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arXiv 2004.01183quant-phphysics.comp-ph
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马尔可夫过程多重实现的经典和量子观测量平均的Dyson方程方法

A Dyson equation approach for averaging of classical and quantum observables on multiple realizations of Markov processes

Simone Sturniolo

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14 pages, 7 figures
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摘要：

在实验技术如核磁共振（NMR）和μ子自旋弛豫（muSR）中，时间相关信号通常是许多微观动力学过程的系综平均的结果。虽然有许多函数用于拟合这些信号，但它们通常仅在特定的状态下有效，并且几乎从未正确描述过“频谱扩散”状态，其中动态发生在与系统的特征频率相当的时间尺度上。这些问题的全面处理将需要对含时哈密顿量的动力学的所有可能实现进行路径积分。在本文中，我们提出了一种数值方法，可以潜在地用于解决这样的时间演化问题，我们对同一问题的蒙特卡罗模拟基准。该方法可以用于任何类型的动态，但对于任何可以近似为马尔可夫过程的动态特别强大，其中每一步的动态仅取决于系统的先前状态。该方法被用来平均经典和量子观测值，在后一种情况下，一个形式主义利用刘维和密度矩阵。

英文摘要：

Time dependent signals in experimental techniques such as Nuclear Magnetic Resonance (NMR) and Muon Spin Relaxation (muSR) are often the result of an ensemble average over many microscopical dynamical processes. While there are a number of functions used to fit these signals, they are often valid only in specific regimes, and almost never properly describe the "spectral diffusion" regime, in which the dynamics happen on time scales comparable to the characteristic frequencies of the system. Full treatment of these problems would require one to carry out a path integral over all possible realizations of the dynamics of the time dependent Hamiltonian. In this paper we present a numerical approach that can potentially be used to solve such time evolution problems, and we benchmark it against a Monte Carlo simulations of the same problems. The approach can be used for any sort of dynamics, but is especially powerful for any dynamics that can be approximated as Markov processes, in which the dynamics at each step only depend on the previous state of the system. The approach is used to average both classical and quantum observables; in the latter case, a formalism making use of Liouvillians and density matrices is used.

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