I theoretically demonstrate the collective population evolution of two-level atoms, based on the mean-field theory. Optical materials with abrupt changes of electromagnetic local densities of states enable steady-state population inversion of two-level atoms which is generally considered impossible. In previous studies, details of the population evolution of a single two-level atom were discussed. However, behaviors of the population evolution of collective two-level atoms are very different from those of a single two-level atom. In this paper, temporal behaviors of the collective population evolution for input electric fields with constant and time-dependent amplitudes are discussed in detail. Moreover, I develop the effective method to calculate the collective spontaneous emission for non-Markovian dynamics. Finally, I discuss the validity of the mean-field theory for a large number of two-level atoms in spontaneous emission using the Markovian approximation.
A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from
N identical two-level atoms is investigated further. It is shown that the equation of motion for the
p(<N)-atom density matrix is coupled to the (
p+1)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time
t can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of
N1. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of
2N coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.
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 radi ation field-essential to its coherence properties-which generalizes Bonifacio, Schwendimann and Haake's adiabatic corresponding theorem. § 1. Introduction. The co-operative spontaneous emission (so-called superradiance) from an as sembly of many two-level atoms first discussed by Dicke in 1954n has been studied by many authors with different methods, roughly speaking, classicall/> and quantum mechanically. 3>-5> The classical approach, though simple, has a special significance for the dynamics of a large number of atoms participating in phase, and has succeeded except when the population of the atoms is completely inverted. The failure of the simple classical treatment in the case of complete inversion is due to the fact that the effect of spontaneous emission is not considered at all, so that emission is impossible from such a state as with no dipole moment.2>.s> Therefore, if one takes into account adequately the quantum fluctuation of the tdtal dipole moment which exists even when the atoms are fully excited, one should be able to describe the spontaneous emission from a classical point of view. This idea was suggested by Senitzky6> in a rather intuitive argument. In the present paper an attempt is made to formulate the above problem from a quasi-probability point of view. To this end we make here a particular choice of the representation of the quantum density operators (or quasi-probability. distri bution functions abbreviated by QPDF hereafter) among others7l-Io> by which the quantum mechanical equation of motion may be transformed into a form in an . associated c-number phase space, i.e., the yvell-known P-representation 9> of Glauber IS taken for the field variables and the so-~alled Q~representation based on the