On the Discretization Error of the Discrete Generalized Quantum Master Equation

TL;DR

This paper analyzes the discretization errors in the transfer tensor method (TTM) for quantum dynamics, clarifies the origin of initial-time corrections, and validates the relationship between discrete and continuous memory kernels through numerical simulations.

Contribution

It provides a detailed analysis of TTM discretization, clarifies the initial-time correction, and establishes a consistent link between discrete and continuous quantum memory kernels.

Findings

Validated the convergence of reconstructed memory kernels.

Demonstrated accurate dynamical evolution as time step approaches zero.

Clarified the origin of the initial-time correction in TTM.

Abstract

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 2014, 112, 110401] can be considered a discrete-time formulation of the Nakajima-Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 2025, 21, 5037] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time \( t=0 \). This Communication presents a detailed analysis of the discretization structure of TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel \( K_N \), and the continuous-time NZ-QME kernel \( \mathcal{K}(N\Delta t) \). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as \( \Delta t \to 0 \). While TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri's work. The relative performance of various schemes for either computing accurate \( \mathcal{K}(N\Delta t) \) from exact dynamics, or obtaining accurate dynamics from exact \( \mathcal{K}(N\Delta t) \), warrants further investigation.