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Journal of Physics A Mathematical and General

Operational number-phase probability distribution

Masashi Ban

Published Sep 11, 1998• 0 citations
Publisher

Abstract

An operational probability distribution for number and phase is introduced and the properties are investigated. This operational probability distribution is obtained by replacing the displacement operator of position and momentum in operational phase-space probability distribution with the displacement operator of number and phase. It is shown that the operational number-phase probability distribution is represented by the convolution of two Wigner functions for number and phase.

DOI
10.1088/0305-4470/31/36/001

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An operational probability distribution for number and phase is introduced and the properties are investigated. This operational probability distribution is obtained by replacing the displacement operator of position and momentum in operational phase-space probability distribution with the displacement operator of number and phase. It is shown that the operational number-phase probability distribution is represented by the convolution of two Wigner functions for number and phase.
We propose a unified method for obtaining the number-phase distribution functions in the extended Fock space and apply it to various distribution functions. A physical number-phase distribution function is easily obtained; it is the expectation value of an extended Wigner operator in a physical state. This method corresponds to Cohen's method for the position-momentum Wigner functions. The properties of the number-phase distribution functions and their relations are presented in a unified manner. Also, through the distribution functions, the correspondence between classical functions and quantum operators is obtained explicitly.
Phase properties of the displaced number states are studied. Exact analytical formulae describing phase distributions based on different phase approaches are obtained and illustrated graphically. It is shown that the Pegg-Barnett phase distribution P( theta ) and the phase distribution P w ( theta ) associated with the Wigner function are very close to each other, while the phase distribution P Q ( theta ) associated with the Q function carries less phase information. The results have clear interpretation in terms of the area of overlap in phase space.
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.