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Low Temperature Physics

Ferromagnetism methods in the theory of superradiance

Alexander I. Sokolovsky, Sergiy F. Lyagushyn

Published Mar 1, 2025• 1 citation
Publisher

Abstract

Superradiance, both optical and acoustic, is one of the most important methods of generating the coherent pulses and obtaining the information about emitters via resonant phenomena. The process of ordering in a system of excited emitters, which results in a powerful short radiation pulse, is described with correlation functions of emitter and field parameters. Their behavior is problem of interest, especially in a spatial system. The known theoretical approaches to such questions are based on using some ways of correlator decoupling, at the same time the validity and exactness of applied operations remain without evaluation. The authors propose a way to solve this problem in the framework of the Bogolyubov reduced description method in the Peletminskii–Yatsenko formulation. Additionally to the usual small parameter of the method (the interaction constant), a new small parameter (the difference between correlation functions in the theory using it as an independent reduced description parameter and the process picture with one less parameter) is introduced. The method is developed for a concentrated Dicke model. The conventional application of quasispin operators for two-level objects allows to describe the system of emitters as a spin system and use the perfect methods of calculating the averages of spin operator combinations, which were developed in the ferromagnetism theory.

DOI
10.1063/10.0035835

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Superradiance, both optical and acoustic, is one of the most important methods of generating the coherent pulses and obtaining the information about emitters via resonant phenomena. The process of ordering in a system of excited emitters, which results in a powerful short radiation pulse, is described with correlation functions of emitter and field parameters. Their behavior is problem of interest, especially in a spatial system. The known theoretical approaches to such questions are based on using some ways of correlator decoupling, at the same time the validity and exactness of applied operations remain without evaluation. The authors propose a way to solve this problem in the framework of the Bogolyubov reduced description method in the Peletminskii–Yatsenko formulation. Additionally to the usual small parameter of the method (the interaction constant), a new small parameter (the difference between correlation functions in the theory using it as an independent reduced description parameter and the process picture with one less parameter) is introduced. The method is developed for a concentrated Dicke model. The conventional application of quasispin operators for two-level objects allows to describe the system of emitters as a spin system and use the perfect methods of calculating the averages of spin operator combinations, which were developed in the ferromagnetism theory.
Spontaneous ordering in a system of excited emitters with a powerful short radiation pulse predicted by Dicke attracted the great attention of scientists for decades as a bright example of the nontrivial behavior of a nonequilibrium system. The Bogolyubov reduced description method has some advantages and provides new possibilities in the theoretical investigation of the Dicke process. The paper presents the authors’ results of modeling such processes concerning both optical ones in the presence of crystal lattice influence and acoustic superradiance, which is possible in solids. The right choice of reduced description parameters allows the account of atom motions (phonons), cavity influence, and excitations in the emitter system (excitons) in the electromagnetic radiation process. The mathematical analogy between the Dicke system and a spin system in a magnetic field opens the way to applying the reduced description technique to acoustic phenomena previously investigated with the boson variable elimination method. Such a new approach to superradiance in crystals with paramagnetic impurities and with competing mechanisms of relaxation (Wagner model) is outlined. Acoustic superradiance is discussed in connection with the problem of structural phase transitions.
The reduced description method proceeds from Bogolyubov’s idea that at large times the non-equilibrium system evolution can be described with the limited number of parameters and principle of correlation weakening. The way to the right choice of such parameters and constructing the time evolution equations for them was opened by the works of Kharkiv school in statistical physics. A quarter of a century ago at Dnipropetrovsk National University the developed method was first applied to the Dicke system of two-level emitters interacting via electromagnetic field. The paper presents a short review of the results obtained. They include the analysis of Dicke Hamiltonian structure, the choice of reduced description parameters satisfying the Peletminskii – Yatsenko scheme conditions, the realization of this scheme for concentrated and prolonged Dicke systems resulting in the temporal equation for emitter subsystemstatistical operator. The reduced description method possibilities are illustrated with using the effective Hamiltonian concept, in particular taking into account emitter motion and higher correlations. Reformulating the theory in terms of electromagnetic field allowed to construct the electrodynamics of medium formed by two-level emitters with considering binary correlations of field variables and to put forward the problem of correlation development picture. A new method of investigating correlations in quasispin subsystem with using ferromagnetism theory approach was also proposed.
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.
Dicke model provides describing the superradiance effect when the spontaneous emission of a great number of excited two-level atoms is running in a correlated way due to their interaction via field. The process is identical to a giant dipole emission. Since the ordering takes place, it is interesting to study the possibility for using the terms of phase transition theory in such physical situation. Equilibrium properties of Dicke type models were studied for a long time because of their relevance for explaining some phenomena in matter-field interaction. The existence of an ordered phase in the emitter subsystem together with macroscopic photon mode filling is established, wherein the correct solution requires the Bogolyubov concept of quasiaverages to exclude the phase uncertainty of quasispin component structures <R+> and <R-> (for the concentrated model). In this paper, the behavior of similar order parameters for the non-equilibrium process description is analyzed and using the order parameter <R+R-> associated with the squared electric dipole moment of the emitter complex is substantiated. Such fact justifies <R3> application in the literature devoted to the superradiance theory. The situation with quasiaverages is compared with the problem of uniformity breaking in the numerical modeling.
A complex of issues related to using the Peletminsky–Yatsenko model for the study of non-equilibrium processes is investigated. The problems are associated with the inability of performing the calculation of necessary averages when system states are described by a new, wider set of reduced description parameters, in particular, fluctuations. A matter system interacting with a non-equilibrium system of bosons is considered. The substance is described by two reduced description parameters η1 , η2 and calculations with the quasi-equilibrium statistical operator corresponding to these parameters are technically impossible; at the same time, the calculations can be performed with such operator in the case of the only parameter η1 . Overcoming this problem by considering only small values δη2 of the other reduced description parameter is proposed. An exact definition of δη2 is given, and the theory of perturbations by powers δη2 is developed for the calculation of the initial quasi-equilibrium statistical operator with accuracy up to second-order contributions. Since all further averages are calculated only with the named operator including only η1 , the simplified calculation procedure is carried out for it.On this basis, the theory of non-equilibrium processes in the system of two-level emitters interacting with the boson field is developed. The emitter system is described by excitation degree η1 and the value of emitter correlations η2. The system is considered as a system of spins. The mathematically identical Dicke model describing superradiance and Wagner model introduced for the analogous acoustic phenomenon description are studied. The quasi-equilibrium statistical operator calculation is constructed.
The process of self-organization in a system of excited emitters studied by Dicke can be the basis of coherent emission generation without stimulated emission phenomena. Conventional analytic investigations in this field were concentrated on emitter subsystem behavior for obtaining the parameters of electromagnetic pulse. We propose the research into field characteristics in the process of pulse formation on the basis of the Bogolyubov method of reduced description by including into its parameters electromagnetic field strengths and binary correlation functions. We mean the macroscopic picture in a finite quantity of points, which provide the possibility of numerical modeling. Now we discuss the structure of numerical equations and the availability of physical parameters for such simulations.