Theory of Control of Quantum Systems

Abstract

We investigate the problem of optimal control of mixed-state quantum systems using a quantum statistical mechanics model and a Liouville space formulation of Hamiltonian and dissipative dynamics. The problem of optimal control is formulated as a problem of maximization of the ensemble average of an observable of the system, subject to certain constraints. In chapter two, we address the question of kinematical constraints on the evolution of the system and derive bounds on the expectation value of arbitrary observables for mixedstate quantum systems. The issue of dynamical realizability of the kinematical bounds is discussed and results on controllability of quantum systems are summarized in chapter three. In chapter four, we present an efficient, rapidly convergent feedback algorithm for constructing optimal controls numerically and prove its convergence properties. Finally, we apply our results on kinematical . bounds and controllability, ~s well as the algorithm presented in chapter four, to several optimal control problems, including maximization of the vibrational energy of a molecular bond, maximization of the top-level population for a three-level system with and without dissipation, and maximization of the energy for systems consisting of non-interacting subsystems, and discuss the results.