Wigner distribution on SU(2)

A. B. Klimov, S. M. Chumakov

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

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.

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

We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.

A. D. Jannussis, Antony Streklas, D. Sourlas +1 more

A. B. Klimov, S. M. Chumakov

We prove that the Wigner-Stratonovich-Agarwal operator that defines the quasi-probability distribution on the sphere [for the SU(2) dynamical group] can be written as an integral of the SU(2) (irreducible unitary) representation element with respect to a single variable that labels the orbits in the coadjoint representation. This allows us to consider contractions of the SU(2) quasi-probability distribution to the cases of the Heisenberg-Weyl group and the two-dimensional Euclidean group.

K. Schram, B.R.A. Nijboer