The first part of this thesis focuses on the problem of visualization of quantum systems. We develop a method to visualize any operators in these systems, based on a generalization of Wigner representations. We demonstrate its intuitive character by illustrating a series of concepts in NMR spectroscopy for systems consisting of two spin-1/2 particles.
The second part is concerned with a problem of optimal control, where the problem of creating a propagator in the shortest amount of time is considered.
The first part of this thesis focuses on the problem of visualization of quantum systems. We develop a method to visualize any operators in these systems, based on a generalization of Wigner representations. We demonstrate its intuitive character by illustrating a series of concepts in NMR spectroscopy for systems consisting of two spin-1/2 particles.
The second part is concerned with a problem of optimal control, where the problem of creating a propagator in the shortest amount of time is considered.
In this thesis, we introduce new methodologies for the visualization, evaluation, and optimal control of spin systems based on four publications appended in Appendix A -D. Essential parts are established on the DROPS representation, a phase-space method which exploits generalized Wigner functions, denoted as droplets, to represent and visualize quantum systems.In the first publication "Wigner tomography of multispin quantum states" in Appendix A, a general sampling theorem for spherical functions is derived.This is evolved to a novel Wigner quantum state tomography: A technique to reconstruct the surface of the droplets representing quantum states by measuring expectation values of rotated axial tensor operators.This concept is extended in the second publication "Wigner process tomography: Visualization of spin propagators and their spinor properties" in Appendix B to experimentally recover the phase space representation of propagators by mapping these operators to density matrices of an augmented system with an additional ancillar qubit and by applying a modified form of the Wigner quantum state tomography.Both techniques are experimentally demonstrated for quantum states of systems up to three qubits and for standard 1 qubit gates using nuclear magnetic resonance spectroscopy.In doing so, the droplets are associated to measurable quantities, which were originally the mere result of an abstract mathematical mapping, and an interpretation in form of multipoles is provided.Those tomography schemes can be regarded as universally applicable validation methods for quantum systems.They are particularly well suited for the characterization of spin operators, pulse sequence elements, or entire pulse sequences in spectroscopy and in quantum information processing.In a third publication "Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems" in Appendix C, the concept of the DROPS representation, which is in its original form restricted to systems of up to four spins 1/2 due to computational challenges, is generalized to any number of coupled spins 1/2 and to two coupled spins with arbitrary spin numbers.These obstacles are circumvented by extending and introducing two different approaches based on explicit projection operators and coefficients of fractional parentage.This results in a powerful and versatilely applicable visualization tool for the analysis of highly diverse spin topologies and paves the way for the Wigner tomography schemes in such quantum systems.This thesis is complemented by proposing novel optimal control strategies, which, combined with the previous techniques, provide a rich toolbox for the control, analysis, and verification of spin systems.In a last publication "Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond" in Appendix D, efficient and robust control strategies are introduced by linking the free rotation of a classical rigid body to the control of two-level quantum systems using external electromagnetic fields.In the case of state to state transfer and gate implementation, a new class of optimal controls is derived.This relies on two free parameters and allows for a variable efficiency, time, and robustness of the transfer process.An experimental demonstration is presented using techniques of nuclear magnetic resonance.
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.
We study the tomography of propagators for spin systems in the context of finite-dimensional Wigner representations, which completely characterize and visualize operators using shapes assembled from linear combinations of spherical harmonics. The Wigner representation of a propagator can be experimentally recovered by measuring expectation values of rotated axial spherical tensor operators in an augmented system with an additional ancilla qubit. The methodology is experimentally demonstrated for standard one-qubit quantum gates using nuclear magnetic resonance spectroscopy. In particular, this approach provides a direct and compelling visualization of the spinor property of the propagators corresponding to the rotation of a spin-1/2 particle.
An efficient and accurate computational framework for solving control problems governed by quantum spin systems is presented. Spin systems are extremely important in modern quantum technologies such as nuclear magnetic resonance spectroscopy, quantum imaging and quantum computing. In these applications, two classes of quantum control problems arise: optimal control problems and exact-controllability problems, with a bilinear con- trol structure. These models correspond to the Schrödinger-Pauli equation, describing the time evolution of a spinor, and the Liouville-von Neumann master equation, describing the time evolution of a spinor and a density operator. This thesis focuses on quantum control problems governed by these models. An appropriate definition of the optimiza- tion objectives and of the admissible set of control functions allows to construct controls with specific properties. These properties are in general required by the physics and the technologies involved in quantum control applications. A main purpose of this work is to address non-differentiable quantum control problems. For this reason, a computational framework is developed to address optimal-control prob- lems, with possibly L1 -penalization term in the cost-functional, and exact-controllability problems. In both cases the set of admissible control functions is a subset of a Hilbert space. The bilinear control structure of the quantum model, the L1 -penalization term and the control constraints generate high non-linearities that make difficult to solve and analyse the corresponding control problems. The first part of this thesis focuses on the physical description of the spin of particles and of the magnetic resonance phenomenon. Afterwards, the controlled Schrödinger- Pauli equation and the Liouville-von Neumann master equation are discussed. These equations, like many other controlled quantum models, can be represented by dynamical systems with a bilinear control structure. In the second part of this thesis, theoretical investigations of optimal control problems, with a possible L1 -penalization term in the objective and control constraints, are consid- ered. In particular, existence of solutions, optimality conditions, and regularity properties of the optimal controls are discussed. In order to solve these optimal control problems, semi-smooth Newton methods are developed and proved to be superlinear convergent. The main difficulty in the implementation of a Newton method for optimal control prob- lems comes from the dimension of the Jacobian operator. In a discrete form, the Jacobian is a very large matrix, and this fact makes its construction infeasible from a practical point of view. For this reason, the focus of this work is on inexact Krylov-Newton methods, that combine the Newton method with Krylov iterative solvers for linear systems, and allows to avoid the construction of the discrete Jacobian. In the third part of this thesis, two methodologies for the exact-controllability of quan- tum spin systems are presented. The first method consists of a continuation technique, while the second method is based on a particular reformulation of the exact-control prob- lem. Both these methodologies address minimum L2 -norm exact-controllability problems. In the fourth part, the thesis focuses on the numerical analysis of quantum con- trol problems. In particular, the modified Crank-Nicolson scheme as an adequate time discretization of the Schrödinger equation is discussed, the first-discretize-then-optimize strategy is used to obtain a discrete reduced gradient formula for the differentiable part of the optimization objective, and implementation details and globalization strategies to guarantee an adequate numerical behaviour of semi-smooth Newton methods are treated. In the last part of this work, several numerical experiments are performed to vali- date the theoretical results and demonstrate the ability of the proposed computational framework to solve quantum spin control problems.
Optimal control of complex quantum systems in theory and experiment is the focus of this thesis. The theoretical aspect of our work was concerned with the development of algorithms that are capable of precisely tailoring the shapes of control pulses to meet various control goals while respecting all physical constraints, both fundamental and problem specific. Having these effective tools at hand, we computed control inputs for a range of physical systems using simulation-based time propagation. We extended the dressed chopped random basis (dCRAB) optimization method, previously developed in our group, such that boundary conditions and other constraints that arise from experimental setups of quantum systems can be regarded. Therefore, we wrote a software suite that enables experimentalists to improve their control pulses by directly interfacing the lab with our control algorithm.