Quantum optics of localized light in a photonic band gap

Abstract

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 ordered 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 \ensuremath{\Delta}\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\sim}5%, the photon localization length ${\ensuremath{\xi}}_{\mathrm{loc}}$\ensuremath{\ge}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 ${10}^{\mathrm{\ensuremath{-}}6}$ at the vacuum 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 emission 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 resonance dipole-dipole interaction which vanishes exponentially for R>${\ensuremath{\xi}}_{\mathrm{loc}}$. This leads to the formation of a narrow photonic impurity band within the gap when a finite density of atoms is present. This impurity band exhibits a different kind of nonlinear optical properties. When two neighboring atoms are both excited, single-photon spontaneous emission at \ensuremath{\sim}2\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{0}$ occurs by a third-order electromagnetic process with rate \ensuremath{\Gamma}\ensuremath{\sim}${\mathrm{\ensuremath{\alpha}}}^{3}$${\mathrm{\ensuremath{\omega}}}_{0}$(${\mathit{a}}_{0}$/R${)}^{8}$, where ${\mathit{a}}_{0}$ is the atomic Bohr radius and \ensuremath{\alpha} is the fine-structure constant.