Analytic controllability of time-dependent quantum control systems

The question of controllability is investigated for a quantum control systemin which the Hamiltonian operator components carry explicit time dependencewhich is not under the control of an external agent. We consider the generalsituation in which the state moves in an infinite-dimensional Hilbert space, adrift term is present, and the operators driving the state evolution may beunbounded. However, considerations are restricted by the assumption that thereexists an analytic domain, dense in the state space, on which solutions of thecontrolled Schrodinger equation may be expressed globally in exponential form.The issue of controllability then naturally focuses on the ability to steer thequantum state on a finite-dimensional submanifold of the unit sphere in Hilbertspace -- and thus on analytic controllability. A relatively straightforwardstrategy allows the extension of Lie-algebraic conditions for strong analyticcontrollability derived earlier for the simpler, time-independent system inwhich the drift Hamiltonian and the interaction Hamiltonia have no intrinsictime dependence. Enlarging the state space by one dimension corresponding tothe time variable, we construct an augmented control system that can be treatedas time-independent. Methods developed by Kunita can then be implemented toestablish controllability conditions for the one-dimension-reduced systemdefined by the original time-dependent Schrodinger control problem. Theapplicability of the resulting theorem is illustrated with selected examples.Comment: 13 page