Analytic controllability of time-dependent quantum control systems

Abstract

Abstract-The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert spaceand thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.

Finite Controllability of Infinite-Dimensional Quantum Systems

“…[30] presents a scheme that is well-known in atomic physics-when an infinite-dimensional quantum system has unequally spaced bound state energy levels, single resonant fields can be used to transfer population (albeit very slowly). [31] provides a prescription for determining whether there exists a submanifold that is strongly analytic controllable, but this prescription does not identify the submanifolds. [32] presents an adiabatic method of controlling a sequentially connected system in which the transition couplings get weaker as one moves away from the ground state.…”

Control problems in quantum systems

“…The controllability issue was first attacked by Tarn et al in the early 1980s with respect to quantum systems with finite-dimensional controllability Lie algebras; later this was extended to time-dependent systems [47] and systems with infinite-dimensional controllability Lie algebras [48]. A special class of such systems was also extensively studied in the control of molecular systems, where quantum systems were assumed to possess at first approx-imation a finite number of levels.…”

Limitations on the control of Schrödinger equations

“…For more details, see [10,37] and the literature therein. (For alternate choices of the operators H j see also [14,24].) A large amount of literature exists on the controllability of finite-dimensional quantum systems.…”

“…(A number of important references may be found in [39], for example.) Specific infinitedimensional quantum systems are treated in [35] and [24] which deal with the control of the quantumharmonic oscillator and the hydrogen atom, respectively. (ii) The particle-particle interaction in many-body quantum systems is often modelled by a nonlinear, self-consistent, potential, which leads to the Hartree equation; the corresponding control problem with dipole term for these models takes the form…”