Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 612-645, January 2020.
The dynamics of a closed quantum system is often studied with the direct evolution of the Schrödinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant to physical observables) of the Schrödinger equation can be generally nonoptimal for numerical simulation. This can limit, and in some cases severely limit, the time step size. We find that the optimal gauge choice is given by a parallel transport formulation. This parallel transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The parallel transport dynamics can be derived from a Hamiltonian structure and is thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the parallel transport dynamics in the context of the singularly perturbed linear Schrödinger equation and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrödinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse.

中文翻译：

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平行传输规的量子动力学

2020年1月，《多尺度建模与仿真》，第18卷，第2期，第612-645页。
通常通过Schrödinger方程的直接演化来研究封闭量子系统的动力学。在本文中，我们建议对于数值模拟，Schrödinger方程的量规选择（即与物理可观观测值无关的自由度）通常可能不是最佳的。这可能会限制时间步长，有时甚至会严重限制时间步长。我们发现最佳量规的选择是由平行运输公式给出的。可以将这种并行传输动力学简单地解释为由残差矢量驱动的动力学，类似于与时间无关的设置中的特征值问题中定义的那些。并行传输动力学可以从哈密顿结构导出，因此适合使用辛和隐式时间离散化方案（例如隐式中点法则）求解，这允许使用较大的时间步长并确保长时间的数值稳定性。我们在奇摄动线性Schrödinger方程的背景下分析了平行的运输动力学，并证明了其在绝热状态下的优越性能。我们使用线性和非线性Schrödinger方程的数值结果以及由超短激光脉冲驱动的苯分子中电子的时变密度泛函理论（TDDFT）计算，证明了我们方法的有效性。