Diagrammatic technique for calculating matrix elements of collective operators in superradiance

Abstract

Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an $N$-atom system in terms of those for an ($N\ensuremath{-}1$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an $m$-dependent factor and an $m$-independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number $\mathcal{r}$ is close to $\frac{N}{2}$, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of $\mathcal{r}$-conserving and $\mathcal{r}$-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of $N$ two-level atoms, spread over large regions, interacting with a multimode radiation field.