General connection between time-local and time-nonlocal perturbation expansions

There exist two canonical approaches to describe open quantum systems by a time-evolution equation: the Nakajima-Zwanzig quantum master equation, featuring a time-nonlocal memory kernel K, and the time-convolutionless equation with a time-local generator G. These key quantities have recently been shown to be connected by an exact fixed-point relation [Phys. Rev. X 11, 021041 (2021)]. Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel K to be translated directly into a corresponding series for the more complicated generator G. This leads to an elegant way of computing the generator using well-developed, standard memory-kernel techniques for strongly interacting open systems. Moreover, it allows for an unbiased comparison of time-local and time-nonlocal approaches independent of the particular technique chosen to calculate expansions of K and G (Nakajima-Zwanzig projections, real-time diagrams, etc.). We illustrate this for leading and next-to-leading-order calculations of K and G for the single impurity Anderson model using both the bare expansion in the system-environment coupling and a more advanced renormalized series. We compare the different expansions obtained, quantify the legitimacy of the generated dynamics (complete positivity) and benchmark with the exact result in the noninteracting limit.

Automated summary

There are two canonical approaches to describe open quantum systems, the Nakajima-Zwanzig quantum master equation with a time-nonlocal memory kernel K, and the time-convolutionless equation with a time-local generator G. A recent study has revealed a fixed-point relation connecting these key quantities, allowing for a recursive relation between their perturbative expansions. This provides an elegant way to compute the generator using standard memory-kernel techniques for strongly interacting open systems and allows for an unbiased comparison of time-local and time-nonlocal approaches.