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IEEE Lasers and Electro-Optics Society Annual Meeting (LEOS)

Local field effects in nonlinear and quantum optics

C. M. Bowden, Michael E. Crenshaw, Jagdish Rai

Published Jan 1, 1997• 0 citations
Publisher

Abstract

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.

DOI
10.1109/leos.1997.645345

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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.
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
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.
The propagation of the electromagnetic field of a laser through a dense Bose gas is examined and nonlinear operator equations for the motion of the center of mass of the atoms are derived. The goal is to present a self-consistent set of coupled Maxwell-Bloch equations for atomic and electromagnetic fields generalized to include the atomic center-of-mass motion. Two effects are considered. The ultracold gas forms a medium for the Maxwell field which modifies its propagation properties. Combined herewith is the influence of the dipole-dipole interaction between atoms which leads to a density-dependent shift of the atomic transition frequency. It is expressed in a position-dependent detuning and is the reason for the nonlinearity. This results in a direct and physically transparent way from the quantum field-theoretical version of the local-field approach to electrodynamics in quantum media. The equations for the matter fields are general. Previously published nonlinear equations are obtained as limiting cases. As an atom-optical application the scattering of a dense beam of a Bose gas is studied in the Raman-Nath regime. The main conclusion is that for increasing density of the gas the dipole-dipole interaction suppresses or enhances the scattering depending on the sign of the detuning.
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 nnn of the host medium. However, an enhancement factor of (n2+2)∕3({n}^{2}+2)∕3(n2+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.
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.
We present the rigorous microscopic quantum theory of the interaction of ultracold Bose and Fermi gases with the electromagnetic field of vacuum and laser photons. The main attention has been paid to the consistent consideration of dynamical dipole-dipole interactions. The theory developed is shown to be consistent with the general principles of the canonical quantization of electromagnetic field in a medium. Starting from the first principles of QED we have derived the general system of Maxwell-Bloch equations for atomic creation and annihilation operators and the propagation equation for the laser field which can be used for the self-consistent analysis of various linear and nonlinear phenomena in atom optics at high densities of the atomic system. All known equations which are used for the description of the behaviour of an ultracold atomic ensemble in a radiation field can be obtained from our general system of equations in a low-density limit.