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Journal of Mathematical Physics

Correlation functions and higher order coherence in inelastically scattered quantum radiation

Andrei N. Weiszmann

Published Feb 1, 1978• 2 citations
Publisher

Abstract

The concept of coherence is analyzed in a system of interacting radiation and matter. Using the projection technique, the reduced field density operator is found, and with it the first and higher order field correlation functions are computed. It is proved that the inelastically scattered field is, for the most part, incoherent in any order at any time interval after collision with an atomic ensemble, except some specific time intervals and positions, where the order of coherence is determined by the atomic correlations considered.

DOI
10.1063/1.523676

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The concept of coherence is analyzed in a system of interacting radiation and matter. Using the projection technique, the reduced field density operator is found, and with it the first and higher order field correlation functions are computed. It is proved that the inelastically scattered field is, for the most part, incoherent in any order at any time interval after collision with an atomic ensemble, except some specific time intervals and positions, where the order of coherence is determined by the atomic correlations considered.
Considering the coherence properties of an inelastically scattered quantum radiation by an ensemble of N identical three level atoms in a solid-like array, we addressed the question whether this process is actually mediated by a series of multipole interactions and/or a series of virtual intermediate-exciton transitions and how the type of the considered mediation is reflected in the higher order coherence properties of the scattered radiation. Using a projector technique the reduced field density operator, the atomic and field correlation functions were found, the higher order coherence properties were studied and the arrival-time and phase correlation conditions of the light quanta were obtained and analysed. The results suggest that not only the arrival-time and phase correlation conditions are mathematically independent from the type of the considered mediating process, but that the multipole interaction approximation of a resonant Raman scattering, in the above considered system, is physically equivalent with the virtual intermediate-exciton transitions too.
Light scattering of monochromatic radiation by atoms and dilute vapours is reviewed. The relative contributions of incoherent and coherent scattering are calculated for weak incident fields. It is shown that, to lowest order in the intensity of the driving field, there can be several inelastic components in the scattered spectrum. The positions and widths of these components are related to light shifts and optical pumping of the atoms in the field. The complications that one encounters in scattering by two atoms, moving atoms, and atomic ensembles are discussed in qualitative terms.
The evolution of a quantized electromagnetic field in a thermally excited dispersion medium is determined by two scattering channels. The coherent channel is formed exclusively by the elastic scattering of quanta. The incoherent channel, along with elastic scattering processes, necessarily contains inelastic scattering processes, including induced radiation. Interference between the channels is absent because of the orthogonality of the wave functions of the medium in its final states, which correspond to different scattering channels. Therefore, in an excited medium, interference processes that are not described by its refractive index may arise. An interference pattern of this kind can be formed, in particular, as a result of the superposition of the resonance radiation incident on an excited medium and the radiation reflected from this medium. In this case, the conventional perturbation theory proves to be inadequate.
The problem of the interaction of a beam of radiation in a coherent state and an atomic system is investigated. A complete set of states of the field is introduced, called displaced stationary states, which are found to be particularly convenient in dealing with first- and second-order processes such as dispersion, Raman scattering, single and double emission, and absorption. In this formalism the resonant conditions for the occurrence of processes which have their counterpart in ordinary first- and second-order perturbation theory (where the state of the field is assumed to be a stationary state) are found in a natural way. In addition, a description is indicated of physical situations which are peculiar to coherent field dynamics and thus have no such counterpart. The coherence properties of the displaced stationary states are analyzed. It is shown under which experimental conditions (such as appropriate photon correlation methods) it would appear to be possible to verify whether the displaced stationary states indeed provide a good description of the underlying dynamics. The formalism can readily be extended to higher-order processes.
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.
Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.