This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.
This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.
This article presents the application of continuous Floquet theory in solid-state NMR. Continuous Floquet theory extends traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first and second-order effective Hamiltonians, streamlining integration with traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of Continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.
An accurate theoretical treatment of periodically driven quantum systems is crucial for various fields in the exact sciences, for instance Nuclear Magnetic Resonance (NMR) spectroscopy. Conventionally, either average Hamiltonian theory or Floquet theory is used to predict or describe experimental outcomes, such as the time evolution or the spectra yielding the information of the sample under study. A detailed analysis of the equivalence of these two approaches with an emphasis on applications in NMR will help to improve the theoretical understanding of NMR experiments.
In this work, we identify the Floquet-Magnus expansion as essential to prove the mathematical equivalence of Floquet theory and average Hamiltonian theory. We advocate a calculation scheme which is less prone to algebraic mistakes because explicit integration is avoided. On this basis, we provide the first four orders of both theories. We further examine their applicability to some experiments in NMR. As examples, we investigate the Bloch-Siegert shift and dipolar coupled spin systems under magic-angle spinning.
Based on our analysis, we recommend the use of the Floquet-Van Vleck approach including both the effective Hamiltonian and the kick operator. The consistent separation of secular and non-secular contributions appears to be especially advantageous for numerical robustness. Its accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence.
Our findings provide important insights into the theoretical background of Floquet theory and average Hamiltonian theory. This includes the extent of their algebraic and perturbative equivalence, with an emphasis on how these findings are of relevance in the analysis of solid-state NMR experiments.
Quantum mechanical Floquet theory, a powerful tool for the solution of Schrödinger equations with a periodically time-dependent Hamiltonian is formalized by the systematic usage of the Floquet Hilbert space, the Floquet density operator, Floquet detection operators, and of the Floquet Liouville space. A concise description is obtained that is useful in analytical calculations and for computer simulations. It is implemented in the context of the general computational environment ‘GAMMA’. Simplified expressions are derived for single-axis and double-axis sample-spinning experiments in NMR.
The work in this thesis presents an operator-based Floquet formalism which can be used to describe a Hamiltonian with multiple time dependencies, i.e. multiple basic frequencies in solid-state Nuclear Magnetic Resonance (NMR) experiments.In solid-state NMR, the time dependence can be caused for example by magic-angle spinning (MAS) where the sample is rotated about a specific (magic) angle with respect to the static field, thus averaging all anisotropic spin interactions.Radio-frequency irradiation, which is
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.
The Floquet-Magnus and Fer expansion schemes were introduced in solid-state nuclear magnetic resonance (NMR) in 2011 and 2006, respectively. Key features of the Floquet magnus expansion are its ability to account for the calculations developed in a finite-dimensional Hilbert space instead of an infinite-dimensional space within the Floquet theory as well as its use of its distinguishable function from other theories such as average Hamiltonian theory, Floquet theory, and Fer expansion, which facilitates the evaluation of the spin behavior in between the stroboscopic observation points. This article focuses on revisiting the Floquet-Magnus and Fer expansion approaches and applying both methods to calculate the effective Hamiltonians and propagators, which control the spin system evolution during the Triple Oscillating Field Technique radiation experiment (TOFU). The TOFU pulse sequence is an important technique that was shown to avoid the dipolar truncation problem and form a new basis for accurate distance measurement by solid-state NMR. We take advantage of the interaction frequencies and the time modulation arising from the TOFU pulse sequence allowing selective recoupling of specific terms in the Hamiltonian that fulfill determined specific conditions. The work presented unifies and generalizes existing results of the Floquet-Magnus and Fer expansions and delivers illustrations of novel springs that boost previous applications that are based on the classical information. We believe that the revisited approaches of this work and the derived expressions can serve as useful information and numerical tools for time evolution in time-resolved spectroscopy, quantum control, and open system quantum dynamics.