08-Jan-2009, Chemical Physics Seminars, Tel Aviv University <br>
<b>Simulation and control of quantum systems</b><br>
Dr. Ilan Degani, Applied Math Department, Bergen University, Norway<br>
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<i>Abstract:</i><br>
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We briefly overview some examples of quantum control problems appearing in different fields, ranging from existing technology
(NMR/MRI) to research (control of nano devices, laser control of molecules). In all these examples the same type of problem emerges: to control a bilinear flow on a unitary (or orthogonal) group with piecewise constant controls. We review the existing optimal control approach, and point out some of its problems. Namely, the standard extremal control equations can not generally hold for piecewise constant controls; the only structural requirement from the controls is to have small $L_2$ norm. We then discuss our approach which derives the correct extremal equations, and which can prefer controls with desired structure. Particularly, we are able to compute optimal controls belonging to a desired subspace of control functions. Thus our methods may hopefuly be useful for computing controls which are producible by realistic laboratory lasers.

Time permitting, we shall also discuss an efficient discretization scheme for Schrodinger equations based on Monge-Ampere PDEs. We compute coordinate transformations, from a computational domain to the physical one, which can zoom in or out of desired regions. In this way computational effort is concentrated where it is needed.
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<i>References</i><br>
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1) I. Degani, A. Zanna, Lene Saelen, and Raymond Nepstad, Quantum control with piecewise constant control functions, submitted (2008).
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2) I. Degani, Monge-Ampere grids and the multidimensional mapped Fourier method, journal of chemical physics, 128 (2008) 164108.