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Journal of Mathematical Physics

Wigner distribution function for finite systems

Natig M. Atakishiyev, S. M. Chumakov, Kurt Bernardo Wolf

Published Dec 1, 1998• 77 citations
Publisher

Abstract

We construct a Wigner distribution function for finite data sets. It is based on a finite optical system; a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry. The dynamical group for this model is SU(2) and the wave functions are sets of N=2l+1 data points. The Wigner distribution function assigns classical c-numbers to the operators of position, momentum, and wave guide mode.

DOI
10.1063/1.532636

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We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.
Euclidean systems include poly- and monochromatic wide-angle optics, acoustics, and also infinite discrete data sets. We use a recently defined Wigner operator and (quasiprobability) distribution function to set up and study the phase-space evolution of these models, subject to differential and difference equations, respectively. Infinite data sets and two-dimensional monochromatic (Helmholtz) fields are thus shown by their Wigner function on a cylinder of direction and location; the Wigner function for polychromatic wavefields has ` c -number' coordinates of (two-dimensional) wavenumber and position.
A phase-space approach to finite-dimensional systems is developed from basic principles. For a system describable by a Hilbert space of dimension d we define a one-to-one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d . The properties fulfilled by this correspondence and its uniqueness are examined. This formalism is applied to the number difference and phase difference of a two-mode field. This case is compared with the marginal distribution for these variables arising from a two-mode Wigner function for number and phase.
We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. In this note we discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behaviour under symmetry groups, and informational completeness.