高维Lindblad和Riccati方程数值模拟的低秩近似
Low rank approximation for the numerical simulation of high dimensional Lindblad and Riccati equations
摘要：
本文描述了一种系统的数值方法来逼近高维林德布拉德方程。它是基于密度算子的确定性秩m近似，秩m是唯一需要调整的参数。从一个已知的初始值，这种秩m近似给出了在每个时间步的密度算子的最大的m个本征值及其本征向量的估计。提出了一种数值格式。它的数值效率的情况下，秩12近似证明了50个原子与200个光子的轻微阻尼相干量子化场共振相互作用的振荡复苏。该方法可用于其他类似的方程。我们特别展示了如何适应这种低秩近似出现在卡尔曼滤波的Riccati微分方程。
英文摘要：
A systematic numerical approach to approximate high dimensional Lindblad equations is described. It is based on a deterministic rank m approximation of the density operator, the rank m being the only parameter to adjust. From a known initial value, this rank m approximation gives at each time-step an estimate of the largest m eigen-values with their eigen-vectors of the density operator. A numerical scheme is proposed. Its numerical efficiency in the case of a rank 12 approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons. The approach may be employed for other similar equations. We in particularly show how to adapt such low-rank approximation for Riccati differential equations appearing in Kalman filtering.
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