Groups and Singularities in Quantum Control and Nonlinear Design

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

0072415 Ramakrishna The project aims to develop i) numerically inexpensive methods for state preparation in quantum control; and ii) control laws for nonlinear systems in the presence of singularities, and techniques for singular exterior differential systems. Several extant and future technologies call for the control of quantum systems - semiconductor heterostructures, spectroscopy, atomic lasers, quantum information processing, control of molecular dynamics, to name a few. Many of these problems can, after judicious modeling and optimization, be recast as the problem of preparing desired unitary generators, via an external control acting on a quantum system. This leads quickly to the problem of path planning for systems with drift on Lie groups, via controls that are constrained. Path planning for systems with drift, with few and constrained inputs, has not hitherto been studied. The only other alternatives for the control of quantum systems are computationally expensive optimization, and tracking which relies heavily on intuition. In this project the role of intuition and optimization will be limited to reformulating a given physical objective as state preparation for a quantum system. The latter problem will be attacked by developing structured factorization of unitary matrices, the structure being determined by the constraints. The resulting control design will be compared with other techniques such as inversion and optimal control. Applications to lasing without inversion, molecular control, quantum computing and information will be then carried out. Related issues, such as tractable descriptions of reachable sets of quantum systems, will also be studied. The second part of the project stems from the fact that the extant non-linear control schemes are rarely directly applicable in singular situations such as a varying rank decoupling matrix and a change in relative degree. Tools from singularity theory, the theory of foliations, nonsmooth analysis and Lie groups will be developed to address this problem. A key feature will be the use of differential forms instead of vector fields for studying such issues. This should also have payoffs for singular exterior differential systems. Applications to techniques such as the method of intermediate integrals for partial differential equations and equations that change type will be studied from this vantage point. It is worth remarking that both components of the project call for similar mathematics. Successful completion of this project should yield benefits for several current and future technologies. For instance, the fabrication of a quantum computer calls for solutions to precisely the type of problems being studied in the project. Similarly, obtaining a snapshot of chemical reactions at the molecular level also requires the control of quantum processes. Active control of quantum phenomena would also lead to the control of chemical reactions via tuned lasers. This should enhance product specificity in chemical reactions, thereby minimizing the presence of products that are not desired. The control of quantum phenomena is still in its infancy. The project aims to move it along further by linking it concretely to traditional control theory. Conversely, the results of the project should shed light on the control of macroworld systems. This will also have benefits for traditional technologies such as robotics, consumer electronics and electrical networks, since the underlying mathematics is quite similar. From a didactic point of view, the project's contribution will be to show to undergraduates and graduate students alike that advanced mathematics is not an arid field, but is in fact a lively process with palpable connections to real life situations. This should enhance the likelihood of students, especially from under represented groups, to pursue higher education in mathematics, science and technology.