Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G}=\widehat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely different ways of computing the time evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally “samples” it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient or Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces nonsemigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.

中文翻译：

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带有内存的量子演化是如何以局部时间方式生成的

两种广泛使用但截然不同的开放量子系统动力学方法分别是Nakajima-Zwanzig和无时间卷积的量子主方程。尽管两者都描述了具有强大记忆效应的相同量子演化，但第一个使用了时间非局部记忆核$\mathcal{ķ}$，而第二个使用时域生成器实现相同的效果 $\mathcal{G}$。在这里，我们显示两者通过简单但通用的定点关系连接：$\mathcal{G}=\widehat{\mathcal{ķ}}[\mathcal{G}]$。这样一来，就可以提取两种完全不同的时间演变方式之间的平凡关系，并结合它们的优势。我们首先讨论固定发电机，它使Markov逼近对于大类演化既非扰动又完全正。我们表明，该发生器不等于存储器内核的低频极限，而是另外以非零特征频率对其进行“采样”。这阐明了频率依赖性和半群分解在现有的马尔可夫近似策略中的微妙作用。其次，我们证明了定点方程式总结了时域非局部量子主方程的时域梯度或Moyal展开，为记忆效应的产生提供了无扰动的见解。最后，我们表明，定点关系可以从给定的内存内核对固定生成器和瞬态生成器进行直接迭代数值计算。对于瞬态发生器，这会产生非半群逼近，这些逼近约束在每个迭代步骤的初始和渐近精度上均受到约束。