Understanding the global structure of two-level quantum systems with relaxation: Vector fields organized through the magic plane and the steady-state ellipsoid

The understanding of the dynamics of a two-level quantum system with relaxation is one of the fundamental questions of quantum mechanics. We describe geometrically the structure of this system by introducing different vector fields, which give insight into the global dynamics of the system. The vector fields are organized through two basic geometric objects, the magic plane and the steady-state ellipsoid. We show how these latter help analyzing the time-optimal control of two-level dissipative quantum systems.