Bloch sphere like construction of SU(3) Hamiltonians using unitary integration

TL;DR

This paper develops a geometric representation of three-level quantum systems using SU(3) group structures, extending the Bloch sphere concept to facilitate analysis of complex quantum evolutions and phases.

Contribution

It introduces a novel geometrical framework for SU(3) Hamiltonians, representing the evolution as two four-dimensional manifolds analogous to the Bloch sphere.

Findings

Geometrical picture for SU(3) systems using two four-dimensional manifolds.

Application of the framework to time-dependent couplings in three-level systems.

Development of geometrical phases for specific quantum evolutions.

Abstract

The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.