Bloch Sphere of the Qutrit System

TL;DR

This paper introduces a new method to analyze the Bloch space of qutrit systems using SU(3) group theory, revealing complex geometric structures and dynamics, including a splitting of the Bloch sphere into lower-dimensional spheres under resonant conditions.

Contribution

It provides a novel geometric and dynamical analysis of the qutrit Bloch space, including the splitting of the Bloch sphere into $\

Findings

Bloch space norm matches geometric and dynamical calculations.

Under resonance, the Bloch sphere splits into $\

complex Bloch trajectories vary across configurations.

Abstract

We present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere $\mathbb{S}^{7}$ splits into two parts, a four-sphere $\mathbb{S}^{4}$ and a two-sphere $\mathbb{S}^{2}$. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.