitter in the microwave-frequency regime. For a lossless, periodic dielectric material with refractive index 3. 5 containing a fcc lattice of spherical air cavities, an almost complete photonic band gap has been observed when the volume- filling fraction of the high-dielectric material is approxi- mately 0. 15. For a frequency range spanning about 6%%uo of the gap center frequency, propagating electromagnetic modes are absent in all but a few directions. Some very recent studies have suggested that the use of nonspherical scatterers or lattices with a nontrivial basis might be even more effective in producing a complete photonic band gap rather than a pseudogap. ' Band- structure calculations have shown that a complete gap may be produced with refractive index n ~ 2. 0 using a di- amond lattice structure. Furthermore, Yablonovitch and his collaborators have found a complete microwave band gap of width about 20%%uo of the gap-center frequency us- ing cylindrical rather than spherical dielectric micro- structures. The existence of such a photonic band gap in a period- ic dielectric or the related strong localization pseudogap in the corresponding disordered dielectric have important consequences both at the classical and quantum- mechanical levels. In this paper we present a detailed in- vestigation of the quantum-electrodynamic interaction of localized light with atoms and molecules placed within the dielectric medium. The photon-atom bound state predicted previously' is examined by means of two 43 12 772 1991 The American Physical Society