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 $\ensuremath{\rho}$ 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 $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state and $\ensuremath{\rho}$ 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 $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ contains a term that decreases more rapidly than $\mathrm{exp}(\ensuremath{-}{|\ensuremath{\alpha}|}^{2})$ as $|\ensuremath{\alpha}|\ensuremath{\rightarrow}\ensuremath{\infty}$. 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, ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ 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.
The "diagonal"
P representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the
P representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the
P representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the
P representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the
P representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the
P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators
a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the
P representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the
P representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter
s assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for
Res<0, has bounded eigenvalues for
Res=0, and has infinite eigenvalues for
s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter
s is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to
s=+1, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the
P representation is appropriate.
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 pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.
In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) field that is based on the number states, the eigenstates of the Hamiltonian of the EM field. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We first determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce different representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM field or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to find the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the different representations. Most importantly, certain field states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.