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arXiv:1106.1077 · Journal of Physics B Atomic Molecular and Optical Physics

Collective electronic excitation coupling between planar optical lattices using Ewald's method

Hashem Zoubi, Helmut Ritsch

Published Sep 26, 2011• 5 citations
arXivHTMLPDF

Abstract

Using Ewald's summation method, we investigate collective electronic excitations (excitons) of ultracold atoms in parallel planar optical lattices including long-range interactions. The exciton dispersion relation can then be suitably rewritten and efficiently calculated for long-range resonance dipole–dipole interactions. Such in-plane excitons resonantly couple for two identical optical lattices, with an energy transfer strength decreasing exponentially with the distance between the lattices. This allows a restriction of the transfer to neighbouring planes and gives rise to excitons delocalized between the lattices. In general, equivalent results will hold for any planar system containing lattice layers of optically active and dipolar materials.

DOI
10.1088/0953-4075/44/20/205303

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Using Ewald's summation method, we investigate collective electronic excitations (excitons) of ultracold atoms in parallel planar optical lattices including long-range interactions. The exciton dispersion relation can then be suitably rewritten and efficiently calculated for long-range resonance dipole–dipole interactions. Such in-plane excitons resonantly couple for two identical optical lattices, with an energy transfer strength decreasing exponentially with the distance between the lattices. This allows a restriction of the transfer to neighbouring planes and gives rise to excitons delocalized between the lattices. In general, equivalent results will hold for any planar system containing lattice layers of optically active and dipolar materials.
Collective electronic excitations in planar optical lattices exhibit strong modifications of the radiative damping rate and directional emission pattern as compared to a single excited atom. Collective excitations for long wave numbers and polarizations orthogonal to the lattice plane exhibit superradiance with a very short lifetime and a tightly confined emission direction. For shorter wavelength and in-plane polarization, they can possess a long lifetime, which, beyond a critical wave number, tends to infinity. Those excitations thus become metastable and decoupled from the free radiation field. They can store a single energy quantum for a long time and transfer the excitation over long distances. In general the spatial, polarization, and frequency dependence of the emission pattern can provide us with optical and electronic properties of optical lattices.
Ultracold atoms uniformly filling an optical lattice can be treated like an artificial crystal. An implementation including the atomic occupation of a single excited atomic state can be represented by a two-component Bose-Hubbard model. Its phase diagram exhibits a quantum phase transition from a superfluid to a Mott insulator phase. The dynamics of electronic excitations governed by electrostatic dipole-dipole interactions in the ordered regime can be well described by wave-like collective excitations called excitons. Here we present an extensive study of such excitons for a wide range of geometries and dimensionality. Their lifetimes can vary over many orders of magnitude from metastable propagation to superradiant decay. Particularly strong effects occur in one dimensional atomic chains coupled to tapered optical fibers. For an optical lattice within a cavity the excitons are coupled to cavity photons and the resulting collective cavity QED model can be efficiently formulated in terms of polaritons. Their properties are explicitly calculated for different lattices and they constitute a non-destructive monitoring tool for important system properties. Even the formation of molecules in optical lattices manifests itself in modified polariton properties as e.g. an anisotropic optical spectrum. Partial dissipation of the exciton energy in the lattice leads to heating, which can be microscopically understood through a mechanism transferring atoms into higher Bloch bands via a resonant excitation transfer among neighboring lattice sites. The presence of lattice defects like vacancies in the Mott insulator induces a characteristic scattering of polaritons, which can be optically observed to monitor the lattice integrity. Our models can be applied to simulate and understand corresponding collective phenomena in solid crystals, where many effects are often masked by noise and disorder.
We study electronic excitations of a degenerate gas of atoms trapped in pairs in an optical lattice. Local dipole-dipole interactions produce a long-lived antisymmetric and a short-lived symmetric superposition of individual atomic excitations as the lowest internal on-site excitations. Due to the much larger dipole moment the symmetric states couple efficiently to neighbouring lattice sites and can be well represented by Frenkel excitons, while the antisymmetric dark states stay localized. Within a cavity only symmetric states couple to cavity photons inducing long-range interactions to form polaritons. We calculate their dispersion curves as well as cavity transmission and reflection spectra to observe them. For a lattice with aspherical sites bright and dark states get mixed and their relative excitation energies depend on photon polarizations.
We study ultracold atoms trapped in a one-dimensional optical lattice prepared in a Mott insulator state of finite extend and collectively coupled to a single cavity mode. Due to resonant dipole-dipole interactions among the atoms, electronic excitations get delocalized and form excitons . These excitons can be explicitly calculated and divided into two groups: antisymmetric modes which decouple from the cavity mode called dark excitons , and symmetric modes coupled to the cavity mode called bright excitons . In a typical geometry the most uniform exciton is coupled to the cavity photons much stronger than other symmetric bright states and dominates the optical response of the atoms ( superradiant state ). In the strong coupling regime this superradiant state mixes with a cavity photon to form a doublet of polariton states, and the other excitons play only a minor role in the dynamics. We analytically calculate the corresponding collective Rabi splitting including the nearest-neighbor dipole-dipole induced excitonic shifts, which strongly depend on the polarization of the cavity mode with respect to the lattice orientation.
We study the resonant electronic excitation dynamics for ultracold atoms trapped in a deep optical lattice prepared in a Mott-insulator state. Excitons in these artificial crystals are similar to Frenkel excitons in noble atom or molecular crystals. They appear when the atomic excited state linewidth is smaller than the exciton bandwidth generated by dipole-dipole coupling. When the atoms are placed within a cavity the electronic excitations and the quantized cavity mode get coupled. In the collective strong coupling regime excitations form two branches of cavity polaritons with Rabi splitting larger than the atomic and the cavity linewidth. To demonstrate their properties we calculate the transmission, reflection, and absorption spectra for an incident weak probe field, which show resonances at the polariton frequencies.
In a recent paper [J. K. Asboth et al., Phys. Rev. Lett. 98, 203008 (2007)] we have shown that traveling density wavelike collective oscillations can arise in an asymmetrically pumped optical lattice, and by increasing the lattice size or pump asymmetry, these waves can destabilize the structure even in the overdamped limit. The long-range interaction giving rise to collective motion stems from the back-action of the atoms on the field creating the lattice. In this paper we present a detailed description of these phenomena. We derive the force on a disk-shaped cloud of trapped particles including the back-action on the trapping light, and analyze its relation to the standard perturbative approach giving the ``dipole force'' and ``radiation pressure.'' We calculate the self-consistent lattice constant for both red and blue detuned lattices and find that it decreases---by the same amount in the two cases---as the pump asymmetry is increased. We present the detailed derivation of the lattice vibration eigenmodes using the transfer matrix method, which reveals that the instability is enhanced resonantly at certain settings of the asymmetry.