Bloch space structure of cascade, lambda and vee type of three-level systems and qutrit wave function

TL;DR

This paper explores the geometric structure of three-level quantum systems (cascade, lambda, vee types) using SU(3) Hamiltonians, revealing a partition of the Bloch space at resonance and proposing a qutrit wave function representation.

Contribution

It characterizes the Bloch space structure of three-level systems and introduces a qutrit wave function representation, highlighting their geometric and algebraic properties.

Findings

At resonance, the Bloch space splits into two subspaces ${\\mathcal{S}}^2$ and ${\\mathcal{S}}^4$

The systems are described by three different SU(3) Hamiltonians

A possible qutrit wave function representation is provided

Abstract

The cascade, lambda and vee type of three-level systems are shown to be described by three different Hamiltonians in the SU(3) basis. We investigate the Bloch space structure of each configuration by solving the corresponding Bloch equation and show that at resonance, the seven-dimensional Bloch sphere ${\mathcal S}^7$ is broken into two distinct subspaces ${\mathcal S}^2{\times}{\mathcal S}^4$ due to the existence of a pair of quadratic constants. We also give a possible representation of the qutrit wave function and discuss its equivalence with the three-level system.