Canonical form of the Evolution Operator of a Time-Dependent Hamiltonian in the Three Level System

TL;DR

This paper derives a canonical form of the evolution operator for a three-level quantum system with a time-dependent Hamiltonian, involving complex Riccati equations and phase equations based on SU(3).

Contribution

It presents a new canonical form of the evolution operator for three-level systems, expanding understanding of their dynamics with Riccati and phase equations.

Findings

Derived three complex Riccati differential equations for the system

Established a canonical form of the evolution operator

Connected the equations to SU(3) symmetry

Abstract

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on $SU(3)$ and its dimension is $8$, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.