发表机构
University of Tübingen（图宾根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种计算复杂度与粒子数无关的玻色多体量子动力学数值算法，基于Bogoliubov理论构造波函数ansatz，结合Hartree方程和Dirac-Frenkel变分原理，并采用动力学低秩近似与BUG积分器高效求解。
AI 中文摘要
我们提出了一种算法，用于在平均场标度区域内，数值求解N >> 1个玻色子通过两体相互作用势相互作用的含时薛定谔方程。与现有的量子动力学数值算法不同，所提出的算法的计算复杂度与粒子数N无关。该算法基于受Bogoliubov理论启发的波函数ansatz，Bogoliubov理论是适用于大N的渐近理论。在第一部分中，我们遵循Bogoliubov理论，通过Hartree方程演化凝聚体函数，但我们使用Dirac-Frenkel含时变分原理来推导Bogoliubov ansatz中激发函数的非线性运动方程。通过仔细利用玻色子对称性，所得近似波函数的导数被证明有界，且与N无关，除一个以与N成正比频率旋转的含时全局相位因子外。在第二部分中，我们通过复合算法数值近似波函数（模去相位），该算法采用运动方程的适当分裂，并利用动力学低秩近似，将基本两粒子激发函数表示为单粒子函数乘积的线性组合。低秩激发函数通过基更新和Galerkin（BUG）积分器随时间演化，该积分器对系数矩阵的小奇异值具有鲁棒性，并允许秩自适应。提供了数值结果以评估所提出算法的性能。
英文摘要
We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac--Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.
Comments43 pages, 4 figures
