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Physical Review Letters

Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms

Sajeev John, Jian Wang

Published May 14, 1990• 764 citations
Publisher

Abstract

It is shown that in dielectrics exhibiting a complete photonic band gap, quantum electrodynamics predicts the occurrence of bound states of photons to hydrogenic atoms. When the atomic transition frequency lies near a photonic band edge, the excited atomic level experiences an anomalous Lamb shift and splits into a doublet. One member of this doublet exhibits resonance fluorescence whereas the other level is dressed by the emission and reabsorption of near-resonant photons whose amplitude decays exponentially from the vicinity of the atom.

DOI
10.1103/physrevlett.64.2418

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It is shown that in dielectrics exhibiting a complete photonic band gap, quantum electrodynamics predicts the occurrence of bound states of photons to hydrogenic atoms. When the atomic transition frequency lies near a photonic band edge, the excited atomic level experiences an anomalous Lamb shift and splits into a doublet. One member of this doublet exhibits resonance fluorescence whereas the other level is dressed by the emission and reabsorption of near-resonant photons whose amplitude decays exponentially from the vicinity of the atom.
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}$(a0{\mathit{a}}_{0}a0​/R)8{)}^{8})8, where a0{\mathit{a}}_{0}a0​ is the atomic Bohr radius and \ensuremath{\alpha} is the fine-structure constant.
We present a comprehensive quantum electrodynamical analysis of the interaction between a continuum with photonic band gaps (PBGs) or frequency cut-off and an excited two-level atom, which can be either ‘bare’ or ‘dressed’ by coupling to a near-resonant field mode. A diversity of novel features in the atom and field dynamics is shown to arise from the non-Markovian character of radiative decay into such a continuum of modes. Firstly the excited atom is shown to evolve, by spontaneous decay, into a superposition of non-decaying single-photon dressed states, each having an energy in a different PBG, and a decaying component. This superposition is determined by the atomic resonance shift, induced by the spontaneously emitted photon, into or out of a PBG. The main novel feature exhibited by the decaying excited-state component is the occurrence of beats between the shifted atomic resonance frequency and the PBG cut-off frequencies, corresponding to a non-Lorentzian emission spectrum. Secondly the induced decay of a resonantly driven atom into such a continuum exhibits a cascade of transitions down the ladder of dressed states, which are labelled by decreasing photon numbers of the driving mode. Remarkably, this cascade is terminated at the dressed-state doublet, from which all subsequent transitions to lower doublets are forbidden because they fall within the PBG. This doublet then becomes an attractor state for the populations of higher-lying doublets. As a result, the photon-number distribution of the driving mode becomes strongly sub-Poissonian.
We study the resonance Raman scattering of light from a three-level atom embedded in a photonic band-gap material or in a frequency dispersive medium whose photon spectrum exhibits a gap due to photon coupling to medium excitations such as excitons and optical phonons. We demonstrate that the one-particle spectrum of the system consists of either a continuous part with energy lying outside the gap or a single discrete mode with energy lying inside the gap. The discrete mode, which occurs when both of the allowed atomic transitions of the $\ensuremath{\Lambda}$ configuration lie inside the gap, can be treated as a photon-atom bound state in which the radiation is localized in the vicinity of the atom. In the case of the continuous spectrum, the Rayleigh and Stokes lines are shifted as well as narrowed (or broadened) as the corresponding transition frequencies are shifted relative to the upper band edge, providing a distinctive experimental signature of atom-photon interactions near a photonic band edge.
We studied the effects of a coherent monochromatic resonant laser field driving the transition of a two-level atom embedded in a photonic band gap (PBG) material on the emission dynamics of the atom. When the transition frequency of the atom lies outside a PBG and sufficiently far from the band edge that the emission dynamics of the atom is not normally affected by the gap, the dynamic stark splitting of the upper level of the atom into two dressed states by a sufficiently strong resonant driving field can bring the lower dressed state close enough to the band edge or even inside the gap so as to be affected by the gap, providing a switchable means of extending the novel effects of atom–photon interaction near the edge of a PBG for atomic transitions outside the gap and far from the band edge. The net effect is as if a sufficiently strong resonant field driving a transition of an atom embedded in a PBG actively shifts the PBG close enough to the transition so that the gap affects the emission dynamics associated with the transition.
We study spontaneous emission near the edge of a photonic band gap. Instead of a simple exponential decay in the vacuum, spontaneous emission displays an oscillatory behavior. A single photon-atom bound dressed state exhibits a fractional steady-state atomic population on the excited state. For a three-level atom we evaluate the spectral splitting and subnatural linewidth of spontaneous emission. In the presence of N-1 unexcited atoms we show that the collective time scale factor is equal to ${\mathit{N}}^{\mathrm{\ensuremath{\varphi}}}$, where \ensuremath{\varphi}=2/3 for an isotropic band gap and \ensuremath{\varphi}=1 or 2 for anisotropic two-dimensional or three-dimensional band edges, respectively.