This work presents a numerical analysis of a Discontinuous Galerkin (DG) method for a transformed master equation modeling an open quantum system: a quantum sub-system interacting with a noisy environment. It is shown that the presented transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of non-harmonic potentials via DG schemes. Specifics of a Discontinuous Galerkin (DG) numerical scheme adequate for the system of convection-diffusion equations obtained for our Lindblad master equation in position basis are presented. This lets us solve computationally the transformed system of interest modeling our open quantum system problem. The benchmark case of a harmonic potential is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem. Two non-harmonic cases are then presented: the linear and quartic potentials are modeled via our DG framework, for which we show our numerical results.

翻译：本文对一种用于模拟开放量子系统（即与噪声环境相互作用的量子子系统）的变换主方程的间断伽辽金（DG）方法进行了数值分析。研究表明，对于一般非谐振势情况，通过DG格式，所提出的变换主方程与同一系统的Wigner-Fokker-Planck模型相比具有更低的计算成本。本文详细阐述了适用于我们基于位置基的Lindblad主方程所得到的对流-扩散方程组的间断伽辽金（DG）数值格式的具体细节。这使得我们能够通过计算求解模拟开放量子系统问题的目标变换方程组。随后给出了谐振势的基准案例，并将数值结果与该问题的解析稳态解进行了比较。接着展示了两个非谐振案例：通过我们的DG框架对线性势和四次势进行建模，并给出了相应的数值结果。