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Theoretical and Mathematical Physics

Dynamics of superradiant processes in two-level macroscopic systems in a crystal

N. N. Bogolyubov, Eugene K. Bashkirov, Fam Le Kien, A. S. Shumovskiǐ

Published Mar 1, 1987• 1 citation
Publisher
DOI
10.1007/bf01041011

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.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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.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)
Spontaneous decay is a phenomenon that has been described for the first time by Dirac, Wigner and Weisskopf, at the early ages of the development of quantum electrodynamics. The theory is applicable to a multi-body system only when atoms don’t interact during their decay. Dicke introduces in his seminal paper the case where this assumption doesn’t hold. The results he obtains not only tell that the system emits pulses, but the pulse duration now depends on 1/N, where N is the number of participating atoms to the process. The pulses temporal profile is described by a squared hyperbolic secant function. The resulting emission intensity scales as N^2. The atomic system radiates more intensively than the ordinary fluorescent emission, this is the reason why the process is called superradiance, or by a generalization to extended systems, superfluorescence. The superradiant emission has preferential escape directions, depending on the atomic sample shape. Superradiance is a collective process and it takes place when the group of excited atoms is left unper- turbed during the formation of of the macroscopic dipole. A fundamental difference compared to the spontaneous and stimulated emission processes is that all atoms act together as a single entity and it isn’t possible to identify single atoms involved. This remarkable behaviour is manifest when the atomic coherence of the system, while influenced by environmental interactions that cause dephasing in atoms, persists for a sufficient interval of time. Typical dephasing mechanisms, such as collisions in gases or lattice phonon interactions in crystals, make its experimental observation difficult. A possible solution to this problem is the use of optical crystals doped with rare-earths that are characterized by long and narrow levels lifetimes.
Dicke's superradiance from a gas system of atoms and molecules was described in terms of two-level atoms, i.e., as an assembly of half-spin operators. In this system the magnitude of the total spin, which is called a cooperation number, is conserved at the maximum for the whole radiative process. We will answer in this paper how this superradiance is modified in the crystal in which an excitation can propagate from atom to atom through dipolar interactions. As a prototype of a crystal, we choose a linear chain of two-level atoms. The superradiance master equation of this system can be solved in terms of the exact solution of Lieb, Shultz, and Mattis for the spin system. The first effect is that the emitted frequency shows opposite shifts red and blue, respectively, for the first and second halves of the superradiance pulse, or vice versa, depending upon the sign of the transfer-matrix element of the excitation. The second effect is the appearance of a slow component in the emission tail, which originates from mixture of the states with smaller cooperation numbers or smaller oscillator strength under the transfer of the excitations. These results are demonstrated for a mesoscopic system of linear chains.