Abstract. Since the dawn of quantum mechanics ways to visualize spins and their interactions attracted attention of researchers and philosophers of science. Angular momentum probability (AMP) surfaces are known for visualizing density matrices; a plotted surface represents probability of finding maximum projection of angular momentum along any direction in space. While AMP surfaces visually reveal symmetries of density matrices which directly translate to measured properties, they focus solely on one total-angular-momentum manifold, neglecting interaction between different manifolds often encountered in the NMR of multispin systems. In this work, we extend applicability of this visualization method and introduce its extension, angular momentum coherence (AMC) surface approach. Three examples of nuclear spin dynamics in two-spin systems are presented and visualized using the AMP/AMC surfaces: (I) spin-lock induced crossing (SLIC) sequence with initial state being the singlet state between two spins; (II) interplay between chemical exchange and spin dynamics during the high-field signal amplification by reversible exchange (SABRE) experiment; (III) zero- to ultralow-field (ZULF) NMR experiment in the presence of magnetic field applied perpendicular to the sensitive axis of the detector. The presented visualization technique extends applicability of AMP surfaces to coupled multispin systems and will facilitate intuitive understanding of spin dynamics during complex NMR experiments as exemplified here by the considered cases. The temporal sequences (“the movies”) of such surfaces show phenomena like interconversion of polarization moments (Auzinsh et al., 2010) and, as particularly relevant in NMR and demonstrated here, polarization transfer between different spins. Such effects are difficult to grasp by looking at (time-dependent) numerical values of density-matrix elements.
Abstract. Since the dawn of quantum mechanics ways to visualize spins and their interactions attracted attention of researchers and philosophers of science. Angular momentum probability (AMP) surfaces are known for visualizing density matrices; a plotted surface represents probability of finding maximum projection of angular momentum along any direction in space. While AMP surfaces visually reveal symmetries of density matrices which directly translate to measured properties, they focus solely on one total-angular-momentum manifold, neglecting interaction between different manifolds often encountered in the NMR of multispin systems. In this work, we extend applicability of this visualization method and introduce its extension, angular momentum coherence (AMC) surface approach. Three examples of nuclear spin dynamics in two-spin systems are presented and visualized using the AMP/AMC surfaces: (I) spin-lock induced crossing (SLIC) sequence with initial state being the singlet state between two spins; (II) interplay between chemical exchange and spin dynamics during the high-field signal amplification by reversible exchange (SABRE) experiment; (III) zero- to ultralow-field (ZULF) NMR experiment in the presence of magnetic field applied perpendicular to the sensitive axis of the detector. The presented visualization technique extends applicability of AMP surfaces to coupled multispin systems and will facilitate intuitive understanding of spin dynamics during complex NMR experiments as exemplified here by the considered cases. The temporal sequences (“the movies”) of such surfaces show phenomena like interconversion of polarization moments (Auzinsh et al., 2010) and, as particularly relevant in NMR and demonstrated here, polarization transfer between different spins. Such effects are difficult to grasp by looking at (time-dependent) numerical values of density-matrix elements.
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.
Solid-state nuclear magnetic resonance (NMR) has emerged as an important technique for material characterization, finding extensive applications across a diverse range of disciplines including physics, materials science, chemistry, and biology. Its utility stems from the ability to probe the local atomic environments and molecular dynamics within solid materials, which provides information on the composition of the material. In recent years, the scope of solid-state NMR has expanded into the realm of quantum information science and technology, where its abundant many-body interactions pulse control methodologies make it have significant research value and application potential. This paper offers a comprehensive overview of the research objects and theoretical underpinnings of solid-state NMR, delving into the critical nuclear spin interaction mechanisms and their corresponding Hamiltonian forms. These interactions, which include dipolar coupling, chemical shift anisotropy, and quadrupolar interactions, are fundamental to the interpretation of NMR spectra and the understanding of material properties at the atomic level. Moreover, the paper introduces typical dynamical control methods employed in the manipulation of solid-state nuclear spins. Techniques such as dynamical decoupling, which mitigates the effects of spin-spin interactions to extend coherence times, and magic-angle spinning, which averages out anisotropic interactions to yield high-resolution spectra. These methods are essential for enhancing the sensitivity and resolution of NMR experiments, enabling the extraction of detailed structural and dynamic information from complex materials. Then we introduce some recent advancements in quantum control based on solid-state NMR, such as nuclear spin polarization enhancement techniques, which include dynamic nuclear polarization (DNP) and cross polarization (CP), significantly boost the sensitivity of NMR measurements. Additionally, the control techniques of Floquet average Hamiltonians are mentioned, showcasing their role in the precise manipulation of quantum states and the realization of quantum dynamics. Finally, the paper presents a series of seminal research works that illustrate the application of solid-state NMR quantum control technologies in the field of quantum simulation. These studies demonstrate how solid-state NMR can be leveraged to simulate and investigate quantum many-body systems, providing valuable insights into quantum phase transitions, entanglement dynamics, and other phenomena relevant to quantum information science. By bridging the gap between fundamental research and practical applications, solid-state NMR continues to play a crucial role in advancing our understanding of quantum materials and technologies.
