# How quantum evolution with memory is generated in a time-local way

K. Nestmann,
V. Bruch,
M. R. Wegewijs

May 2021

### Abstract

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: $\mathcal{G} = \hat{\mathcal{K}}[\mathcal{G}]$. 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 / 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 non-semigroup approximations which are constrained to be both *initially* and *asymptotically* accurate at each iteration step.

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Publication

Phys. Rev. X **11**, 021041