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Journal of Modern Optics

Density Matrix Elements and Moments for Generalized Gaussian State Fields

Gerhard Adam

Published Jun 1, 1995• 42 citations
Publisher

Abstract

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.

DOI
10.1080/09500349514551141

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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.
For one-mode light described by the Wigner function of a generic Gaussian form with five real parameters, the photon distribution function is obtained explicitly in terms of the Hermite polynomial of two variables. The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. The first and the second statistical moments of the photon distribution function are calculated. The generating function for the photon distribution is discussed. Various special cases, including the shifted thermal states, correlated and squeezed states, coherent states, etc., are considered.
The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.
The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of , in the monomial basis, are shown to be essentially the SU (2) Clebsch–Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n -mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.