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Physical Review A

Coherence of neutron fields

E. Ledinegg, E. Schachinger

Published May 1, 1983• 3 citations
Publisher

Abstract

Glauber's definition of quantum coherence is used for neutron fields under the assumption that the complete occupation number space is a direct product of Fermi subspaces. As a result, completely coherent microfields are obtained which define a density operator in full analogy to Glauber's $P$ representation of boson fields. For better physical significance, a transformation from the $P$ representation to a momentum representation is performed. It is proved that the second-order coherence function in this representation is equivalent to Wolf's second-order coherence function of a classical Dirac field. Finally, the results of the theory are used to calculate explicitly the second-order coherence function and the coherence time of an ideally collimated neutron beam.

DOI
10.1103/physreva.27.2555

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Glauber's definition of quantum coherence is used for neutron fields under the assumption that the complete occupation number space is a direct product of Fermi subspaces. As a result, completely coherent microfields are obtained which define a density operator in full analogy to Glauber's PPP representation of boson fields. For better physical significance, a transformation from the PPP representation to a momentum representation is performed. It is proved that the second-order coherence function in this representation is equivalent to Wolf's second-order coherence function of a classical Dirac field. Finally, the results of the theory are used to calculate explicitly the second-order coherence function and the coherence time of an ideally collimated neutron beam.
The application of Glauber's definition (1963) of quantum coherence allows the introduction of neutron fields of partial coherence under the assumption that the complete occupation number space is a direct product of Fermi subspaces. Excitations are described by 'collective' creation and annihilation operators which span an algebra isomorphic to the sl(2,R) algebra. The associated coherent states are of partial coherence and the finite dimensional representation of the su(2,R) algebra is not regularly coherent. In contrast, the transition to an infinite dimensional representation space results in regular coherent properties of the field. This is demonstrated using the representation space of the su(1,1) algebra which has a real isomorphism to the sl(2,R) algebra. The coherent states calculated from Glauber's condition for coherence are completely coherent, and are, moreover, identical to those found by Barut and Girardello (1971) in starting from a far more abstract argument.
A functional formalism for coherence theory is developed in terms of the (quasi) probability functional of the radiation field. The formalism is closely connected to the P representation of the density operator. For the fields confined within a spatial volume of finite size, this connection is expressed as a multidimensional Fourier integral. A certain type of correspondence, illustrated by explicit examples, between the classical and the quantum mechanical description of the field statistics is established. The recent results of Titulaer and Glauber concerning the (1, 1)th-order coherent fields are analyzed from the point of view of the functional formulation.
The spatial coherence function in all three orthogonal directions has been measured by means of perfect crystal neutron interferometry. It has been demonstrated that the coherence function is given by the Fourier transform of the related momentum distribution, which in turn is determined by the collimation and monochromatization of the beam incident upon and traversing the interferometer. Thus, a measurement of the coherence function can replace a measurement of the momentum distribution in certain cases. Retrieval of the coherence can be accomplished by phase echo and/or postselection methods. A complete retrieval of the coherence is impossible in principle due to unavoidable loss factors.
In this paper we define the normalized coherence function of arbitrary order (m, n), in a manner which seems to be a natural generalization of that defined for the second-order coherence function. Both classical and quantized optical fields are considered and the results are compared. It is shown that for classical fields and also for quantized optical fields having nonnegative definite diagonal coherent state representations of the density operator, the modulus of these normalized coherence functions is bounded by the values 0 and 1. This definition differs from the one recently given by Glauber for quantized optical fields, where the normalized coherence functions may take arbitrarily large values even for fields having nonnegative definite diagonal representations of the density operator. Conditions for ``complete coherence,'' i.e., those under which the modulus of the normalized coherence function attains the limiting value 1, are discussed. Some consequences of stationarity and quasi-monochromaticity are also discussed.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.