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arXiv:1009.4030 · Journal of Modern Optics

Operator formalism for the Wigner phase distribution

T. Subeesh, Vivishek Sudhir

Published Apr 18, 2011• 4 citations
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Abstract

We construct a Hermitian phase operator for the radially integrated Wigner distribution, which is known to be sensitive to phase. We show that this operator is complete and also elucidate a set of complete but non-orthogonal states that seems to be naturally associated with such an operator. Further, we show that our operator satisfies a weak equivalence relation with the Pegg–Barnett operator, thus showing that the essential phase information furnished by both formalisms are the same. It is also shown that this operator gives results which are in agreement with the expected uniform phase distribution of a Fock state.

DOI
10.1080/09500340.2011.569859

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We construct a Hermitian phase operator for the radially integrated Wigner distribution, which is known to be sensitive to phase. We show that this operator is complete and also elucidate a set of complete but non-orthogonal states that seems to be naturally associated with such an operator. Further, we show that our operator satisfies a weak equivalence relation with the Pegg–Barnett operator, thus showing that the essential phase information furnished by both formalisms are the same. It is also shown that this operator gives results which are in agreement with the expected uniform phase distribution of a Fock state.
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.
The phase distribution obtained within the Pegg–Barnett Hermitian phase formalism is compared to the phase distributions obtained from the s -parametrized quasiprobability distributions integrated over the "radial" variable for some real states of the field. Exact analytical formulas for the s -parametrized phase distributions of coherent states, squeezed states, and displaced number states are obtained. A general formula relating the s -parametrized phase distributions to the Pegg–Barnett distribution is derived. Numerical examples illustrating the similarities and differences are presented in a graphical form.
The problem of defining a Hermitian quantum phase operator is nearly as old as quantum mechanics itself. Throughout the years, a number of solutions have been proposed, ranging from abstract operator formalisms to phase-space methods. In this work, we make an explicit connection between two of the most prominent approaches by proving that the probability distribution of phase in the Paul formalism follows exactly from the Pegg-Barnett formalism by combining the latter with the quantum-limited amplifier channel. Our findings suggest that the Paul framework may be viewed as a semiclassical limit of the Pegg-Barnett approach.