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Physical Review A

Two-atom problem and polariton-impurity band in dispersive media and photonic-band-gap materials

Valery I. Rupasov, Mahi R. Singh

Published Jul 1, 1997• 25 citations
Publisher

Abstract

Quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied within the framework of the dipole resonance (rotating-wave) approximation. For finite interatomic distances a polariton-atom bound state lying within the polaritonic gap is shown to be split into a doublet due to an effective atom-atom interaction. Evaluating the splitting as a function of the interatomic separation, we estimate a width of a polariton-impurity band when a system of spatially correlated atoms is present. The results obtained for the case of two atoms are then generalized to the case of an atomic chain (a one-dimensional atomic lattice). The equations describing a propagation of a band polariton are derived and the expressions for the band polariton spectrum and its effective mass are obtained. The model under consideration has quite a general structure and is applied to the case of photonic-band-gap materials as well. Numerical calculations are presented for SiC semiconductors.

DOI
10.1103/physreva.56.898

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Quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied within the framework of the dipole resonance (rotating-wave) approximation. For finite interatomic distances a polariton-atom bound state lying within the polaritonic gap is shown to be split into a doublet due to an effective atom-atom interaction. Evaluating the splitting as a function of the interatomic separation, we estimate a width of a polariton-impurity band when a system of spatially correlated atoms is present. The results obtained for the case of two atoms are then generalized to the case of an atomic chain (a one-dimensional atomic lattice). The equations describing a propagation of a band polariton are derived and the expressions for the band polariton spectrum and its effective mass are obtained. The model under consideration has quite a general structure and is applied to the case of photonic-band-gap materials as well. Numerical calculations are presented for SiC semiconductors.
Quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. Despite a nonlocal effective polariton-polariton coupling, the system is proven to be integrable and is diagonalized exactly by means of the Bethe-ansatz technique. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits some unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons propagate within the gap, while a soliton with an odd number of polaritons is pinned to the atom and forms a many-polariton-atom bound state, in which the radiation and the medium polarization are localized in the vicinity of the atom. The model under consideration has a quite general structure and can also be applied to the case of photonic-band-gap materials. \textcopyright{} 1996 The American Physical Society.
A system of polaritons interacting with a two-level atom placed within a frequency dispersive medium is proven to be integrable and diagonalized exactly by the Bethe ansatz method, despite a nonlocal effective polariton-polariton coupling. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons (``even'' solitons) propagate within the gap, while an ``odd'' soliton is pinned to the atom and forms a many-polariton--atom bound state.
Radiative coupling between identical atoms sharing an excitation is studied in media where dispersion creates photonic band gaps, i.e., forbidden bands of light propagation for all directions and polarizations. Photonic band gaps can exist in media whose dielectric index exhibits strong three-dimensionally periodic modulations (``photonic crystals'') or in bands of polaritonic media. It is shown that the resonant dipole-dipole interaction as well as cooperative fluorescence of two-atom systems can be strongly enhanced or suppressed, to the extent of being essentially eliminated, by one of the following mechanisms: (a) dependence on location of the atoms within a unit cell of a photonic crystal, i.e., sensitivity to the spatial variation of the field normal modes; (b) dependence on the density of normal modes in allowed bands and on the density of virtual (evanescent) modes in band gaps. The ability to suppress the dipole-dipole interactions at interatomic separations characterizing quasimolecules would have far-reaching implications on their dissociative or collisional dynamics, spectroscopy, and rates of energy transfer.
A theory of coherent population trapping (CPT) has been developed in photonic band-gap (PBG) and dispersive polaritonic band-gap (DPBG) materials when doped with an ensemble of five-level atoms. These materials have gaps in their photon energy spectra leading to unusual optical properties. The atoms are prepared as coherent superpositions of the two lower states and interact with a reservoir and two photon fields. The transition between the two lower states of an atom is dipole forbidden. The Schrödinger equation and the Laplace transform method are used to calculate the expressions for the number densities of the atomic states. Numerical simulations are performed for both PBG and DPBG materials with the phase factor between the coherent states chosen such that the number density of the upper state becomes zero when the photon fields have the same intensity. It is found that when the resonance energies lie away from the band edges and within the lower band, the CPT effect is observed in both materials when the fields have identical intensities. Similar results are also found when the resonance energies lie away from the band edges and within the upper band. When one of the resonance energies lies near the lower band edge, the number density vanishes at all intensities of the fields for both materials. This is an effect of the band structure of the materials and is not due to the CPT effect. A similar result is seen when one of the resonance energies lies near the upper band edge of a PBG material. However, for a DPBG material, the number density does not become zero when one of the resonance energies lies near the upper band edge, except due to the CPT effect brought on by the identical intensities of the photon fields. This is a very interesting phenomenon.
Numerical simulations are performed on the narrowing of transient linewidths of the absorption peaks of an ensemble of four-level atoms doped in a dispersive band-gap material. The atoms do not interact with each other. They are, however, interacting with a reservoir, due to which, the top two levels of the atoms decay to a pair of lower levels. A probe laser field is applied to study the absorption characteristics of the system. The density matrix method is used to calculate the photon yield for the absorption peak in order to observe linewidth narrowing. Numerical simulations are performed for SiC which is a dispersive polaritonic band-gap material. We have observed the occurrence of linewidth narrowing in SiC when the resonance energies lie within the bands. But when the resonance energy lies near the lower band edge of the polaritonic crystal, the narrowing disappears. On the other hand, when the resonance energy lies near the upper band edge, the narrowing of the linewidth remains as pronounced as before. Moreover, we have found that the photon yield is very sensitive to the locations of the resonance energies within the band. These unusual properties can be used to make new types of photonic devices such as switches, gates, etc. Similar results are also found in photonic band-gap materials.
We study the resonance Raman scattering of light from a three-level atom embedded in a photonic band-gap material or in a frequency dispersive medium whose photon spectrum exhibits a gap due to photon coupling to medium excitations such as excitons and optical phonons. We demonstrate that the one-particle spectrum of the system consists of either a continuous part with energy lying outside the gap or a single discrete mode with energy lying inside the gap. The discrete mode, which occurs when both of the allowed atomic transitions of the $\ensuremath{\Lambda}$ configuration lie inside the gap, can be treated as a photon-atom bound state in which the radiation is localized in the vicinity of the atom. In the case of the continuous spectrum, the Rayleigh and Stokes lines are shifted as well as narrowed (or broadened) as the corresponding transition frequencies are shifted relative to the upper band edge, providing a distinctive experimental signature of atom-photon interactions near a photonic band edge.