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arXiv:cond-mat/0607277 · Physical Review B

Quantum theory of a resonant photonic crystal

Y. D. Chong, David E. Pritchard, Marin Soljacic A

Published Jun 27, 2007• 13 citations
arXivHTMLPDF

Abstract

We present a quantum model of two-level atoms localized in a three-dimensional lattice based on the Hopfield polariton theory. In addition to a polaritonic gap at the excitation energy, a photonic band gap opens up at the Brillouin zone boundary. Upon tuning the lattice period or angle of incidence to match the photonic gap with the excitation energy, one obtains a combined polaritonic and photonic gap as a generalization of Rabi splitting. For typical experimental parameters, the size of the combined gap is on the order of $25\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, up to ${10}^{5}$ times the detuned gap size. The dispersion curve contains a branch supporting slow-light modes with vanishing probability density of atomic excitations.

DOI
10.1103/physrevb.75.235124

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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.

Physical review. B, Condensed matter•Jun 1, 1991•395 citations
We present a quantum model of two-level atoms localized in a three-dimensional lattice based on the Hopfield polariton theory. In addition to a polaritonic gap at the excitation energy, a photonic band gap opens up at the Brillouin zone boundary. Upon tuning the lattice period or angle of incidence to match the photonic gap with the excitation energy, one obtains a combined polaritonic and photonic gap as a generalization of Rabi splitting. For typical experimental parameters, the size of the combined gap is on the order of 25cm\ensuremath−125\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}25cm\ensuremath−1, up to 105{10}^{5}105 times the detuned gap size. The dispersion curve contains a branch supporting slow-light modes with vanishing probability density of atomic excitations.
Quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied within the framework of the dipole resonance (rotating-wave) approximation. For finite interatomic distances a polariton-atom bound state lying within the polaritonic gap is shown to be split into a doublet due to an effective atom-atom interaction. Evaluating the splitting as a function of the interatomic separation, we estimate a width of a polariton-impurity band when a system of spatially correlated atoms is present. The results obtained for the case of two atoms are then generalized to the case of an atomic chain (a one-dimensional atomic lattice). The equations describing a propagation of a band polariton are derived and the expressions for the band polariton spectrum and its effective mass are obtained. The model under consideration has quite a general structure and is applied to the case of photonic-band-gap materials as well. Numerical calculations are presented for SiC semiconductors.
Radiative coupling between identical atoms sharing an excitation is studied in media where dispersion creates photonic band gaps, i.e., forbidden bands of light propagation for all directions and polarizations. Photonic band gaps can exist in media whose dielectric index exhibits strong three-dimensionally periodic modulations (``photonic crystals'') or in bands of polaritonic media. It is shown that the resonant dipole-dipole interaction as well as cooperative fluorescence of two-atom systems can be strongly enhanced or suppressed, to the extent of being essentially eliminated, by one of the following mechanisms: (a) dependence on location of the atoms within a unit cell of a photonic crystal, i.e., sensitivity to the spatial variation of the field normal modes; (b) dependence on the density of normal modes in allowed bands and on the density of virtual (evanescent) modes in band gaps. The ability to suppress the dipole-dipole interactions at interatomic separations characterizing quasimolecules would have far-reaching implications on their dissociative or collisional dynamics, spectroscopy, and rates of energy transfer.
Introduction The propagation of light in periodically ordered photonic crystals bears a strong analogy to the wave propagation of conduction electrons in atomic crystals. The development of Bloch modes and dispersion are determined by the interference of waves that are diffracted by lattice planes [19, 52]. The periodicity commensurate with the wavelength (a ≈ λ/2) gives rise to Bragg diffraction that is associated with frequency windows for which waves are forbidden to propagate in a particular direction. Such stop gaps have long been known to arise for electromagnetic waves, notably for X-rays in atomic crystals [214]. At optical frequencies, the stop gaps that occur in one-dimensional periodic structures, known as Bragg stacks, are widely used in optical laboratories as broadband highly reflecting dielectric mirrors [556]. The distinguishing feature of three-dimensional (3D) photonic crystals is that a common stop gap can be achieved for all directions and for all polarizations simultaneously: the widely pursued 3D photonic band gap . Historically a 3D band gap has never been considered for the propagation of X-rays in periodic media, since a gap requires a high refractive index contrast of order unity, whereas the refractive index varies by less then 10 −4 in the X-ray range. At frequencies inside the 3D photonic band gap, the density of optical states (DOS) completely vanishes. As the density of states can also be interpreted as the density of vacuum fluctuations [334], a 3D band gap thus serves as an effective shield to these fluctuations. The total absence of optical modes in a photonic band gap has implications beyond classical optics that break the analogy between the behavior of light in photonic crystals and the behavior of electrons in atomic crystals. 3D photonic band gap crystals play an important role in cavity quantum electrodynamics (cQED) [64, 549], where they offer at least five prospects for new physics. Firstly, probably the most eagerly pursued phenomenon is the complete inhibition of spontaneous emission: an excited quantum emitter – such as an atom, molecule, or quantum dot – embedded in a crystal with its transition frequency tuned to within the 3D band gap remains forever excited since it cannot decay to the ground state by emitting a photon.
We study the interaction of a Bose-Einstein condensate in an optical lattice with an additional largely detuned light field propagating through it. If the condensate is in its ground state it acts as a periodic dielectric, and gives rise to photonic band gaps at optical frequencies. The band structure of the combined system of condensed lattice atoms and photons is studied by using the concept of polaritons. If elementary excitations of the condensate are present, they will produce defect states inside the photonic band gaps. The frequency of localized defect states is calculated using the Koster-Slater model.
Quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. Despite a nonlocal effective polariton-polariton coupling, the system is proven to be integrable and is diagonalized exactly by means of the Bethe-ansatz technique. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits some unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons propagate within the gap, while a soliton with an odd number of polaritons is pinned to the atom and forms a many-polariton-atom bound state, in which the radiation and the medium polarization are localized in the vicinity of the atom. The model under consideration has a quite general structure and can also be applied to the case of photonic-band-gap materials. \textcopyright{} 1996 The American Physical Society.
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.