Regularization of thePRepresentation

Abstract

A representation is introduced for the density operator of the electromagnetic field that is suitable for all density operators and that reduces to the coherent-state $P$ representation when the latter exists. It expresses the density operator $\rho$ as the sum of four terms, each of which is a two-dimensional weighted integral over outer products of coherent states. The first integral has the form of the $P$ representation, i.e., the outer products are projection operators. The absence of singularities in this term is achieved by the presence of the three supplementary integrals which vanish when the density operator possesses the $P$ representation. In general, for stationary density operators, only the first two terms of the regularized $P$ representation are necessary. A simple prescription is given for obtaining the four weight functions of this representation from the function $〈\alpha \left|\rho \right|\alpha 〉$, where $|\alpha 〉$ is a coherent state and $\rho$ is the density operator. According to this prescription, the $P$ representation does not exist and one or more of the supplementary, regularizing terms is necessary when the function $〈\alpha \left|\rho \right|\alpha 〉$ contains a term that decreases more rapidly than $\exp (-{\left|\alpha \right|}^2)$ as $\left|\alpha \right|\to \infty$. The regularized $P$ representation affords nonsingular integral expressions for all density operators and for most expectation values, including, when they are finite, those of the normally ordered products of the creation and annihilation operators, $a^{†}$ and $a$. The construction and use of this representation is illustrated with the aid of simple examples in which the density operator does not possess the $P$ representation.