Holonomic quantum control via nonlinear realizations

Geometric phases appear as holonomies in principal bundles over quantum state spaces. In this work, we consider the case when the principal bundle itself is a Lie group and the quantum space of states a homogeneous space of that group. This structure allows the application of the theory of nonlinear realizations of symmetry for the construction non-Abelian geometric phases corresponding to this bundle structure. When the quantum state space is the complex Grassmann manifold U(N)/(U(N-k) × U(k)), we identify the total non-Abelian Aharonov-Anandan phase as the U(k)-valued cocycle of the U(N) action on the Grassmann manifold . We describe generalizations of this result in two cases: 1) the case of isospectral dynamics of mixed states, 2) the case of non-self adjoint dynamics over the Grassmannian.