Master-Equation Approach to Spontaneous Emission. III. Many-Body Aspects of Emission from Two-Level Atoms and the Effect of Inhomogeneous Broadening

Abstract

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 $p(<N)\mathrm{}$-atom density matrix is coupled to the ($p+1\mathrm{}$)-atom density matrix. We first consider the initial excitation of the system to state $|{\theta }_0,{\varphi }_0〉$ (${\theta }_0<\pi$) 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 $\theta \mathrm{}\left(t\right)\mathrm{}$ ($\theta <\pi$), with ${sin}^2\frac{1}{2}\theta$ 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 $〈(S_i^Z-〈S_i^Z〉)(S_j^Z-〈S_j^Z〉)〉$ for any pair of atoms is of the order of $\frac{1}{N}$. The case when ${\theta }_0=\pi$ 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.