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arXiv:0904.1804 · Physical Review A

Fano-Hopfield model and photonic band gaps for an arbitrary atomic lattice

Mauro Antezza, Yvan Castin

Published Jul 21, 2009• 53 citations
arXivHTMLPDF

Abstract

We study the light dispersion relation in a periodic ensemble of atoms at fixed positions in the Fano-Hopfield model (the atomic dipole being modeled with harmonic oscillators). Compared to earlier works, we do not restrict to cubic lattices, and we do not regularize the theory by hand but we renormalize it in a systematic way using a Gaussian cutoff in momentum space. Whereas no omnidirectional spectral gap is known for light in a Bravais atomic lattice, we find that, for a wide range of parameters, an omnidirectional gap occurs in a diamond atomic lattice, which may be realized in an experiment with ultracold atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz (or Clausius-Mossotti) relation for an arbitrary lattice.

DOI
10.1103/physreva.80.013816

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We study the light dispersion relation in a periodic ensemble of atoms at fixed positions in the Fano-Hopfield model (the atomic dipole being modeled with harmonic oscillators). Compared to earlier works, we do not restrict to cubic lattices, and we do not regularize the theory by hand but we renormalize it in a systematic way using a Gaussian cutoff in momentum space. Whereas no omnidirectional spectral gap is known for light in a Bravais atomic lattice, we find that, for a wide range of parameters, an omnidirectional gap occurs in a diamond atomic lattice, which may be realized in an experiment with ultracold atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz (or Clausius-Mossotti) relation for an arbitrary lattice.
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 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 simulators employing cold atoms are among the most promising approaches to tackle quantum many-body problems. Nanophotonic structures are widely employed to engineer the band structure of light and are thus investigated as a means to tune the interactions between atoms placed in their vicinity. A key shortcoming of this approach is that excitations can decay into free photons, limiting the coherence of such quantum simulators. Here, we overcome this challenge by proposing to use a simple cubic three-dimensional array of atoms to produce an omnidirectional band gap for light and show that it enables coherent, dissipation-free interactions between embedded impurities. We show explicitly that the band gaps persist for moderate lattice sizes and finite filling fraction, which makes this effect readily observable in experiment. Our paper paves the way toward analog spin quantum simulators with long-range interactions using ultracold atomic lattices, and is an instance of the emerging field of atomic quantum metamaterials.
We address the general problem of the excitation spectrum for light coupled to scatterers havingquantum fluctuating positions around the sites of a periodic lattice. In addition to providing an imaginarypart to the spectrum, we show that these quantum fluctuations affect the real part of the spectrum, in a waythat we determine analytically. Our predictions may be observed with ultracold atoms in an optical lattice,on a J 1⁄4 0 ! J0 1⁄4 1 narrow atomic transition. As a side result, we resolve a controversy for theoccurrence of a spectral gap in a fcc lattice.
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.
Based on the narrow linewidth ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$ transition of the bosonic 88{}^{88}88Sr atom, we investigate the dispersion relation for light propagating in a three-dimensional (3D) atomic dipolar lattice. Two factors determine the presence of the photonic band gap (PBG): lattice geometry and the atomic polarization. For a two-level atomic system, the former factor plays a major role. An omnidirectional PBG is predicted in the non-Bravais diamond lattice while no PBG exists in the Bravais lattices. However, since a PBG happens around the resonant ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$ line, the probe beam suffers a significant atomic absorption, which is the major drawback in Bragg scattering. We propose applying an external field to couple the atomic ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{D}_{1}$ transition so as to modify the atomic polarization of the ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$ transition. In the three level ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{D}_{1}$ atomic system, the atomic polarization strongly affects the band structure besides the lattice geometry. Dual PBGs can exist in simple cubic and face-centered cubic lattices, but the original PBG, in the diamond lattice composed of the two-level atoms, vanishes. Meanwhile, since two PBGs can be pushed far away from the atomic resonant ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$ line, the atomic absorption is significantly suppressed, which enhances the reflection efficiency in Bragg scattering.
This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.