Geometric phase and controllability of quantum systems

We investigate the problem of controllability of quantum systems. Through suitable adjustment of external forces (controls) one may guide the state of a quantum system. We give criteria when the system is completely controllable, i.e. when we have possibility to reach an arbitrary state in the phase space. We also investigate the structure of reachable sets. Special attention is paid to the symplectic structure which is intimately related to the geometric phase. The geometric phase of Aharonov and Anandan, as a generalization of Berryʹs phase in adiabatic processes, measures the “symplectic area” enclosed during a cyclic evolution in the phase space. We show that when the reachable set is a symplectic submanifold of the phase space, then the geometric phase could be derived via an “intrinsic” formula containing quantities from the reachable set only.