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Diagrammatic technique for calculating matrix elements of collective operators in superradiance. [eigenstates for N two-level atom systems

C. T. Lee

Published Aug 1, 1975• 0 citations
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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.

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Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an NNN-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 mmm-dependent factor and an mmm-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 r\mathcal{r}r is close to N2\frac{N}{2}2N​, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of r\mathcal{r}r-conserving and r\mathcal{r}r-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of NNN two-level atoms, spread over large regions, interacting with a multimode radiation field.
Adopting the so-called genealogical construction, the eigenstates of collective operators can be expressed corresponding to a specified mode for an N-atom system in terms of those for an (N-1)-atom system. Matrix element of a collective operator of an arbitrary mode is presented which 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 was obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups was then introduced. This gave a simple and systematic way of calculating the RME. Results show explicitly the geometry dependence of superradiance and the relative importance of r-conserving and r-nonconserving processes and clears up the chief difficulty encounted in the problem of N two-level atoms, spread over large regions, interacting with a multimode radiation field.