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Physical Review A

Moment problem for a density matrix and a biorthogonal set of operator bases

Ching Tsung Lee

Published Nov 1, 1992• 16 citations
Publisher

Abstract

The practical aspect of the moment problem for a density matrix for single-mode radiation is solved; namely, two simple explicit expressions for the density matrix in terms of its moments are presented. A biorthogonal set of operators is established, so that two bases are available for expressing an arbitrary operator.
DOI
10.1103/physreva.46.6097

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A single-mode radiation field with Gaussian Wigner functions (Gaussian state field) is studied. The single-mode Gaussian-field generating function for both the density matrix elements and the moments of the creation and annihilation operators is calculated. It is shown that the matrix elements as well as the moments of different orders can be expressed in terms of Hermitian polynomials of two variables. Furthermore, the cumulants of the Gaussian state field are calculated. Finally, we generalize the Gaussian state field and compute the corresponding moments and density matrix elements.
In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec­ ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en­ semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com­ mon statistical phenomena, such as the effects of elastic collisions on the in­ duced dipole moments of atoms.
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.