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Annual Meeting Optical Society of America

What replaces the Lorentz-Lorenz formula for a cooled resonance gas?

Alexander V. Ghiner, Gregory I. Surdutovich

Published Jan 1, 1993• 0 citations
Publisher

Abstract

A rarefied gas of neutral atoms cooled and trapped by means of the resonance light pressure forces’ is an example of a system when n λ3 ≤ 1, but Nα ≥1. Here N denotes the density of atoms, λ is the light wavelength, a is the microscopic polarizability. It means that local field effects are essential, whereas Lorentz–Lorenz relation between α and refractive index n already, generally speaking, is not applicable. For rarefied gas even the question of the method integral equation2 applicability in itself is not evident, but in any case its application implies that the Lorentz’s sphere radius must be much larger than wavelength, which is incompatible with the traditional approach. With the choice of a large Lorentz sphere the variables substitution method3 allowed to calculate simultaneously linear and nonlinear medium characteristics and to obtain a connection between local and macroscopic fields when the spatial dispersion is taken into account. These characteristics essentially depend on the interaction law between the particles. In the no-interaction condition (ideal gas) Lorentz-Lorenz formula is valid for arbitrary densities, including the case Nλ3<< 1. It has been revealed that for repulsive forces between the atoms in a case Nα ≥ 1 an effect of the optical transparency (n2− 1 << 1) takes place.

DOI
10.1364/oam.1993.mqq.6

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A rarefied gas of neutral atoms cooled and trapped by means of the resonance light pressure forces’ is an example of a system when n λ3 ≤ 1, but Nα ≥1. Here N denotes the density of atoms, λ is the light wavelength, a is the microscopic polarizability. It means that local field effects are essential, whereas Lorentz–Lorenz relation between α and refractive index n already, generally speaking, is not applicable. For rarefied gas even the question of the method integral equation2 applicability in itself is not evident, but in any case its application implies that the Lorentz’s sphere radius must be much larger than wavelength, which is incompatible with the traditional approach. With the choice of a large Lorentz sphere the variables substitution method3 allowed to calculate simultaneously linear and nonlinear medium characteristics and to obtain a connection between local and macroscopic fields when the spatial dispersion is taken into account. These characteristics essentially depend on the interaction law between the particles. In the no-interaction condition (ideal gas) Lorentz-Lorenz formula is valid for arbitrary densities, including the case Nλ3<< 1. It has been revealed that for repulsive forces between the atoms in a case Nα ≥ 1 an effect of the optical transparency (n2− 1 << 1) takes place.
From a macroscopic viewpoint, atomic vapours appear homogeneous and isotropic. As we have seen during the discussion of the microscopic dynamics that light induces in atomic systems, however, the interaction with light as well as external fields can break the rotational and translational symmetries and make the system both inhomogeneous and anisotropic. This microscopic anisotropy appears directly in the optical properties of the system. This chapter discusses the propagation of light in such a medium and describes experiments that analyse the transmitted light to obtain information about the microscopic state of the medium. Isotropic atoms The Lorentz–Lorenz model Outline The interaction of electromagnetic radiation with matter, in particular light, has inspired philosophers of nature for many centuries and has led to heated debates like the one between Newton and Goethe. The original interest was with the difference in absorption of various materials as a function of the optical wavelength and with the refraction of light, i.e., its dispersion. The first theoretical analysis that connected these macroscopic effects to microscopic properties of the material was the Lorentz–Lorenz theory of dispersion (Lorentz 1880; Lorenz 1881), which was put forward shortly after the Maxwell equations and published in the year after Maxwell's death. (See Figure 6.1.) This theory models the material as a collection of dipoles, driven by the electromagnetic wave. We give here a brief summary of the theory, since its physical content is still the foundation of today's description although the mathematical formalism has changed significantly.
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
We study light propagation through a slab of cold gas using both the standard electrodynamics of polarizable media and massive atom-by-atom simulations of the electrodynamics. The main finding is that the predictions from the two methods may differ qualitatively when the density of the atomic sample $\ensuremath{\rho}$ and the wave number of resonant light kkk satisfy $\ensuremath{\rho}{k}^{\ensuremath{-}3}\ensuremath{\gtrsim}1$. The reason is that the standard electrodynamics is a mean-field theory, whereas for sufficiently strong light-mediated dipole-dipole interactions the atomic sample becomes strongly correlated. The deviations from mean-field theory appear to scale with the parameter $\ensuremath{\rho}{k}^{\ensuremath{-}3}$, and we demonstrate noticeable effects already at $\ensuremath{\rho}{k}^{\ensuremath{-}3}\ensuremath{\simeq}{10}^{\ensuremath{-}2}$. In dilute gases and in gases with an added inhomogeneous broadening the simulations show shifts of the resonance lines in qualitative agreement with the predicted Lorentz-Lorenz shift and ``cooperative Lamb shift,'' but the quantitative agreement is unsatisfactory. Our interpretation is that the microscopic basis for the local-field corrections in electrodynamics is not fully understood.
We derive the dispersion relation for the propagation of quasiresonant light with frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$ in an ultracold gas of bosonic atoms in the dilute regime, i.e., for an atomic density ${\mathrm{\ensuremath{\rho}}}_{0}$\ensuremath{\ll}(${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$/c)3{)}^{3})3. In our calculation, valid up to order 2 in density, two types of corrections to the Lorentz-Lorenz formula for the refractive index appear. The first one is due to the bosonic nature of the atoms and its contribution is related to the two-body correlation function. The second correction originates from multiple scattering of photons within pairs of close atoms, giving rise to the resonant van der Waals interaction. The temperature dependence of the refractive index gives a clear signature of quantum statistical effects, even if the degeneracy threshold for Bose-Einstein condensation is not reached.
Generalized single-atom Maxwell-Bloch equations for optically dense media are derived taking into account noncooperative radiative atom-atom interactions. Applying a Gaussian approximation and formally eliminating the degrees of freedom of the quantized radiation field and of all but a probe atom leads to an effective time-evolution operator for the probe atom. The mean coherent amplitude of the local field seen by the atom is shown to be given by the classical Lorentz-Lorenz relation. The second-order correlations of the field lead to terms that describe relaxation or pump processes and level shifts due to multiple scattering or reabsorption of spontaneously emitted photons. In the Markov limit a nonlinear and nonlocal single-atom density matrix equation is derived. To illustrate the effects of the quantum corrections, we discuss amplified spontaneous emission and radiation trapping in a dense ensemble of initially inverted two-level atoms and the effects of radiative interactions on intrinsic optical bistability in coherently driven systems.