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Journal of Physics and Electronics

Dynamics of a system of two-level emitters in the Dicke model taking into account small correlations between them

Alexander I. Sokolovsky, Sergiy F. Lyagushyn

Published Dec 29, 2023• 2 citations
Publisher

Abstract

On the basis of our previous study of the quantum system of atoms and the electromagnetic field, a system of non-relativistic stationary and non-interacting small one-electron atoms located in a region much smaller in size than the characteristic wavelength of the field is considered. Atoms are considered in the two-level approximation and interact with the field by dipole-electric interaction. We carefully analyze the state space of the atom and the operators in it in the spin s = 1 / 2 formalism, which leads to the Dicke Hamiltonian in terms of the spin formalism. Non-equilibrium states of the system are investigated using the reduced description method and described by the occupation numbers of photon states ηαk , the degree of excitation of the atoms η1 , and the value of correlations between them η2. The basis of our consideration is the Peletminsky–Yatsenko model, the foundations of which and, in particular, the approach to solving the Cauchy problem, based on the notion of effective initial conditions, are thoroughly discussed. Since the Dicke model deals with the dynamics of a system of particles with spin, the technique of calculating the average values of products of spin operator proposed by Vax, Larkin, and Pikin is developed to a certain extent. Averages in a state with a statistical operator of a system of atoms with given excitations are considered. The obtained results are maximally simplified in the case of spin s = 1 / 2 . The impossibility of calculations with a quasi-local statistical operator when describing the state of the system with parameters ηαk , η1, η2 is overcome with using a somewhat simplified approach of our previous work, in which, instead of a parameter η2, the system is described by a small deviation δη2 = η2 - η20 of the parameter η2 from its value η20 when describing the system only by the parameters ηαk, η1. Developed approaches to calculations with spin operators are used to calculate the quasi-equilibrium statistical operator when choosing of the quantities ηαk, η1 , and δη2 as reduced description parameters and the right-hand sides of the time equations for these parameters.

DOI
10.15421/332326

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On the basis of our previous study of the quantum system of atoms and the electromagnetic field, a system of non-relativistic stationary and non-interacting small one-electron atoms located in a region much smaller in size than the characteristic wavelength of the field is considered. Atoms are considered in the two-level approximation and interact with the field by dipole-electric interaction. We carefully analyze the state space of the atom and the operators in it in the spin s = 1 / 2 formalism, which leads to the Dicke Hamiltonian in terms of the spin formalism. Non-equilibrium states of the system are investigated using the reduced description method and described by the occupation numbers of photon states ηαk , the degree of excitation of the atoms η1 , and the value of correlations between them η2. The basis of our consideration is the Peletminsky–Yatsenko model, the foundations of which and, in particular, the approach to solving the Cauchy problem, based on the notion of effective initial conditions, are thoroughly discussed. Since the Dicke model deals with the dynamics of a system of particles with spin, the technique of calculating the average values of products of spin operator proposed by Vax, Larkin, and Pikin is developed to a certain extent. Averages in a state with a statistical operator of a system of atoms with given excitations are considered. The obtained results are maximally simplified in the case of spin s = 1 / 2 . The impossibility of calculations with a quasi-local statistical operator when describing the state of the system with parameters ηαk , η1, η2 is overcome with using a somewhat simplified approach of our previous work, in which, instead of a parameter η2, the system is described by a small deviation δη2 = η2 - η20 of the parameter η2 from its value η20 when describing the system only by the parameters ηαk, η1. Developed approaches to calculations with spin operators are used to calculate the quasi-equilibrium statistical operator when choosing of the quantities ηαk, η1 , and δη2 as reduced description parameters and the right-hand sides of the time equations for these parameters.
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 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.
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 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 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.
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.