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arXiv:quant-ph/9809087 · Physical Review A

Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula

Michael Fleischhauer, Susanne F. Yelin

Published Mar 1, 1999• 73 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.59.2427

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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.
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.
In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with NNN resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.
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This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.
Starting from the first principles of nonrelativistic QED we have developed the quantum theory of the interaction of a two-component ultracold atomic ensemble with the electromagnetic field of vacuum and laser photons. The main attention has been paid to the consistent consideration of dynamical dipole-dipole interactions in the radiation field. Taking into account local-field effects we have derived the system of Maxwell-Bloch equations. Optical properties of the two-component Bose gas are investigated. It is shown that the refractive index of the gas is given by the Maxwell-Garnett formula. All equations which are used up to now for the description of the behavior of an ultracold atomic ensemble in a radiation field can be obtained from our general system of equations in the low-density limit. Raman-Nath diffraction of the two-component atomic beam is investigated on the basis of our general system of equations.