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Physical Review

Density Operators for Coherent Fields

Urbaan M. Titulaer, Roy J. Glauber

Published May 27, 1966• 286 citations
Publisher

Abstract

We study the forms which the conditions for coherence impose upon the density operators for electromagnetic fields. All fields possessing first-order coherence, we show, may be regarded as ones in which only a single mode is excited; this mode need not, however, be a monochromatic one. The higher order coherence conditions may be regarded as specifying certain moments of the distribution of the number of photons in the single excited mode. Full coherence, in particular, is shown to require that the number of photons present follow a Poisson distribution. The class of fields which possess full coherence is shown to be larger than the class of eigenstates of the annihilation operators. The unique character of the eigenstates, on the other hand, is demonstrated by means of a number of simple theorems.

DOI
10.1103/physrev.145.1041

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Physical Review•Jun 15, 1963•4,074 citations
We study the forms which the conditions for coherence impose upon the density operators for electromagnetic fields. All fields possessing first-order coherence, we show, may be regarded as ones in which only a single mode is excited; this mode need not, however, be a monochromatic one. The higher order coherence conditions may be regarded as specifying certain moments of the distribution of the number of photons in the single excited mode. Full coherence, in particular, is shown to require that the number of photons present follow a Poisson distribution. The class of fields which possess full coherence is shown to be larger than the class of eigenstates of the annihilation operators. The unique character of the eigenstates, on the other hand, is demonstrated by means of a number of simple theorems.
The "diagonal" PPP 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 PPP representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the PPP representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the PPP 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.
Simple proofs are given for two properties of a Bose field discovered recently in the quantum-theoretic description of optical coherence. The first is the theorem of Glauber and Titulaer that first-order coherence means that only one mode is excited. The second is the theorem of Aharanov, Falkoff, Lerner, and Pendleton that eigenstates of the annihilation operators are characterized by their ability to factor when the system is divided into two channels. The restriction of the latter to the case of a single excited mode is removed.
Recently Glauber has described the properties of coherent radiation fields, and has constructed the density matrix of the field in two simple cases: (1) The radiating system is a classical radiator and no reaction is considered; (2) the central-limit theorem applies to a collection of radiators. This paper investigates other simple "almost" exactly soluble problems, in which a quantum-mechanical two-level system interacts with a quantized electromagnetic field originally in a pure coherent state in a single mode. The first-order correlation function ${G}^{(1)}=〈{E}^{\ensuremath{-}}{E}^{+}〉$ is compared with $〈{E}^{\ensuremath{-}}〉〈{E}^{+}〉$ at resonance when the stimulating field is initially a pure coherent state and the two-level system is initially in its excited state. The corresponding quantities are also computed for a field whose initial density matrix is a Gaussian superposition of coherent states (e.g., blackbody radiation), as well as for a field which is initially described as having a given number of photons.
A factorization condition which must be satisfied by the first nnn correlation functions for the electromagnetic field operators has been used to define nth-order coherence. The first-order coherence condition has been shown to imply maximum fringe contrast in interference patterns. In the present paper we investigate the mathematical consequences of assuming the condition for maximum fringe contrast. By considering the correlation functions as scalar products and formulating rigorous inequalities for them we are able to show that the assumed condition in turn implies factorization of the first-order correlation function. By extending the same methods we are able to show that all of the higher order correlation functions factorize into forms similar to those required for full coherence, but differing from them through the inclusion of a sequence of constant numerical factors. These coefficients are shown to furnish a convenient description of the higher-order coherence properties of the field. Their values are presented for some typical examples. We derive a number of inequalities satisfied by the coefficients for the case of fields which possess positive-definite weight functions in the PPP representation. Some inequalities obeyed by the correlation functions for such fields are derived as well.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.