# The Wigner flow on the sphere

## TL;DR

This paper derives a continuity equation for the SU(2) Wigner function's evolution under nonlinear Kerr effects, highlighting quantum-classical differences in the Wigner current's structure and stagnation lines.

## Contribution

It provides explicit expressions for the quantum Wigner current and analyzes the quantum-classical transition in the Wigner flow on the sphere.

## Key findings

- Quantum Wigner current differs significantly from the classical one.
- Classical limit shows distinct stagnation line structures.
- Explicit formulas for Wigner current under Kerr evolution.

## Abstract

We derive a continuity equation for the evolution of the SU(2) Wigner function under nonlinear Kerr evolution. We give explicit expressions for the resulting quantum Wigner current, and discuss the appearance of the classical limit. We show that the global structure of the quantum current significantly differs from the classical one, which is clearly reflected in the form of the corresponding stagnation lines.