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Physical Review A

Photonic band structure of the three-dimensional 88Sr atomic lattice

Deshui Yu

Published Oct 20, 2011• 10 citations
Publisher

Abstract

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 ${}^{88}$Sr 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.

DOI
10.1103/physreva.84.043833

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In this paper, the probe absorption spectrum of an atom in a double-band photonic crystal have been studied. In the modes, we assume that one of the two atomic transitions in a ?-type atomic system is interacting with free vacuum modes and another transition is interacting with free vacuum modes, isotropic photonic band gap (PBG) modes and anisotropic PBG modes, separately. The effects of the fine structure of the atomic lower levels on the probe absorption spectrum are investigated in detail in the three cases. The most interesting thing is that the two (four) transparencies at one (two) probe absorption peak(s), caused by the fine structure of the lower levels of an atom, are predicted in the case of isotropic PBG modes.
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.
A calculation of the optical band structure of a three dimensional lattice of resonant two-level atoms in the dipole approximation is presented. The formation of band gaps is exhibited and confirmed by a calculation of the density of states. The band structure can be characterized by two dimensionless parameters. We find a longitudinal polarization mode as well as a class of vacuum modes that are unaltered by the interaction with matter. Numerical calculations are performed for a face centered cubic lattice; other lattices can be evaluated as easily.
We introduce the `decay rate' terms into the density matrix equations of an atom embedded in a photonic band gap (PBG) reservoir successfully. By utilizing the master equations, the probe absorption spectra and the refractivity properties of a three-level atom in the PBG reservoir are obtained. The interaction between the atom and the PBG reservoir as well as the effects of the quantum interference on the absorption of the atom has also been taken into account. It is interesting that two different types of the anomalous dispersion relations of refractivity are exhibited in one dispersion line. The methodology used here can be applied to theoretical investigation of quantum interference effects of other atomic models embedded in a PBG reservoir.
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 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.