Computational Techniques for Optimal Control of Quantum System

The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time—dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite—dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite—dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2— versus H1—based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. vi A matrix—free cascadic BFGS algorithm is introduced in the L2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov—Newton scheme applied to a norm—preserving discrete model of a dipole quantum control problem. The use of second—order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first— and second—order optimality conditions. Criteria for the discretization of the non—convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust line- search procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high—energy state transitions. Control of bound—to—bound and bound—to—continuum transitions in open quantum systems and vector field control of two—dimensional systems is presented. An efficient space—time spectral discretization of the time—dependent Schrödinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined

Optimal control of number squeezing in trapped Bose-Einstein condensates

We theoretically analyze atom interferometry based on trapped ultracold atoms, and employ optimal control theory in order to optimize number squeezing and condensate trapping. In our simulations, we consider a setup where the confinement potential is transformed from a single to a double well, which allows to split the condensate. To avoid in the ensuing phase-accumulation stage of the interferometer dephasing due to the nonlinear atom-atom interactions, the atom number fluctuations between the two wells should be sufficiently low. We show that low number fluctuations (high number squeezing) can be obtained by optimized splitting protocols. Two types of solutions are found: in the Josephson regime we find an oscillatory tunnel control and a parametric amplification of number squeezing, while in the Fock regime squeezing is obtained solely due to the nonlinear coupling, which is transformed to number squeezing by peaked tunnel pulses. We study splitting and squeezing within the frameworks of a generic two-mode model, which allows us to study the basic physical mechanisms, and the multi-configurational time dependent Hartree for bosons method, which allows for a microscopic modeling of the splitting dynamics in realistic experiments. Both models give similar results, thus highlighting the general nature of these two solution schemes. We finally analyze our results in the context of atom interferometry.Comment: 17 pages, 21 figures, minor correction

On the treatment of distributed uncertainties in PDE‐constrained optimization

Most physical phenomena are significantly affected by uncertainties associated with variations in properties and fluctuations in operating conditions. This has to be reflected also in the design and control of real-application systems. Recent advances in PDE constrained optimization open the possibility of realistic optimization of such systems in the presence of model and data uncertainties. These emerging techniques require only the knowledge of the probability distribution of the perturbations, which is usually available, and provide optimization solutions that are robust with respect to the stochasticity of the application framework. In this paper, some of these methodologies are reviewed. The focus is on PDE constrained optimization frameworks where distributed uncertainties are modeled by random fields and the structures in the underlying optimization problems are exploited in the form of multigrid methods and one-shot methods. Applications are presented, including control problems with uncertain coefficients and erodynamic design under geometric uncertainties (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim