Diagrammatic technique for calculating matrix elements of collective operators in superradiance
Abstract
Adopting the socalled "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an Natom system in terms of those for an (N1)atom system. Using these Dicke states as bases and using the WignerEckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an mdependent factor and an mindependent 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 is close to N2, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of rconserving and rnonconserving processes. This clears up the chief difficulty encountered in the DickeSchwendimann approach to the problem of N twolevel atoms, spread over large regions, interacting with a multimode radiation field.
 Publication:

Physical Review A
 Pub Date:
 August 1975
 DOI:
 10.1103/PhysRevA.12.575
 Bibcode:
 1975PhRvA..12..575L
 Keywords:

 Asymptotic Methods;
 Atomic Energy Levels;
 Eigenvectors;
 Matrix Theory;
 Operators (Mathematics);
 Quantum Mechanics;
 Branching (Mathematics);
 Circular Cylinders;
 Eigenvalues;
 Hamiltonian Functions;
 Permutations;
 Recursive Functions;
 Topology;
 Transition Probabilities;
 Vector Spaces;
 Atomic and Molecular Physics