In this work, we overview time-reversal nuclear magnetic resonance (NMR) experiments in many-spin systems evolving under the dipolar Hamiltonian. The Loschmidt echo (LE) in NMR is the signal of excitations which, after evolving with a forward Hamiltonian, is recovered by means of a backward evolution. The presence of non-diagonal terms in the non-equilibrium density matrix of the many-body state is directly monitored experimentally by encoding the multiple quantum coherences. This enables a spin counting procedure, giving information on the spreading of an excitation through the Hilbert space and the formation of clusters of correlated spins. Two samples representing different spin systems with coupled networks were used in the experiments. Protons in polycrystalline ferrocene correspond to an 'infinite' network. By contrast, the liquid crystal N-(4-methoxybenzylidene)-4-butylaniline in the nematic mesophase represents a finite proton system with a hierarchical set of couplings. A close connection was established between the LE decay and the spin counting measurements, confirming the hypothesis that the complexity of the system is driven by the coherent dynamics.
Spin dynamics in pulsed Nuclear Magnetic Resonance (NMR) spectroscopy are studied theoretically using spin density operator formulation. The theory exploits the use of irreducible spherical tensor (multipole) operators of spin angular momentum in developing alternative treatments to spin dynamical problems. Applications in spin decoupling and soft pulse dynamics are considered, giving new analytical solutions for the complete density matrix. The focus of the thesis, however, is on the development of two distinctly new methods of solution to the quantum Liouville equation for NMR field modulation. The first method derives a linear integral equation for the magnetization and provides a rapidly converging integral series solution which is useful in explaining many pulse shapes. The second method uses the Wei-Norman Lie algebra approach$ sp{ rm 96}$ which reduces the complete arbitrary spin I problem to a Riccati equation. Novel exact analytical solutions for a class of temporal field modulations are obtained. In contrast to all other existing formulations, the theory provides the first known examples of exactly soluble nonrectangular pulse shapes for all resonance off-set, all field amplitudes, all time and valid for all spin I $ geq$ 1/2. In addition, a new series expansion for the complete spin density matrix is obtained. The theory is applied to producing spin inversion profiles which show applicability to selective excitation.
This thesis is concerned with the design of numerical methods for quantum simulation and the development of improved models for quantum relaxation. Analysis is presented for the treatment of quantum systems using the density matrix formalism. This approach has been developed from the early days of quantum mechanics as a tool to describe from a statistical point of view a large number of identical quantum ensembles. Traditional methods are well established and reliable, but they perform poorly for practical simulation as the system size is scaled up. Ad hoc schemes for nuclear spin dynamics appearing in the literature can be shown to fail in certain situations. The challenge is therefore to identify efficient reduction methods for the quantum system which are also based on a rigorous foundation. The method presented in the thesis, for the time–independent Hamiltonian case, combines a quantum density matrix formalism with a procedure based on Chebyshev polynomials; application of the method to Nuclear Magnetic Resonance (NMR) spectroscopy is considered, and it is shown that the new technique outperforms existing alternatives in term of computational costs. The case of a time–dependent Hamiltonian in NMR simulation is studied as well and some splitting methods are presented. To the author’s knowledge this is the first time such methods have been applied within the NMR framework, and the numerical results show a better error–to–cost rate than traditional methods. In a separate strand of research, formulations for open quantum systems are studied and new dynamical systems approaches are considered for this problem. Motivations This thesis work is mainly focused on nuclear spin dynamics. Nuclear spin dynamics constitutes the basis for NMR, which is a very powerful spectroscopy technique that exploits the interaction between nuclear spins and magnetic fields. The same technique is used to reveal the presence of hydrogen atoms in the blood for Magnetic Resonance Imaging (MRI). Within this framework the role of simulations is extremely important, as it provides a benchmark for studies of new materials, and the development of new magnetic fields. The main computational issue is that with current software for NMR simulation it is extremely expensive to deal with systems made of more than few (7–10) spins. There is therefore a strong need to develop new algorithms capable of simulating larger systems. In recent years NMR simulations have been found to be one of the most favorable candidates for quantum computing. There are two reasons for this: nuclear quantum states maintain extremely long coherences, and it is possible to attain a very strong control on the quantum state via the application of sequences of pulses. In order to develop a proper quantum computer it is fundamental to understand how the entangled states lose coherence and relax back to equilibrium by means of external interactions. This process is described as relaxation in an open quantum system. The theory for such systems has been available for 50 years but there are still substantial limitations in the two main approaches. There are also relatively few numerical approaches for the simulation of such systems, for this