Time evolution of coupled spin systems in a generalized Wigner representation

TL;DR

This paper develops a comprehensive Wigner function formalism for finite-dimensional coupled spin systems, enabling visualization and prediction of their quantum state evolution, with explicit results for systems of up to three spins 1/2.

Contribution

It introduces a complete, self-contained Wigner formalism for coupled spins, extending phase-space methods to finite-dimensional multi-partite quantum systems.

Findings

Derived equations of motion for coupled spins using Wigner functions

Explicitly calculated time evolution for up to three spins 1/2

Provided visualizable examples for experimental scenarios

Abstract

Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. Continuous phase spaces have also been studied for finite-dimensional quantum systems such as spin systems. However, much less is known for finite-dimensional, coupled systems, and we present a complete theory of Wigner functions for this case. In particular, we provide a self-contained Wigner formalism for describing and predicting the time evolution of coupled spins which lends itself to visualizing the high-dimensional structure of multi-partite quantum states. We completely treat the case of an arbitrary number of coupled spins 1/2, thereby establishing the equation of motion using Wigner functions. The explicit form of the time evolution is then calculated for up to three spins 1/2. The underlying physical principles of our Wigner representations for coupled spin systems are illustrated with multiple examples which are easily translatable to other experimental scenarios.