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The Journal of Chemical Physics

Resonant excitation transfer: A quantum electrodynamical study

Davıd L. Andrews, Brad S. Sherborne

Published Apr 1, 1987• 106 citations
Publisher

Abstract

A new quantum electrodynamical (QED) method is presented, based in the Schrödinger representation, for the calculation of the rate of energy transfer between identical molecules. In contrast to existing methods in this representation, the new treatment gives explicitly causal and energy-conserving results. By returning to perturbation theory the correct, complex form for the electric dipole–electric dipole interaction tensor is obtained, without recourse to the physical, ‘‘outgoing wave’’ arguments of quantum scattering theory necessary if the Fermi rule is used. This method also allows a new interpretation for the role of the time-ordered diagrams involved, which may be useful in the rigorous treatment of higher order cooperative processes. The QED treatment uses virtual photon coupling, and incorporates both the Coulombic, R−6 dependence, and the R−2 dependence characteristic of two-step radiative transfer.

DOI
10.1063/1.451910

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A new quantum electrodynamical (QED) method is presented, based in the Schrödinger representation, for the calculation of the rate of energy transfer between identical molecules. In contrast to existing methods in this representation, the new treatment gives explicitly causal and energy-conserving results. By returning to perturbation theory the correct, complex form for the electric dipole–electric dipole interaction tensor is obtained, without recourse to the physical, ‘‘outgoing wave’’ arguments of quantum scattering theory necessary if the Fermi rule is used. This method also allows a new interpretation for the role of the time-ordered diagrams involved, which may be useful in the rigorous treatment of higher order cooperative processes. The QED treatment uses virtual photon coupling, and incorporates both the Coulombic, R−6 dependence, and the R−2 dependence characteristic of two-step radiative transfer.
We have visited the nonradiative electron transfer process between optically active ions in luminescent materials. In contrast to the earlier formulations using multipolar expansion of instantaneous coulomb interaction between electrons, we have utilized a quantum electrodynamics (QED) approach to study the process of energy transfer from an excited ion to another. Such energy transfers occur in both sensitized luminescence and cross relaxations in the excited state. We have here investigated the nonradiative energy transfer processes specifically mediated by electric dipole moment-magnetic dipole moment and magnetic electric dipole moment-magnetic dipole moment interactions using a Feynman diagrammatic method. We have shown how experimentally relevant transfer rates could be obtained using a QED formulation involving discrete energy levels and spectral line shape function for emission and absorption profiles of interactive ions. Additional features of the dependence of the energy transfer rates on the interionic distance arising from retarded interaction potential have been pointed out.
Using the electromagnetic response function of an electric dipole located within a dielectric geometry, we derive the mathematical equivalence between the classical response and quantum mechanical resonant dipole–dipole interaction between two quantum objects (atoms, quantum dots, etc). Cooperative spontaneous emission likewise emerges from this equivalence. We introduce a practical numerical technique using finite difference time domain for calculating both dipole–dipole interaction and collective spontaneous emission in confined dielectric structures, where strong light–matter coupling might arise. This method is capable of obtaining resonant dipole–dipole interaction over a wide range of frequencies in a single run. Our method recaptures the results of quantum mechanical second order perturbation theory for weak light–matter coupling. In strong coupling situations such as near a photonic band edge, second order Rayleigh–Schrödinger perturbation theory leads to divergences, and instead Brillouin–Wigner perturbation theory is required. This is equivalent to the use of a variational wavefunction to describe the exciton transfer between initial and final states. We introduce a system of coupled classical oscillators, that describes resonant dipole–dipole interaction and vacuum Rabi splitting in the strong-coupling regime, and that provides an effective numerical scheme based on the finite difference time domain method. This includes the effects of quantum entanglement and the correlation of quantum fluctuations. We discuss the crossover to Forster energy transfer when quantum correlations between the dipoles are damped by strong environmental interactions.
