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arXiv:0805.2134 · Physical Review A

Comparison of quantum and classical local-field effects on two-level atoms in a dielectric

Michael E. Crenshaw

Published Nov 26, 2008• 37 citations

Abstract

The macroscopic quantum theory of the electromagnetic field in a dielectric medium interacting with a dense collection of embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-field condition. Specifically, macroscopic quantum electrodynamics predicts that the Lorentz redshift of the resonance frequency of the atoms will be enhanced by a factor of the refractive index $n$ of the host medium. However, an enhancement factor of $({n}^{2}+2)∕3$ is derived using the Bloembergen procedure in which the classical Lorentz local-field condition is applied to the optical Bloch equations. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. Microscopic quantum electrodynamics confirms the classical local-field-based results. Then the application of macroscopic quantum electrodynamic theory to embedded atoms is proved false by a specific example in which both the correspondence principle and microscopic theory of quantum electrodynamics are violated.

DOI: 10.1103/physreva.78.053827

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The macroscopic quantum theory of the electromagnetic field in a dielectric medium interacting with a dense collection of embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-field condition. Specifically, macroscopic quantum electrodynamics predicts that the Lorentz redshift of the resonance frequency of the atoms will be enhanced by a factor of the refractive index

n

of the host medium. However, an enhancement factor of

({n}^{2}+2)∕3

is derived using the Bloembergen procedure in which the classical Lorentz local-field condition is applied to the optical Bloch equations. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. Microscopic quantum electrodynamics confirms the classical local-field-based results. Then the application of macroscopic quantum electrodynamic theory to embedded atoms is proved false by a specific example in which both the correspondence principle and microscopic theory of quantum electrodynamics are violated.

Summary form only given. The dynamics of a medium consisting of a dense collection of two-level atoms interacting via the electromagnetic field has been derived from first principles. In dense media and condensed matter, the microscopic local field that couples with a single atom is related to the macroscopic field and volume propagation by the Lorentz-Lorenz relation written for an isotropic homogeneous medium. The dynamical equations for the medium, coupled self-consistently with the Maxwell inhomogeneous wave equation, are given in the rotating wave and dipole coupling approximations. The microscopic development is used within the framework of full quantization, and the explicit effects of spontaneous relaxation for the host as well as the atoms are treated, together with corresponding fluctuations. Particular effects upon intrinsic adiabatic inversion, self-induced transparency, and lasing without inversion, are presented and discussed. In particular, the equilibrium, stationary, and dynamical effects of the host medium, and medium and field fluctuations upon local field effects are discussed.

The recent first-principles derivation of Langevin-Bloch equations of motion for near-resonant atoms embedded in a linear dielectric host leads to a clear, direct physical reinterpretation of fundamental physical principles as applied to one of the most basic problems of quantum mechanics: the renormalization of the spontaneous emission rate of an atom in a dielectric host. Here we show the role of the Lorentz local field, the fluctuation-dissipation theorem, the wave number and wave vector, the density of states, Fermi's golden rule, the electromagnetic energy density, canonical quantization, and the classical constitutive relations in this context.

In the frames of quantum electrodynamics, the behavior of spontaneous decay is investigated for a single two-level atom embedded into a microscopic object (cluster). The transition frequency of cluster atoms is supposed to be considerably detuned from the transition frequency of decaying atom. The consideration is performed on the base of fully microscopic quantum-electrodynamical approach with simultaneous solution of Heisenberg equations for atomic and quantum field operators. The special attention is paid to the investigation of renormalization of spontaneous emission rate due to proper account of discrete structure of medium. The formula for decay rate is obtained without using the notion of refractive index. It is obtained that spontaneous decay strongly depends on the geometrical arrangement of atoms in microstructure. For some form of clusters, decay rate is found to be considerably large compared to its vacuum value that is not the case for macroscopically large dielectrics.

It is well known that, for static fields, the local microscopic electric field EL, that couples to the atomic dipole moment p, is related to the macroscopic field E and volume polarization P, in a dense dielectric medium, by the Lorentz-Lorenz (LL) relation.1 For dense media, this relation implies a nonlinear relationship between the microscopic molecular polarizability α and the macroscopic susceptibility χ, for a material volume density N. The LL relation also holds for arbitrary time-dependent fields in a nonlinear medium.2 What this implies is that all of quantum optics that uses an external electric field rather than the local field is internally inconsistent and will yield only approximate results for an optically dense medium. We consider the effect of the local field interaction on a simple three-level A system predicted to exhibit lasing without inversion and a high index of refraction at zero absorption.3 A threshold transition allows us to choose a value of this parameter that enhances inversionless gain and absorptionless dispersion by at least two orders of magnitude over that foundin usual lasing without inversion.4