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Physica Scripta

Ordering of "ladder" operators, the Wigner function for number and phase, and the enlarged Hilbert space

A. Lukš, Vlasta Peřinová

Published Jan 1, 1993• 51 citations
Publisher

Abstract

A suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field. The extended Wigner function for number and phase in the enlarged Hilbert space has been used for the derivation of the Wigner function for number and phase in the original Hilbert space.

DOI
10.1088/0031-8949/1993/t48/014

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A suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field. The extended Wigner function for number and phase in the enlarged Hilbert space has been used for the derivation of the Wigner function for number and phase in the original Hilbert space.
The general Weyl–Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd-dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number–phase Wigner function in quantum optics. Analogous results for the “almost” symmetric ordering in an even-dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.
It is shown that the number-phase (or rotational) Wigner function is obtained from the Weyl symmetrization rule for the correspondence between classical functions and quantum operators. In spite of the complicated form of the commutator for the number and phase operators, the Weyl symmetrization rule is similar to that in the case of the position and momentum operators. In addition, it is found that the ordering of the number and phase operators also has the same structure as that for the position-momentum pair.
A class of positive quantum phase-space distributions based on antinormal ordering of the squeezed-photon annihilation and creation operators is introduced. This class of distributions, referred to as the generalized antinormally ordered distributions, is shown to be generated when the Wigner distribution is smoothed with a squeezed Gaussian wave packet. The distribution function associated with generalized antinormal ordering can thus be identified as the Husimi distribution function. The usefulness of the distributions in studies of the quantum particle dynamics is illustrated with simple examples.
On the basis of the phase states, we present the correct integral expressions of the two number-phase Wigner functions discovered so far. These correct forms are derived from those defined in the extended Fock space with negative number states. The analogous conditions to Wigner's original ones cannot lead to the number-phase function uniquely. To show this fact explicitly, we propose another function satisfying all these conditions. It is also shown that the ununiqueness of the number-phase Wigner function result from the phase-periodicity problem.