Ray space Riccati evolution and geometric phases for N-level quantum systems

TL;DR

This paper derives matrix Riccati equations for N-level quantum systems' reduced dynamics, explores their relation to geometric phases, and links quantum evolution to classical Hamiltonian dynamics.

Contribution

It provides a simplified derivation of Riccati equations for N-level systems and connects quantum geometric phases with classical Hamiltonian evolution.

Findings

Riccati equations govern reduced quantum dynamics on coset spaces

Connection established between quantum geometric phases and classical phase space

Reformulation of Riccati equations as classical Hamiltonian evolution

Abstract

We present a simple derivation of the matrix Riccati equations governing the reduced dynamics as one descends from the group U(N) describing the Schroedinger evolution of an N-level quantum system to the various coset spaces, Grassmanian manifolds, associated with it. The special case pertaining to the geometric phase in N-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.