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Physical Review A

Diagrammatic technique for calculating matrix elements of collective operators in superradiance

Ching Tsung Lee

Published Aug 1, 1975• 10 citations
Publisher

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.

DOI
10.1103/physreva.12.575

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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−1N\ensuremath{-}1N\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.
The cooperative spontaneous emission from a system of NNN identical two-level atoms, excited arbitrarily and interacting with a single mode of radiation, is studied by using the Bonifacio-Schwendimann-Haake approach. The evolution of the probability distribution of the atomic system is considered as a Markov process described by a time-evolution matrix (TEM). The exact analytic expressions for the elements of this TEM are obtained by solving the superradiance master equation. It is proved that the TEM satisfies the normalization condition and the Chapman-Kolmogorov equation as required. The exact general formulas for the expectation values of various physical observables, expressed in terms of an arbitrary initial distribution, are obtained. Finally, these results are applied to a specific example, the superradiant initial state.
In order to study the cooperative emission of radiation from an extended system of NNN two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled jjj and kkk via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the zzz axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the zzz dependence predominates, in the opposite case the (x,yx, yx,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the zzz dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the zzz axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
Dicke's superradiance from a gas system of atoms and molecules was described in terms of two-level atoms, i.e., as an assembly of half-spin operators. In this system the magnitude of the total spin, which is called a cooperation number, is conserved at the maximum for the whole radiative process. We will answer in this paper how this superradiance is modified in the crystal in which an excitation can propagate from atom to atom through dipolar interactions. As a prototype of a crystal, we choose a linear chain of two-level atoms. The superradiance master equation of this system can be solved in terms of the exact solution of Lieb, Shultz, and Mattis for the spin system. The first effect is that the emitted frequency shows opposite shifts red and blue, respectively, for the first and second halves of the superradiance pulse, or vice versa, depending upon the sign of the transfer-matrix element of the excitation. The second effect is the appearance of a slow component in the emission tail, which originates from mixture of the states with smaller cooperation numbers or smaller oscillator strength under the transfer of the excitations. These results are demonstrated for a mesoscopic system of linear chains.
A collective description of density matrix is presented for identical multi-level atoms, which are either excited initially, driven coherently or pumped incoherently. The density matrix is defined as expectation value of projection or transition operators in a basis of atom's product states. The identical matrix elements are identified with several integers, which specify uniquely the involved operators. To remove the redundancy, these identical elements are treated as single quantity and the equation for this quantity is dervied by mapping the transition or projection operators to a single vector specified with these integers. As a result, the number of computed elements increases polynomially rather than exponentially with the number of atoms. As an example, we carry out exact simulation of hundreds of two-level atoms and demonstrate the different conditions for observing superradiance and superfluorescence.