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arXiv:quant-ph/0208095 · Journal of Optics B Quantum and Semiclassical Optics

A number–phase Wigner function

H. M. Moya-Cessa

Published Jun 1, 2003• 16 citations
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Abstract

One of the most prominent quasiprobability functions in quantum mechanics is the Wigner function, which gives the correct marginal probability functions if integrated over position or momentum. Here we depart from the definition of the position–momentum Wigner function to, by analogy, construct a number–phase Wigner function that, if summed over photon numbers, gives the correct phase distribution and, if integrated over phase, gives the correct photon distribution.

DOI
10.1088/1464-4266/5/3/367

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One of the most prominent quasiprobability functions in quantum mechanics is the Wigner function, which gives the correct marginal probability functions if integrated over position or momentum. Here we depart from the definition of the position–momentum Wigner function to, by analogy, construct a number–phase Wigner function that, if summed over photon numbers, gives the correct phase distribution and, if integrated over phase, gives the correct photon distribution.
We define a quasiprobability distribution SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the photon number and phase analogies of the properties of Wigner's original function, which describes the position and momentum observables. For example, the marginals of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the continuous phase and the discrete photon-number probability distributions. We give examples of the SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) representation of various states and show, in particular, that SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) displays the quantum interference associated with Schr\"odinger cat states. We also describe how SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) can be determined from quantities that are, in principle, measurable.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.
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.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.