Wigner representation, its reconstruction, and optimal control of spin systems with applications in NMR spectroscopy, quantum technology, and beyond
Year: 2018
Type: dissertation
Abstract: 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.
Author: David Leiner
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funders: European Commission, Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche, Centre National de la Recherche Scientifique, Technische Universität München, Fondo de Cooperación Internacional en Ciencia y Tecnología, Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México, Elitenetzwerk Bayern
OpenAlex ID: W2910071208