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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

On Glauber's representation for general quantum-mechanical operators

R. Bonifacio, Lorenzo M. Narducci, E. Montaldi

Published Feb 1, 1967• 12 citations
Publisher
DOI
10.1007/bf02771403

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The positive-P representation of Drummond and Gardiner expresses the density operator as {rho} = {integral} d{sup 2}{alpha} {integral} d{sup 2}{beta} P({alpha},{beta}) {vert bar}{alpha}> is a Glauber state for a particular mode of the radiation field. The generalization to a finite number of modes is straightforward. The situation for amplified spontaneous emission (ASE) is more complex, since there are no mirrors to define a discrete set of modes. Thus we must replace the positive-P representation for a single mode by a functional form. In particular, it is necessary to define generalized Glauber states for arbitrary field configurations. We spent most of our time in solving this problem and in obtaining the simplest consequences of the definition of functional Glauber states. In this paper we define generalized annihilation operators and functional Glauber states and exhibit their properties. We also use the generalized operators to define functional generalizations of the conventional phase space methods.
It is proved that every density matrix is the limit, in the sense of weak operator convergence, of a sequence of operators each of which may be represented as an integral over projection operators onto coherent states (in the sense of Glauber) with a square-integrable weight function. This result is a special case of one that holds for all operators with trace and for overcomplete families of states other than just the coherent states. We prove our more general result, at no cost of complexity, within the more general framework of continuous-representation theory. The significance of our results for representing traces of operators is indicated.