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arXiv:0710.5448 · New Journal of Physics

Exciton–polariton scattering as signature of defects in cold atom optical lattices

Hashem Zoubi, Helmut Ritsch

Published Feb 6, 2008• 13 citations
arXivHTMLPDF

Abstract

We study the effect of defects in the Mott insulator phase of ultracold atoms in an optical lattice on the dynamics of resonant electronic excitations. These excitations can be described analogous to Frenkel excitons in solids and form polaritons, when coupled to an optical resonator. Defects, which are empty sites in a singly occupied Mott insulator state or singly occupied sites for a filling factor two, change the exciton dynamics. We show that vacancies behave like hard sphere scatterers, while singly occupied sites in a doubly filled region generate either attractive or repulsive interaction potentials. We suggest cavity polaritons as tools which detect such defects and show how the scattering can be controlled by changing the exciton–photon detuning. In the case of aspherical individual lattice sites, we calculate the effective scattering potential as a function of the cavity photon polarization direction which exhibits a crossover from a repulsive into an attractive potential and should give a clearly observable signal.

DOI
10.1088/1367-2630/10/2/023001

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We study the effect of defects in the Mott insulator phase of ultracold atoms in an optical lattice on the dynamics of resonant electronic excitations. These excitations can be described analogous to Frenkel excitons in solids and form polaritons, when coupled to an optical resonator. Defects, which are empty sites in a singly occupied Mott insulator state or singly occupied sites for a filling factor two, change the exciton dynamics. We show that vacancies behave like hard sphere scatterers, while singly occupied sites in a doubly filled region generate either attractive or repulsive interaction potentials. We suggest cavity polaritons as tools which detect such defects and show how the scattering can be controlled by changing the exciton–photon detuning. In the case of aspherical individual lattice sites, we calculate the effective scattering potential as a function of the cavity photon polarization direction which exhibits a crossover from a repulsive into an attractive potential and should give a clearly observable signal.
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 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.
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 derive an effective Bose-Hubbard model that predicts a phase transition from Bose-Einstein condensate to Mott insulator in two different systems subject to applied periodic potentials: microcavity exciton polaritons and indirect excitons. Starting from a microscopic Hamiltonian of electrons and holes, we derive an effective Bose-Hubbard model for both systems and evaluate the on-site Coulomb interaction UUU and hopping transition amplitudes ttt. Experimental parameters required for observing a phase transition between a Bose-Einstein condensate and a Mott insulator are discussed. Our results suggest that strong periodic potentials and polaritons with a very large excitonic component are required for observing the phase transition. The form of the indirect exciton interaction is derived including direct and exchange components of the Coulomb interaction. For indirect excitons, the system crosses over from a Bose-Hubbard model into a double layer Fermi-Hubbard model as a function of increasing bilayer separation. The Fermi-Hubbard model parameters are calculated, and the criteria for the location of this crossover are derived. We conjecture that a crossover between a Bose Mott insulator to a Fermi Mott insulator should occur with increasing bilayer separation.
We study the coupling of resonant electronic excitations to vibrational center-of-mass motion of ultracold atoms in an optical lattice, when ground- and excited-state optical potentials differ. On-site excitation-vibration coupling can be approximately treated by renormalizing the effective transition energy, which appears, e.g., in lattice clock setups. In addition the dipole-dipole coupling between neighboring sites mediates emission and absorption of vibrational quanta in conjunction with optical transitions. This mechanism for excitation of vibrations can lead to significant heating of the lattice atoms motion during resonant light interaction. We calculate estimates of the corresponding parameters for a gas in the Mott-insulator state from a perturbation expansion in small atomic displacements.