Floquet理论与平均哈密顿理论再探：等价性、收敛性及其在NMR中的应用
Floquet Theory and Average Hamiltonian Theory Revisited: Equivalence, Convergence and Applications to NMR
- TU Dortmund University（多特蒙德工业大学）
- ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明Floquet理论与平均哈密顿理论在数学上等价，提出基于Floquet-Magnus展开的计算方案，推荐Floquet-Van Vleck方法，其精度比平均哈密顿理论高约三倍，并应用于NMR实验分析。
AI中文摘要：
对周期性驱动量子系统进行精确的理论处理对于精确科学中的多个领域至关重要，例如核磁共振（NMR）波谱学。传统上，人们使用平均哈密顿理论或Floquet理论来预测或描述实验结果，如时间演化或提供所研究样品信息的谱图。详细分析这两种方法的等价性并强调其在NMR中的应用，将有助于提高对NMR实验的理论理解。\n在本工作中，我们确定Floquet-Magnus展开是证明Floquet理论与平均哈密顿理论数学等价性的关键。我们提倡一种计算方案，该方案因避免了显式积分而不易出现代数错误。在此基础上，我们提供了这两种理论的前四阶。我们进一步考察了它们对NMR中某些实验的适用性。作为例子，我们研究了魔角旋转下的Bloch-Siegert位移和偶极耦合自旋系统。\n基于我们的分析，我们推荐使用包含有效哈密顿量和踢算符的Floquet-Van Vleck方法。对久期项和非久期贡献的一致分离似乎对数值稳健性特别有利。尽管它们在形式上等价，但其精度比平均哈密顿理论提供的精度高出约三倍。\n我们的发现为Floquet理论和平均哈密顿理论的理论背景提供了重要见解。这包括它们代数和微扰等价性的程度，并着重于这些发现如何与固态NMR实验的分析相关。
英文摘要：
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.
