Fano-Hopfield model and photonic band gaps for an arbitrary atomic lattice
📝 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 cut-off 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 ultra-cold atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz (or Clausius-Mossotti) relation for an arbitrary lattice.
💡 Analysis
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 cut-off 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 ultra-cold atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz (or Clausius-Mossotti) relation for an arbitrary lattice.
📄 Content
arXiv:0904.1804v2 [cond-mat.quant-gas] 22 Jul 2009 Fano-Hopfield model and photonic band gaps for an arbitrary atomic lattice Mauro Antezza and Yvan Castin Laboratoire Kastler Brossel, ´Ecole Normale Sup´erieure, CNRS and UPMC, 24 rue Lhomond, 75231 Paris, France (Dated: May 31, 2018) 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 cut-offin 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 ultra-cold atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz (or Clausius-Mossotti) relation for an arbitrary lattice. PACS numbers: 42.50.Ct, 67.85.-d, 71.36.+c I. INTRODUCTION The determination of the spectrum of light in a pe- riodic ensemble of atoms is a fundamental problem that still raises intriguing questions. After the seminal work of Hopfield [1] (see also Agranovich [2]), based on the Fano oscillator model for the atomic dipole [3], many theo- retical works have been performed [4]. Most of them, in- spired by the typical context of condensed matter physics considered in [1], focus on the long-wavelength limit where Lorentz-Lorenz (or Clausius-Mossotti) type rela- tions may be derived [5, 6]. Recently, this problem was extended to the whole Brillouin zone in the case of cubic lattices [7, 8], which allows to address the presence or the absence of an omnidirectional spectral gap for light. This problem of light spectrum in atomic lattices is no longer a purely theoretical issue. Recent experiments with ultracold atoms, having led to the observation of a Mott phase with one atom per lattice site [9], have indeed opened up the possibility to investigate the propagation of light in an atomic lattice, taking advantage of the large variety of optical lattices that may be realized to trap the atoms [10], all this in the regime where the lattice spacing is of the order of the optical wavelength, so that a probing of the whole Brillouin zone can be envisaged. In the photonic crystals in solid state systems, made of dielectric spatially extended objects (rather than point- like atoms), after the pioneering work of [11] for diamond lattices of dielectric spheres, many configurations are now known to lead to a spectral gap, with a variety of appli- cations to light trapping and guiding [12, 13]. On the contrary, for atomic lattices, no omnidirectional spectral gap was found, neither in cubic atomic lattices [8] nor in several less symmetric Bravais lattices [14]. Two factors determine the presence of an omnidirec- tional gap: crystal geometry and details of the light- matter scattering process, both can separately close a gap. Indeed, in photonic crystals materials, character- ized by a macroscopic modulation of the refractive in- dex, the same lattice geometry can lead or not to an omnidirectional gap depending on the modulation of the refractive index, as is the case for the simple cubic (sc) or body centered cubic (bcc) lattices [15]. Differently from photonic crystals materials, in the physics of ultracold atoms, one can realize periodic structures with a single atom per site [9] rather than a macroscopic number, re- alizing an ideal crystalline structure with a variety of possible geometries [10]. Atoms are scatterers character- ized by a strong resonant and point-like interaction with light, so that the features of the light propagation in an atomic lattice cannot be straightforwardly extrapolated from known results in solid state photonic crystals. Also in the atomic case, it is not possible, only by geometric considerations, to predict the presence of an omnidirec- tional photonic band gap, the details of the light-matter scattering process do matter [8]. Here we develop, within the Fano-Hopfield model, a self-consistent theory for the elementary excitation spec- trum of the light-atom field in atomic periodic systems, which is valid not only for cubic symmetry atomic lattices [8] but also for any Bravais lattice, and, even more, also for periodic non-Bravais lattices (i.e. for crystals with sev- eral atoms per primitive unit cell). Our theory includes the light polarizations degrees of freedom, and is based on the introduction of a Gaussian momentum cut-offwhich allows to eliminate all divergences (even in a periodic infinite system of atoms) by a systematic renormaliza- tion procedure. We then use our theory to address the existence of an omnidirectional spectral gap for light in atomic lattices. In particular, we show that the diamond atomic lattice, which is a non-Bravais lattice composed of two identi
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