In this paper a quantum electrodynamical theory of cooperative Raman scattering in fluids is developed. The process is one in which pairs of molecules undergo concerted Raman transitions via an intermolecular energy transfer mechanism. The formalism is also extended to chromophore pairs in which intramolecular energy transfer is involved. Using the Power-Zienau-Woolley multipolar Hamiltonian, intensity equations of the Kramers-Heisenberg type are derived which are valid over all regions of molecular separation. In the near zone it is illustrated how the intensity of cooperative scattering is simply that obtained by consideration of the instantaneous response of one molecule to the scalar field generated by another. Within this regime it is shown that the rate of scattering has the familiar ${R}^{\mathrm{\ensuremath{-}}6}$ dependence on intermolecular separation R associated with interacting induced dipoles. In the far zone it is demonstrated that the fully retarded and causal interaction between molecules is mediated through virtual photon coupling, and that this leads to a result for the intensity that dies off with ${R}^{\mathrm{\ensuremath{-}}2}$. This long-range-interaction contribution to the overall scattering intensity is generally ignored in collisional treatments of intermolecular interactions, yet is likely to be of more significance than higher-order multipole terms in near-zone calculations. Finally it is shown how bands in a Raman spectrum due to cooperative scattering effects may be identified by their unique pressure characteristics, or by observation of light scattering that occurs outside the interaction volume of the laser beam. In the case where the scattering occurs between molecules or chromophores held in a fixed orientation with respect to one another, it is demonstrated how detection may be facilitated through the manifestation of a differential scattering effect.
In this paper it is demonstrated how the phenomena of cooperative Raman scattering and cooperative two-photon absorption may be described in a unified manner, through canonically transforming the system Hamiltonian. The unique features associated with the virtual photon coupling between centres are discussed. In particular it is illustrated how in the near-zone, close proximity leads to two-body two-photon events involving interaction-induced dipoles. In the far-zone, it is shown how the coupling between molecules may be understood in terms of a radiative dipole effect. Attention is focused upon the behaviour of this coupling for distributive and cooperative mechanisms in fluids, and the relative importance of each process discussed, particularly with regard to their very different range dependences. A detailed analysis of the coupling reveals that in two-photon absorption the distributive mechanism dominates, whilst in Raman scattering cooperative energy transfer is favoured. Finally recent progress in the field of synergism is reviewed along with the characteristics associated with synergism in Raman scattering and two-photon absorption.
In this paper quantum electrodynamics is used to investigate the effects of radiation field-induced energy transfer as a mechanism for the manifestation of frequency shift and both homogeneous and inhomogeneous line broadening. Using resolvent operator methods and the Power–Zienau–Woolley multipolar Hamiltonian to derive an effective Hamiltonian, it is illustrated how all coherent and incoherent field-induced line shift and line broadening including Förster energy transfer, dispersion, and cooperative Rayleigh scattering can be described solely in terms of a single multipolar interaction. The detailed analysis reveals that a description of intermolecular interactions in terms of virtual photon coupling leads a detailed range dependence that is not only different to nonretarded semiclassical theory, but also complex. This paper therefore illustrates for the first time that all intermolecular processes contribute to both line broadening and energy shift, and that the imaginary contributions to virtual photon coupling lead to significant retardation effects in condensed media. Finally by applying the results within a density matrix framework important aspects regarding the statistical averaging of these additional terms in the Hamiltonian are discussed, and it is illustrated how intermolecular coupling leads to non-Lorentzian, redshifted, asymmetric line profiles.
The calculation of the retarded fourth-order resonance interaction between identical molecules with one shared quantum of excitation energy is described. The interaction is calculated in the electric dipole approximation for molecular transitions with the help of time-ordered graphs and a reduced Hamiltonian which treats molecule-photon and molecule-molecule interactions on the same footing. Interactions involving spins, valence forces and permanent electric or magnetic multipoles are not considered. In the near and wave zones the leading terms of the retarded interaction are proportional to R -6 and R -2 respectively, where R is the intermolecular separation.