Adaptive Magnus Integrators for Quantum Systems

In the project `Adaptive Magnus Integrators for Quantum Systems we will study large time- dependent systems of ordinary differential equations of Schrödinger type. The models considered are relevant, among others, for the simulation of a new type of transistor, a new class of solar cells, and the control of quantum systems, i.e., the optimization of a system with respect to a suitable criterion. The understanding of the underlying physical processes strongly relies on numerical simulations by computer based methods. To advance the numerical approximation in time, Magnus- type integrators are put forward, which approximate the solution as a sequence of exponentials of fixed matrices. The main focus will be on the construction and analysis of error estimators for the adaptive choice of the time-steps, as adaptivity is the key to the realization of large-scale simulations of real-world application problems. We will construct and analyze high-order schemes equipped with a posteriori error estimators as the basis for the adaptive selection of the time steps. In each step, matrix exponentials have to be computed. These will be approximated in low-dimensional spaces by the Lanczos method. A defect-based error indicator will replace the common strategies to control the exponentiation algorithm which are viable only when the approximation is already close to the true solution. The new time integrators will hence enable a more accurate, efficient and reliable integration of the differential equations and moreover provide a posteriori information on the error, thereby enabling solid predictions regarding the properties of the investigated problems. The applications which will be investigated with the new numerical time propagators arise in solid state physics and the control of quantum systems. A new type of transistor, a Mott transistor, will be explored via numerical simulation, and a new class of solar cells, both based on transition metal oxide heterostructures. The Mott transistor has an ideal switching characteristic between on (metal) and off (insulator).The solar cells on the other hand are ideally suited to overcome the Schockley- Queisser limit of 33% efficiency, bearing good prospects for solar cells of the highest efficiency. An important aspect in technical applications is the control of quantum systems by manipulating electric and magnetic fields by laser pulses. This introduces the necessity to solve a large number of problems of the type treated by the numerical approaches we will provide, making efficiency of numerical time propagators an important issue. All the mentioned project goals cannot be realized with the required accuracy and efficiency by available numerical methods. The adaptive high-order methods of this project will hence also enable us to reach new frontiers regarding the proposed applications.

In the project "Adaptive Magnus Integrators for Quantum Systems", we studied mathematical models of a new type of oxide-based solar cells which promise a significant gain in the efficiency of energy conversion as compared to conventional Silicium-based solar cells. The study of the physical properties of solar cells based on such novel designs can at the first stage be based on the computer solution of a mathematical description of the physical process. The resulting models are solved numerically on a computer, which implies a huge demand on computing resources. To speed up this process, sophisticated numerical algorithms are required. Furthermore, it is important to estimate the quality of the obtained approximation as compared to the exact solution of the mathematical model. To reach the goals of enhanced efficiency and reliable accuracy, the numerical approximation can be equipped with adaptive error control. In our project, new mathematical approximation methods in conjunction with reliable error estimators and adaptive improvement of the solution process were devised and implemented on massively parallel computers. This served the goal to study the physical processes relevant to the design of novel oxide solar cells such as impact ionization and double occupation, both of which may serve to enhance the efficiency of energy conversion in solar cells. The results obtained in the project also have implications in other research areas such the study of superconductors.