Using an SU(3)/U(2) Wigner Function to Represent Noisy Spin Ensembles

TL;DR

This paper introduces a novel SU(3) Wigner function representation for noisy spin ensembles, enabling visualization and analysis of states affected by local noise outside the symmetric subspace.

Contribution

It extends the Wigner function framework from SU(2) to SU(3) to effectively represent noisy spin ensembles affected by local noise sources.

Findings

Physical noise constraints reduce parameters to three, interpretable as polar, azimuthal, and radial components.

The solid spin Wigner function is visualized on a solid ball, providing intuitive insights.

The method captures state dynamics outside the symmetric subspace due to noise.

Abstract

The SU(2) Wigner function represents a quantum state of a spin-$J$ as a real-valued function on the surface of a 2-sphere. For an ensemble of $N$ spin-1/2 particles, this representation is useful when the dynamics is restricted to a single SU(2) irrep, e.g., the symmetric subspace with $J=N/2$. Physically relevant noise sources tend to be local, such as spontaneous emission, depolarizing, and incoherent optical pumping, all of which transfer the state outside of the initial irrep, and as such the SU(2) Wigner function is no longer a useful representation. In this work, we address this issue by encoding a noisy spin ensemble in an SU(3) irrep, and evaluating the SU(3) Wigner function for that irrep. We find that physical constraints enforced by the noise eliminate all but three real parameters from the input to the Wigner function, which can then be interpreted as a polar, azimuthal, and radial component. This interpretation leads us to refer to the resulting Wigner function as the solid spin Wigner function, visualized on a solid ball rather than a hollow sphere.