Sixth-order time-convolutionless master equation and beyond: Late-time resummations, two types of divergences, and the limits of validity

TL;DR

This paper develops a resummation technique for the time-convolutionless master equation to address late-time divergences in open quantum systems, establishing limits of validity and capturing decoherence effects.

Contribution

It introduces a Hadamard trick-based resummation method that incorporates time integrals into the spectral density, extending the applicability of TCL master equations.

Findings

Resummation suppresses late-time divergences in TCL equations.

The method captures decoherence and spectral overlap effects.

Validity is limited by a maximum expansion order nmax.

Abstract

Perturbative master equations are essential for modeling open quantum systems but often exhibit late-time divergences when environmental correlations decay algebraically. In this work, we analyze the time-convolutionless (TCL) master equation, expanded to order 2n and demonstrate that, while van Kampens cumulants suppress early-time secular growth, they ultimately diverge at long times. To overcome this, we introduce a resummation technique based on the Hadamard trick, which incorporates time integrals directly into the bath spectral density via element-wise multiplication. This approach establishes a maximum expansion order, nmax, and defines a precision limit of the asymptotic states. The resummed master equation features renormalized Bohr frequencies that capture decoherence and spectral overlap effects. In the unbiased spin-boson model, this results in secular inflation of the generator at a temperature-independent rate equal to the decoherence rate and a finite validity time. For exponentially decaying correlations, the method recovers a proper Markovian limit below a critical coupling threshold.