Ray space `Riccati' evolution and geometric phases for e?‘?-level quantum systems

摘要

We present a simple derivation of the matrix Riccati equations governing the reduced dynamics as one descends from the group $mathbb{U}(N)$ describing the Schr?μdinger evolution of an e?‘?-level quantum system to the various coset spaces and Grassmanian manifolds associated with it. The special case pertaining to the geometric phase in e?‘?-level systems is described in detail. Further, we show how the matrix Riccati equation thus obtained can be reformulated as an equation describing Hamiltonian evolution in a classical phase space and establish correspondences between the two descriptions.