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Holonomic quantum control via nonlinear realizations
  • David Bar Moshe
David Bar Moshe
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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.