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Canadian Journal of Physics

Superradiant Intensity Fluctuations in Neoclassical Radiation Theory

Andrzej Zardecki

Published Dec 15, 1974• 1 citation
Publisher

Abstract

The fluctuation properties of a system of two level atoms are investigated by introducing a random precession term into the equations of motion of the atomic super Bloch vector. Starting from the assumptions of neoclassical radiation theory a general formula for the dipole field of a small system of atoms is derived. The intensity fluctuations are calculated in the narrowing limit for two extreme cases of very large and very small atomic to thermal energy ratios. The former case, describing a true superradiant emission, involves asymptotic solutions to the Fokker-Planck equation. The latter case, treated by perturbation theory, corresponds to spontaneous emission in the presence of incoherent pumping.

DOI
10.1139/p74-320

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Progress of Theoretical Physics•Sep 1, 1975•7 citations
The fluctuation properties of a system of two level atoms are investigated by introducing a random precession term into the equations of motion of the atomic super Bloch vector. Starting from the assumptions of neoclassical radiation theory a general formula for the dipole field of a small system of atoms is derived. The intensity fluctuations are calculated in the narrowing limit for two extreme cases of very large and very small atomic to thermal energy ratios. The former case, describing a true superradiant emission, involves asymptotic solutions to the Fokker-Planck equation. The latter case, treated by perturbation theory, corresponds to spontaneous emission in the presence of incoherent pumping.
The fluctuating properties of superradiant and superfluorescent systems are investigated by supplementing the equations of motion for the expectation values of quantum observables with random force terms. In the case of a small atomic system the time evolution of the radiated intensity as well as the intensity fluctuations are computed. We also find the time dependence of the average dipole moment in the presence of chirping. For a pencil-shaped geometry, a generalized system of Maxwell–Bloch equations is analysed. The inhomogeneous broadening is then included. We compare the semiclassical equation for the probability distribution of the atomic variables with the quantum equations based both on normal and antinormal ordering. With the exception of the region in the vicinity of the poles of the Bloch sphere, a close correspondence between the semiclassical and quantum theories is established. The oscillatory superfluorescence is treated similarly to quantum analysis by employing a new method of Lugiato for open systems.
The problem of the evolution in time of a system of two-level atoms that are coupled through their electromagnetic fields, is studied and solved in the framework of semiclassical radiation theory. The atoms may be in any initial states and the radiation reaction is fully taken into account. Both superradiance and time-dependent frequency shift or chirping effects are found. The assumption that all the atoms see the same field is shown to place severe constraints on the evolution of the system, in the form of constants of the motion. These constraints are most easily pictured in terms of the Bloch-vector representation of the atomic system, and they lead to explicit solutions for the time development of each atom. It is shown that the evolution of the complex system is describable by means of a collective super Bloch vector, whose behavior is similar to the behavior of the Bloch vector for a single isolated atom. The constraints on the motion also imply that the system cannot radiate all its energy coherently; some of it remains trapped, to be dissipated ultimately by incoherent processes. Some curves are presented to illustrate the behavior of the system in special cases.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
The effect of field fluctuations on the motion of a two-level atom in resonant laser radiation is investigated theoretically. The internal motion of the atom is treated quantum mechanically (optical Bloch equations); the translational motion, which is coupled to the internal motion, is treated classically (Ehrenfest's theorem); and the fluctuating radiation is treated as a classical electromagnetic wave with stochastic phase (phase-diffusion model). An expression for the radiation force and equations of motion for the ensemble-averaged Bloch vector are derived and used to calculate (i) the radiation force in a plane running wave, (ii) the steady-state radiation force in a general standing wave, and (iii) the dissipative force in a plane standing wave in the limit of strong phase fluctuations. In case (iii) it is found that the atomic velocity is damped when the radiation is tuned above resonance, which is opposite to the case of a weak coherent standing wave, in which damping occurs for radiation tuned below resonance. It is shown that fluctuations have a substantial effect on the radiation force only when the spectral width of the radiation approaches or exceeds the spontaneous emission rate.
In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with NNN resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.
The dynamics of the co-operative spontaneous emission from many two-level atoms is investigated by introducing a specific form of quasi-probability distribution which apparently excludes the vacuum fluctuation of the radiation field but includes the fluctuation of the atomic dipole moment. It is shown that the equation of motion for the random variables associated with the atom-field coupled system can be written in a quite classical form. As a result, the formula is derived for computing the multi-time correlation functions of the radiation field–essential to its coherence properties–which generalizes Bonifacio, Schwendimann and Haake's adiabatic “corresponding theorem”.