Mathematical and Numerical Aspects of Quantum Dynamics

New optimal control problems for quantum systems

Gero Friesecke

Technische Universität München

Optimal control theory (OCT) for quantum systems is traditionally considered with simple ``cost functionals'' (e.g., a goal term promoting closeness to a favoured subspace plus a space-time L2 norm for regularization.) But the general OCT setting offers great freedom to design and compute controls and states with desired features. As a first example (joint with Felix Henneke und Karl Kunisch, AIMS Math. Control and Related Fields 2018, 8(1): 155-176) we show how one can design frequency-sparse quantum controls, by means of penalizing a suitable time-frequency representation of the control field and employing an L1 or measure form with respect to frequency. Simulations show, and a rigorous proof confirms, that the control fields uses only a finite number of frequencies, even for infinite-dimensional Schroedinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. As a second example (joint with Michael Kniely, arXiv 2018) we formulate various desirable properties of photovoltaic materials as mathematical control goals for excitations in the Kohn-Sham equations, show that the resulting problems are well posed, and present illustrative numerical simulations of optimal doping profiles and the resulting excitations for 1D finite nanocrystals.