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DYNAMICS OF SUPERRADIANT PROCESSES IN TWO-LEVEL

N. N. Bogolyubov, Eugene K. Bashkirov, A. S. Shumovskiǐ

Published Jan 1, 1987• 0 citations
OpenAlex

Abstract

.By partial elimination of the boson variables an exact hierarchy of kinetic equations for a two-level crystal system interacting with an electromagnetic field is obtained and investigated. The scattering of photons by phonons is taken into account. It is shown that the presence of scattering by phonons leads to a reduction in the delay time of the superradiant pulse. The radiation intensities at the fundamental, sum, and difference frequencies are determined. The influence of radiation processes on the dynamics of the phonon subsystem is studied. Superradiance in two-level macroscopic systems is usually investigated by means of models of rigidly fixed emitters [1,2] or by representing the emitters as a Maxwellian gas [3,4]. However, in real crystals the emitters themselves vibrate. There have been various recent experiments on the generation of superradiance in impure crystals [5-8]. In connection with the interpretation of these experiments, it is of interest to consider the influence of Mandelstam-Brillouin scattering on the parameters of superradiant systems. Such a problem is also of interest in connection with the development of a superradiant laser with ferroelectric working medium [9], in particular for the detection of gravitational waves [I0]. In the theory of superradiance, Thompson [Ii, 12] has constructed a well-known model in which s contribution due to scattering of the light by phonons is determined in the first order of an expansion of the density with respect to the relative displacements of the ions. In [11-12], only the equilibrium properties of such a model were considered. In our earlier paper [13], the method of partial elimination of the boson variables proposed in [14] was developed to describe the dynamical properties of such a system. In the present paper, we investigate in detail the process of collective radiation in such a system, and we also consider the influence of the radiation processes on the dynamics of the phonon subsystem in the crystal. The Hamiltonian of Thompson's model has the form H=H.~+HF+HMF, HM= ~ ~cooRfz + ~ ~qbq+bq, Hv=Z ?~oA.a~+a~: , r q ~ (I)

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Resonant Scattering of Photons from an Atom-Lattice System

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We investigate the resonant scattering of photons from an atom in the thermal equilibrium with a phonon bath. A two-level model of the atom is discussed in detail. In the model problem, the atom is coupled directly to the lattice through a diagonal interaction which is bilinear in the lattice quanta. Relaxation effects are introduced indirectly via correlation functions. In the limit in which the relaxation time is much shorter than the radiative lifetime, the frequency spectrum of the scattered radiation consists of three parts: (1) a coherent, elastic term weighted by the factor ${tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$, (2) an incoherent quasielastic term weighted by the factor [$1\ensuremath{-}{tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$], (3) a fluorescent term characterizing emission at the natural frequency ${\ensuremath{\omega}}_{0}$, which arises from an effective two-photon-two-phonon interaction. The extension of the analysis to the problem of resonant scattering from paramagnetic ions in crystals is outlined with particular emphasis on the influence of neighboring ionic levels and the relative importance of the one- and two-phonon processes.

