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International Conference on Mathematical Methods in Electromagnetic Theory

Electromagnetic field state description with correlation functions of different nature

Sergiy F. Lyagushyn, Alexander I. Sokolovsky

Published Sep 1, 2010• 2 citations
Publisher

Abstract

Modern electromagnetic theory based on quantum conceptions supposes turn to averages of operator dynamic variables for describing the field state and analyzing experimental results with computational methods. Theoretical analysis of such non-equilibrium phenomenon as Dicke superfluorescence leads to an algebra of variables which are necessary for building a set of evolution equations. Rather different correlation functions can be determined in measurements according to the coincidence scheme using Glauber's quantum detectors. A natural way of connecting various correlation functions is expressing them via the photon density matrix and Wigner function. The problem of comparing the theoretical and experimental approaches is analyzed in the present paper.

DOI
10.1109/mmet.2010.5611381

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Modern electromagnetic theory based on quantum conceptions supposes turn to averages of operator dynamic variables for describing the field state and analyzing experimental results with computational methods. Theoretical analysis of such non-equilibrium phenomenon as Dicke superfluorescence leads to an algebra of variables which are necessary for building a set of evolution equations. Rather different correlation functions can be determined in measurements according to the coincidence scheme using Glauber's quantum detectors. A natural way of connecting various correlation functions is expressing them via the photon density matrix and Wigner function. The problem of comparing the theoretical and experimental approaches is analyzed in the present paper.
The problem of parameters, which are necessary for nonequilibrium electromagnetic field description, is investigated on the basis of the Bogolyubov reduced description method establishing the necessity of binary correlations in the minimal set of parameters taken into account in evolution equations. The corresponding theory can be built in terms of one-particle density matrices, Wigner distribution functions, simultaneous correlation functions of field operators, and generating functionals. The obtained results can be analyzed with the help of approaches elaborated in quantum optics. Various methods in theoretical and experimental research into field correlations are compared.
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.
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 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.
Pzz is a real but non-positive one.Nevertheless, the field state description with the Glauber-Sudarshan P-distribution remains the most demonstrative and consumable.For example, a proposed definition of non-classical states of electromagnetic field (Bogolyubov (Jr.) et al., 1988) uses the expression (1) for the statistical operator.A state is referred to as non-classical one if one of two requirements is obeyed: either average number of photons in a mode is less than 1, or P-function is not positively determined or has singularity that is higher than the  -function.www.intechopen.com
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.