New wigner function for number and phase

Joan A. Vaccaro, Amitabh Joshi

David K. Ferry, Xavier Oriols, Josef Weinbub

K. Kakazu

Employing the extended phase states, we show that six conditions analogous to Wigner's original ones cannot lead uniquely to the number-phase Wigner function. To show this fact explicitly, we propose a new example of the Wigner function satisfying all these conditions. The nonuniqueness of the number-phase Wigner function results from the 2π-periodicity of the phase. It is also shown that the two number-phase Wigner functions obtained by several authors have an endpoint problem, which leads to an ambiguous result. Their correct integral forms are derived from those defined in the extended Fock space with negative number states.

Maciej Przanowski, Przemysław Brzykcy, Jaromir Tosiek

William K. Wootters

John A. Vaccaro, David Thomas Pegg

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.