Physical Review•Jun 15, 1967•76 citations
.By partial elimination of the boson variables an exact hierarchy of kinetic equations for a two-level crystal system interacting with an electromagnetic field is obtained and investigated. The scattering of photons by phonons is taken into account. It is shown that the presence of scattering by phonons leads to a reduction in the delay time of the superradiant pulse. The radiation intensities at the fundamental, sum, and difference frequencies are determined. The influence of radiation processes on the dynamics of the phonon subsystem is studied. Superradiance in two-level macroscopic systems is usually investigated by means of models of rigidly fixed emitters [1,2] or by representing the emitters as a Maxwellian gas [3,4]. However, in real crystals the emitters themselves vibrate. There have been various recent experiments on the generation of superradiance in impure crystals [5-8]. In connection with the interpretation of these experiments, it is of interest to consider the influence of Mandelstam-Brillouin scattering on the parameters of superradiant systems. Such a problem is also of interest in connection with the development of a superradiant laser with ferroelectric working medium [9], in particular for the detection of gravitational waves [I0]. In the theory of superradiance, Thompson [Ii, 12] has constructed a well-known model in which s contribution due to scattering of the light by phonons is determined in the first order of an expansion of the density with respect to the relative displacements of the ions. In [11-12], only the equilibrium properties of such a model were considered. In our earlier paper [13], the method of partial elimination of the boson variables proposed in [14] was developed to describe the dynamical properties of such a system. In the present paper, we investigate in detail the process of collective radiation in such a system, and we also consider the influence of the radiation processes on the dynamics of the phonon subsystem in the crystal. The Hamiltonian of Thompson's model has the form H=H.~+HF+HMF, HM= ~ ~cooRfz + ~ qbq+bq, Hv=Z ?oA.a+a: , r q ~ (I)
The adequate description of the dynamic properties of macroscopic two-level systems interacting with the electromagnetic field is of particular interest in connection with the problem of choosing the working regime and active medium for the realization of single-pass resonator-free superradiative lasers. Since the derivation of an exact kinetic equation for such systems is a very complicated mathematical problem, different simplifying assumptions of physical nature are generally employed. There is, for example, the approach based on a Markov master equation developed in a series of papers by Bonifacio et al. [1-6]. The dynamics of the photon system with allowance for processes of incoherent pumping was studied on the basis of Zwanzig's approach [7] by Haake and Mandel [8, 9]. The Bogolyubov-Krylo v method of solution of nonlinear equations was used by Willis [10J. Some important estimates for the conversion times and other physical characteristics relating to the problem of creating a gamma laser were obtained by Khokhlov et al. [11-14] (see also the review [15]). Recently, exploitation of the deep analogies in the mathematical formulation of the model problems in the theory of superradiance and in polaron theory made it possible to obtain an exact generalization of the kinetic equation for superradiative systems [161 on the basis of the method of eliminating the boson variables developed in [17, 18]. It proved possible to obtain a number of physical results, in particular, to determine the magnitudes of the constant and collective frequency shift in the presence of inhomogeneous Lorentz broadening under conditions of cooperative spontaneous radiation. In following papers [19, 20], which were based on the generalized kinetic equation of [16], an equation for the single-particle distribution function was constructed. Comparison of this equation with Pauli's master equation made it possible to determine the characteristic relaxation times. There is however a simpler and more direct way of determining the probability F_ of transition of a radiator to an excited state with absorption of a photon and the probability F+ of transition to the de-excited state with emission of a photon (x~-F~ -', z are the characteristic relaxation times). It is associated with the use of the method of two-time temperature-depe ndent Green's functions. This method, which was introduced into statistical mechanics by Bogolyubov and Tyablikov [21], plays an important part in the theory of nonequilibrium processes (see, for example, [22, 23]). In the present paper, we investigate the dynamical properties of superradiative systems by means of two-time temperature-depe ndent Green's functions and, in particular, we obtain relations for the characteristic relaxation times. For clarity~ we consider the simplest model, characterized by the Hamiltonian
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 chain of kinetic equations for a three-level extended macroscopic system interacting with an electromagnetic field is obtained on the basis of the method of elimination of the boson variables taking into account the coherent pumping processes. On its basis the conventional superradiance of the system is examined in the markovian limit. The relations between the parameters of the superradiant pulse and those of the pumping pulse of arbitrary duration are obtained. The possibility of description of the triggering superradiance is discussed.
In order to study the cooperative emission of radiation from an extended system of NNN two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled jjj and kkk via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the zzz axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the zzz dependence predominates, in the opposite case the (x,yx, yx,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the zzz dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the zzz axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
We develop the theory of dynamical superradiance - the collective exchange of energy between an ensemble of initially excited emitters and a single-mode cavity - for organic materials where electronic states are coupled to vibrational modes. We consider two models to capture the vibrational effects: first, vibrations treated as a Markovian bath for two-level emitters, via a pure dephasing term in the Lindblad master equation for the system; second, vibrational modes directly included in the system via the Holstein-Tavis-Cummings Hamiltonian. By exploiting the permutation symmetry of the emitters and weak U(1) symmetry, we develop a numerical method capable of exactly solving the Tavis-Cummings model with local dissipation for up to 140 emitters. Using the exact method, we validate mean-field and second-order cumulant approximations and use them to describe macroscopic numbers of emitters. We analyze the dynamics of the average cavity photon number, electronic coherence, and Bloch vector length and show that the effect of vibrational mode coupling goes beyond simple dephasing. Our results show that superradiance is possible in the presence of vibrational mode coupling; for negative cavity detunings, the vibrational coupling may even enhance superradiance. We identify asymmetry of the photon number rise time as a function of the detuning of the cavity frequency as an experimentally accessible signature of such vibrationally assisted superradiance.
We investigate the resonant scattering of photons from an atom in the thermal equilibrium with a phonon bath. A two-level model of the atom is discussed in detail. In the model problem, the atom is coupled directly to the lattice through a diagonal interaction which is bilinear in the lattice quanta. Relaxation effects are introduced indirectly via correlation functions. In the limit in which the relaxation time is much shorter than the radiative lifetime, the frequency spectrum of the scattered radiation consists of three parts: (1) a coherent, elastic term weighted by the factor ${tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$, (2) an incoherent quasielastic term weighted by the factor [$1\ensuremath{-}{tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$], (3) a fluorescent term characterizing emission at the natural frequency ${\ensuremath{\omega}}_{0}$, which arises from an effective two-photon-two-phonon interaction. The extension of the analysis to the problem of resonant scattering from paramagnetic ions in crystals is outlined with particular emphasis on the influence of neighboring ionic levels and the relative importance of the one- and two-phonon processes.