Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice-versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.

中文翻译：

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局部和非局部主方程之间的相互作用：精确和近似动力学

主方程是描述开放量子系统演化的有用工具。为了表征动力学的数学特征和物理起源，考虑同一系统的不同类型的主方程通常很有用。在这里，我们推导出时域和积分微分描述之间的精确联系，重点是交换动力学类。阻尼基形式主义的使用使我们能够设计一个通用程序，从一个主方程到另一个主方程，反之亦然，只需使用时间函数及其拉普拉斯变换。我们进一步分析了时域和积分微分主方程的 Lindbladian 形式，其中我们解释了不同组 Lindbladian 算子的出现。此外，我们研究了一种类似 Redfield 的近似，它通过时间上的粗粒度将精确的积分微分方程转换为时间局部方程。除了将生成的主方程的结构与那些与精确动力学相关的结构相关联之外，我们还研究了近似对马尔可夫性的影响。特别是，我们表明，与预期相反，时间上的粗粒度可能会引入记忆效应，导致违反动力学的可分性。