Ture Taner M, Jang Seogjoo J
Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, United States.
PhD Program in Chemistry, Graduate Center of the City University of New York, New York, New York 10016, United States.
J Phys Chem A. 2024 Apr 11;128(14):2871-2882. doi: 10.1021/acs.jpca.3c07866. Epub 2024 Apr 2.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time quantum dynamics, and open system quantum dynamics.
马格努斯展开（ME）提供了一种通用方法，用于在指数形式内展开含时哈密顿量的实时传播子，从而在任何阶次下都满足幺正性。我们利用这一特性，并在每个时间间隔内对含时哈密顿量进行拉格朗日插值公式的显式积分，推导出了对于一般含时哈密顿量的微分时间演化算符保持幺正性的近似表达式。所得的二阶近似与使用两个时间端点处哈密顿量的平均值相同。我们确定了三种涉及不同时刻哈密顿量对易子的四阶近似，并推导了一个六阶表达式。对这些近似以及其他适用于具有正弦时间依赖性的两态含时哈密顿量的可用表达式进行测试，提供了关于这些近似相对性能的信息，并表明所推导的表达式可作为时间分辨光谱学、量子控制、量子传感、实时量子动力学和开放系统量子动力学中时间演化的有用数值工具。
