Sixth-order time-convolutionless master equation and beyond: Late-time resummations, two types of divergences, and the limits of validity
arXiv:2406.11088 · doi:10.1103/PhysRevA.111.042214
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.
58 pages, 9 figures. Error in Eqs. F31-F145 corrected. Figs. 8 and 9 added to numerically validate the TCL6 generator. An addendum to Phys. Rev. A 111, 042214 (2025) will provide details of the error and its correction
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