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Low Temperature Physics

To the numerical simulation of the correlation picture in Dicke superradiance

Sergiy F. Lyagushyn

Published Jan 26, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1063/10.0009287

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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 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.
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.
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 self-organization process in a Dicke model is studied with taking into account electromagnetic field states. The interaction between two-level emitters provided by the field results in generating field waves of increasing amplitudes. The picture of superradiance development in a prolonged system can be described in terms of electromagnetic field in continuous medium formed by emitters. The generalized induction in the Dicke system is introduced on the basis of the material equation for charge current obtained using the Bogolyubov reduced description method. The dispersion equation for waves and its solutions are derived and analyzed. Non-singular expressions for material coefficients are proposed and used for calculating the wave characteristics.
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.
Our research into Dicke superradiance process is directed to building the connected picture of the behavior of emitter and field subsystems in order to elucidate the correlation properties of generated field. The interaction between subsystems has a resonant nature, so we can start from the fixed operation frequency of emitters. There are phenomenological methods for taking into account the frequency broadening of different natures. In our approach based on the Bogolyubov's functional hypothesis allowing the macrostate description with a limited number of parameters, some concepts of electrodynamics of continuous media are used and evolution equations of Maxwell type should be supplemented with material equations dependent on the operation frequency and field mode ones. Singularity in material coefficients disappears, if nonresonant interaction is considered. The manifestation of the role of nonresonant modes in field kinetics is a goal of this investigation.