Publication Date

9-16-2014

Abstract

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.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Evangelos A. Coutsias

Second Committee Member

Sanjay Krishna

Third Committee Member

Stephen Lau

Fourth Committee Member

Pavel Lushnikov

Fifth Committee Member

Louis A. Romero

Language

English

Keywords

Optimal control, quantum mechanics, partial differential equations, numerical methods

Document Type

Dissertation

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