Geometric Analysis and Optimal Control of Quantum Systems in the KP Configuration; Generalizations to nonlinear Systems with Symmetries

Iowa State University, Ames IA

Investigators

Abstract

The precise and efficient control of the state of quantum mechanical systems, such as atoms, nuclei and electrons, is a requirement in most applications of these systems. Such a control is typically obtained through the interaction with external, appropriately shaped, electromagnetic fields. Moreover, one often wants not only to drive the state to a desired value but also to optimize the available resources. The minimization of time is especially important. In applications to computation, fast dynamics result in the speed-up of the implemented algorithms. Furthermore, in general, the evolution has to occur within the time frame during which the mathematical model can be considered valid, before the effect of un-modeled dynamics becomes relevant. In this context, this research will accomplish three inter-related objectives: 1) It will provide explicit optimal control design algorithms for a large class of quantum mechanical systems very common in important applications. 2) It will provide an in depth mathematical analysis of the role of symmetries in quantum control systems and how these symmetries can be used to simplify mathematical models of quantum systems, thus considerably extending the existing theory. 3) It will validate the mathematical results through experiments in quantum optics and nuclear magnetic resonance via the collaboration with experimental laboratories. The overall result will be a rich toolbox for the optimal manipulation of quantum mechanical systems to be used in applications in secure communication, powerful quantum computing, design of measurement devices, medical diagnostics and, in general, every device which uses quantum systems. The activities will involve an interdisciplinary research team composed of engineers, physicists and mathematicians with the objective of developing a common language and science. The resulting knowledge will be the basis of a new area of engineering and a curriculum in Quantum Engineering, a field that will become very important in the future as the applications of quantum mechanics in everyday life continue to expand. The main mathematical tools used and developed in this research come from the field of differential geometry and in particular Riemannian and sub-Riemannian geometry. The starting point are the so-called KP mathematical models which are models whose state varies on a Lie group and whose dynamical equations correspond to a Cartan decomposition of the associated Lie algebra. The corresponding optimal control problems are, on one hand, very common in applications, and, on the other hand, explicitly solvable. In the project they serve as a test-bed to investigate properties of quantum control systems in general. These involve the role of symmetries, the qualitative behavior of optimal trajectories (geodesics) and the geometry of the reachable sets. A key technical ingredient of the mathematical approach is the use of symmetry reduction as a tool to analyze the control problem on a lower dimensional quotient space. On this space a simpler control problem can be posed and often explicitly solved. This procedure will substantially enlarge the existing toolbox in quantum control, which is frequently restricted to small dimensional systems. The experimental implementations of the resulting control design will be for systems in quantum optics and nuclear magnetic resonance, which are among the most promising candidates for the construction of quantum computers. Furthermore, the mathematical analysis will require the introduction of elements from the theory of singular and stratified spaces, which is important in other areas of applications of control besides quantum mechanics. In this context, this project will contribute to the development of control theory for classical systems as well.