reason it is important to develop numerical alternatives for the description of open quantum systems. Thesis Outline The thesis is organized as follow: the first two chapters provide background material to familiarize the reader with fundamental concepts of both quantum mechanics and nuclear spin dynamics; in this part of the thesis no new results are presented. The first chapter introduces the concept of quantum systems and the mathematical environment with which we describe those systems. We also present the main equations we need to solve to determine the dynamics of a quantum system in a statistical framework. In the second chapter we introduce the nuclear spin system, that is the physical system that has been the main reference frame in this work, for both tests and practical applications of the new algorithms. We describe how nuclear spin systems are at the basis of very important applications like NMR spectroscopy and MRI. We present in some detail the physical features of the NMR technique and the equations we need to solve to describe the dynamics of a spin system; we also focus on the relevance of numerical simulations for these systems, and consequently which must be the interest in developing new algorithms, and the major obstacles which must be overcome. In the third chapter we investigate the numerical challenges that arise in simulation of quantum systems, we describe some of the methods that have been developed in the literature, focusing on the performances and the computational costs of them, setting the new developments of this thesis in the proper research frame. We discuss one of the major issues: the evaluation of the matrix exponential. We also present the analysis we have done of a recent method called Zero Track Elimination (ZTE) that has been developed specifically for NMR simulations. This analysis shows the limitations of this method but also gives a mathematical explanation of why–and in which cases–it works. In the fourth chapter we present the main result of the thesis, the development of a new method that directly evaluates the expectation values for a quantum simulation via a different application of the well known Chebyshev expansion. We have proved that this new method can provide an excellent boost in terms of performance, with computational costs that can be reduced by a factor ten in common cases. (The results of this chapter and the new method have been presented in international conferences and recently they have been submitted for publication). We also present some attempts we have made in the application of splitting methods for the evolution of the system in a time dependent environment. To our knowledge this is the first time splitting methods have been used for NMR simulations. The results of this approach are as follows: for a particular splitting technique combined with a Lanczos iteration method it is possible to speed up the calculation by a third if compared with a Lanczos type method whilst keeping the error below a critical threshold. This last approach is still a work in progress especially in terms of developing clever ways to split the Hamiltonian. The last chapter of this thesis deals with simulation of quantum systems interacting with an external environment. After presenting the main theoretical approaches for the description of such systems we then survey several the techniques that are currently used for the numerical implementation of such theories. As a work in progress we present a considerably different new approach we have been developing aiming to overcome some of the issues that arise when treating this kind of system within usual frameworks. This is somewhat speculative work that gives rise to some new directions in the development of a numerical description for open quantum systems. We also present some numerical results. (The main core of this chapter has been presented in international conferences).
Abstract : With the production of polar molecules in the quantum regime, long-range dipolar interactions are expected to facilitate understanding of strongly interacting many-body quantum systems and to realize lattice spin models for exploring quantum magnetism. In ordinary atomic systems, where contact interactions require wavefunction overlap, effective spin interactions on a lattice can be mediated by tunnelling, through a process referred to as superexchange; however, the coupling is relatively weak and is limited to nearest-neighbour interactions. In contrast, dipolar interactions exist even in the absence of tunnelling and extend beyond nearest neighbours. This allows coherent spin dynamics to persist even for gases with relatively high entropy and low lattice filling. Measured effects of dipolar interactions in ultracold molecular gases have been limited to the modification of inelastic collisions and chemical reactions. Here we use dipolar interactions of polar molecules pinned in a three-dimensional optical lattice to realize a lattice spin model. Spin is encoded in rotational states of molecules that are prepared and probed by microwaves. Resonant exchange of rotational angular momentum between two molecules realizes a spin-exchange interaction. The dipolar interactions are apparent in the evolution of the spin coherence, which shows oscillations in addition to an overall decay of the coherence. The frequency of these oscillations, the strong dependence of the spin coherence time on the lattice filling factor and the effect of a multipulse sequence designed to reverse dynamics due to two-body exchange interactions all provide evidence of dipolar interactions. Furthermore, we demonstrate the suppression of loss in weak lattices due to a continuous quantum Zeno mechanism. Measurements of these tunnelling-induced losses allow us to determine the lattice filling factor independently.