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arXiv:2603.25512

Nonperturbative Resummation of Divergent Time-Local Generators

Dragomir Davidovic

quant-phhep-thmath-phSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations

v1 · Mar 26, 2026 v2 · Jun 30, 2026 v3 · Sep 27, 2026

Abstract

Time-local generators of open quantum systems are generically divergent at long times, even though the reduced dynamics remains regular. We construct, by analytic continuation, nonperturbative dynamical maps consistent with these generators. For the weak-coupling unbiased spin--boson model, this construction yields an explicit dynamical map that nonperturbatively resums the TCL generator and exposes how the divergences signal the approach to a singular time at which the reduced dynamics becomes noninvertible. The reconstructed map is validated against TEMPO simulations at short times and the exactly solvable rotating-wave model at all times. In the full spin--boson model, the same continuum mechanism produces both an early-time anisotropy, with a measurable phase shift that provides a signature of the environmental correlation and the pointer direction, and a late-time singularity at which the reduced dynamics becomes noninvertible. By contrast, in the rotating-wave model the map approaches this point without reaching it and remains invertible at all times. These results establish a nonperturbative framework for reconstructing reduced dynamics from divergent time-local generators, diagnosing the onset of noninvertibility, and identifying experimentally accessible early-time signatures of environment-induced anisotropy.

Comments: 28 pages 7 figures

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Time-local generators of open quantum systems are generically divergent at long times, even though the reduced dynamics remains regular. We construct, by analytic continuation, nonperturbative dynamical maps consistent with these generators. For the weak-coupling unbiased spin--boson model, this construction yields an explicit dynamical map that nonperturbatively resums the TCL generator and exposes how the divergences signal the approach to a singular time at which the reduced dynamics becomes noninvertible. The reconstructed map is validated against TEMPO simulations at short times and the exactly solvable rotating-wave model at all times. In the full spin--boson model, the same continuum mechanism produces both an early-time anisotropy, with a measurable phase shift that provides a signature of the environmental correlation and the pointer direction, and a late-time singularity at which the reduced dynamics becomes noninvertible. By contrast, in the rotating-wave model the map approaches this point without reaching it and remains invertible at all times. These results establish a nonperturbative framework for reconstructing reduced dynamics from divergent time-local generators, diagnosing the onset of noninvertibility, and identifying experimentally accessible early-time signatures of environment-induced anisotropy.

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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

{\mathrm{TCL}}_{2n}

and demonstrate that, while van Kampen's 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's spectral density via elementwise multiplication. This approach establishes a maximum expansion order, ${n}_{\text{max}}=\ensuremath{\lceil}s+1\ensuremath{\rceil}$, and defines a precision limit of the asymptotic states of $O({\ensuremath{\lambda}}^{2\ensuremath{\lceil}s\ensuremath{\rceil}})$, where

s

is the power-law exponent and $\ensuremath{\lambda}$ is the weak-coupling constant. The resummed master equation features renormalized Bohr frequencies that capture decoherence and spectral overlap effects. In the unbiased spin-boson model, this results in an inflation of the generator at a temperature-independent rate equal to the decoherence rate ${\ensuremath{\nu}}_{2}$ and a finite validity time ${t}_{L}\ensuremath{\approx}(s+1)/{\ensuremath{\nu}}_{2}$. For exponentially decaying correlations, the method recovers a proper Markovian limit below a critical coupling threshold.

The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.

We introduce a systematic framework to derive the effective Hamiltonian governing the coherent dynamics of non-Markovian open quantum systems. By applying the minimal dissipation principle, we uniquely isolate the coherent contribution to the time-local generator of the reduced dynamics. We derive a general expression for the effective Hamiltonian and develop a recursive perturbative expansion that expresses it in terms of system-bath interaction terms and bath correlation functions. This expansion provides a systematic tool for analyzing energy renormalization effects across different coupling regimes. Applying our framework to paradigmatic spin systems, we reveal how environmental correlations influence energy shifts and eigenbasis rotations, offering insights into strong-coupling effects and non-Markovian quantum thermodynamics.

The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.

Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.