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International Conference on Mathematical Methods in Electromagnetic Theory

Correlation properties of electromagnetic field generated by emitters with random orientation

Sergiy F. Lyagushyn, Yu.M. Salyuk, Alexander I. Sokolovsky

Published Jun 1, 2008• 4 citations
Publisher

Abstract

The necessity of more thorough description of electromagnetic field is substantiated for the problems of quantum optics. An attempt to construct the correlation functions of emission generated by a superfluorescent system of Dicke type is done. Operator evolution equations for a system of two-level emitters interacting with electromagnetic field lead to quantum analogues of Maxwell equations. Reduced description method applied to the superfluorescent process allows to find parameters characterizing the field fluctuations and some generalizations of matter equations.

DOI
10.1109/mmet.2008.4580964

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The necessity of more thorough description of electromagnetic field is substantiated for the problems of quantum optics. An attempt to construct the correlation functions of emission generated by a superfluorescent system of Dicke type is done. Operator evolution equations for a system of two-level emitters interacting with electromagnetic field lead to quantum analogues of Maxwell equations. Reduced description method applied to the superfluorescent process allows to find parameters characterizing the field fluctuations and some generalizations of matter equations.
The paper discusses some approaches to elucidating the picture of electromagnetic field correlations emerging and developing in the Dicke system of two-level atoms interacting via field. Based on the reduced description method, a set of differential equations for the reduced description parameters including field correlation functions is constructed using different quantum evolution pictures and regarding field-matter interaction as a small parameter. Calculations are based on the electrodynamics of continuous media. Meaningful results are obtained in the second-order in interaction. The material equations are found in the class of generalized functions and the material coefficients are estimated in the dependence of coordinates for applying to the spatial description of correlations. Initial and border conditions are discussed. A simplified form of the set of equations is proposed for the initial stage of the process with the supposition about the fixed orientation of emitters acceptable for a one-dimensional system.
The paper presents mathematical apparatus allowing to calculate the correlation properties of electromagnetic field generated by the Dicke system of two-level emitters. Evolution equations are constructed on the basis of Bogolyubov reduced description method for the system macrostate depicted with energy density of emitters and field amplitudes and binary correlation functions with using ideas of electrodynamics of continuous media. Modern computer calculation tools make it possible to simulate and analyze field states for various space arrangements of emitters; the corresponding mathematical problem is formulated. Glauber correlation functions should be used for diagnosing emission characteristics.
The process of cooperative spontaneous emission in a system of two-level emitters interacting via electromagnetic field is analyzed using the Bogolyubov reduced description method, which provides the possibility of studying the state of the generated field. Field macrostates are described with electric and magnetic field amplitudes and binary correlation functions of their components. Since the previously obtained set of differential equations describing the Dicke system evolution with considering field correlations is very cumbersome, we must find some way of simplifying the model to make it accessible for numerical modeling. The general picture of correlation development can be elucidated with using one- and two-dimensional models. The paper presents the set of evolution equations for Dicke system with fixed orientation of dipole moments of emitters forming a two-dimensional structure. The material equations and equilibrium correlations are discussed. The transition to the Cauchy problem for ordinary differential equations is proposed.
In optical experiments electromagnetic field manifests itself as a quantum-statistical object. Its state should be described with a statistical operator or with correlation functions calculated with it. A rather full description of field is provided with field amplitudes and simultaneous binary functions. Incidentally Glauber functions serve as a tool for diagnostics of field properties. The correlation functions are regarded as averaged macroscopic values and can be calculated as electrodynamic parameters of medium. The paper presents a scheme of such calculations for a system of two-level emitters interacting via field (Dicke model). Evolution equations are built on the basis of Bogolyubov reduced description method. Solving them can be put in life with modern computational technologies such as GPU.
The process of self-ordering in the famous Dicke model was studied in the framework of eliminating the boson variables. But the reduced description method enables us to obtain also the picture of electromagnetic field evolution provided field amplitudes and correlation functions are included into the number of reduced description parameters. In the Dicke Hamiltonian structure the interaction term includes the operators of emitter dipole moments or dipole moment density (polarization) since a spatial system is under consideration. Thus operator evolution equations are based on using such operators and their derivatives. The chain of evolution equations for averaged field amplitudes and binary correlation functions are obtained with using the statistical operator calculated in a perturbation theory in quasispin-photon interaction assumed to be small. The problem of chain decoupling does not arise since at any step we have a closed set of equations. The sets should be solved on the basis of material equations for current density and their generalizations for more complicated correlation functions. The way to constructing such equations and estimating the material parameters which are necessary for the numerical modeling of the development of correlations is discussed in the paper.