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arXiv:2411.02409

Reply to "Comment on "Unified Framework for Open Quantum Dynamics with Memory""

Felix Ivander, Lachlan P. Lindoy, Joonho Lee

Published Oct 18, 2024• 0 citations
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Abstract

We present our response to the commentary piece by Makri {\it et al.} [arXiv:2410.08239], which raises critiques of our work [Nat. Commun. 15, 8087 (2024)]. In our paper, we considered various settings of open-quantum system dynamics, including non-commuting, non-diagonalizable system-bath coupling, and bosonic/spin/fermionic baths. For these, we showed a direct and explicit relationship between the discrete-time memory kernel ($\mathcal K$) of the generalized quantum master equation (GQME) and the discrete-time influence functions ($I$) of the path integrals. As an application of this, we showed one can construct $\mathcal K$ without projection-free dynamics inputs that conventional methods require, and we also presented a quantum sensing protocol that characterizes the bath spectral density from reduced system dynamics. As the Comment focused on the relationship between ($\mathcal K$) and $I$ in one specific setup (i.e., commuting, diagonalizable system-bath coupling with a bosonic bath), we focus on that aspect in this response. In summary, we could not find a set of equations that explicitly connect $I$ and $\mathcal K$ from Makri's 2020 paper [J. Chem. Theory Comput. 16, 4038 (2020)]. Furthermore, while our analysis is specific to the choice of discretization of path-integral and GQME, we have not found issues with the GQME discretization employed. As per critiques on citations, in our paper, we note that we had acknowledged Makri's driven SMatPI work and Wang and Cai's tree-based SMatPI work for the number of Dyck paths needed for the computation of the memory kernel.

DOI
10.48550/arxiv.2411.02409

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We present our response to the commentary piece by Makri {\it et al.} [arXiv:2410.08239], which raises critiques of our work [Nat. Commun. 15, 8087 (2024)]. In our paper, we considered various settings of open-quantum system dynamics, including non-commuting, non-diagonalizable system-bath coupling, and bosonic/spin/fermionic baths. For these, we showed a direct and explicit relationship between the discrete-time memory kernel (K\mathcal KK) of the generalized quantum master equation (GQME) and the discrete-time influence functions (III) of the path integrals. As an application of this, we showed one can construct K\mathcal KK without projection-free dynamics inputs that conventional methods require, and we also presented a quantum sensing protocol that characterizes the bath spectral density from reduced system dynamics. As the Comment focused on the relationship between (K\mathcal KK) and III in one specific setup (i.e., commuting, diagonalizable system-bath coupling with a bosonic bath), we focus on that aspect in this response. In summary, we could not find a set of equations that explicitly connect III and K\mathcal KK from Makri's 2020 paper [J. Chem. Theory Comput. 16, 4038 (2020)]. Furthermore, while our analysis is specific to the choice of discretization of path-integral and GQME, we have not found issues with the GQME discretization employed. As per critiques on citations, in our paper, we note that we had acknowledged Makri's driven SMatPI work and Wang and Cai's tree-based SMatPI work for the number of Dyck paths needed for the computation of the memory kernel.
A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. We elaborate on these three critiques in this Comment.
Abstract The dynamics of quantum systems coupled to baths are typically studied using the Nakajima-Zwanzig memory kernel ( K{{{\bf{{{{\mathcal{K}}}}}}}}K K ) or the influence functions (I), particularly when memory effects are present. Despite their significance, formal connections between the two have not been explicitly known. We establish their connections by examining the system propagator for a N-level system linearly coupled to Gaussian baths with various types of system-bath coupling. For a certain class of problems, we devised a non-perturbative, diagrammatic approach to construct K{{{\bf{{{{\mathcal{K}}}}}}}}K K from I for (driven) systems interacting with Gaussian baths, bypassing conventional projection-free dynamics inputs. Our work provides a way to interpret approximate path integral methods in terms of approximate memory kernels. Moreover, it offers a Hamiltonian learning procedure to extract the bath spectral density from reduced system trajectories, opening new avenues in quantum sensing and engineering. The insights we provide advance our understanding of non-Markovian dynamics and will serve as a stepping stone for future theoretical and experimental developments in this area.
This thesis describes the development of practical and efficient computational approaches to the quantum dynamics of complex systems. Most of the work presented here relies on the generalized quantum master equation (GQME) formalism, which provides a simple equation of motion of reduced dimensionality for a set of dynamical quantities, e.g., nonequilibrium averages and equilibrium time correlation functions. The reduced dimensionality of the GQME comes at a cost: the introduction of the memory kernel, which accounts for the influence of all ``excluded'' degrees of freedom. Focusing first on the second-order perturbative treatment of the memory kernel known as Redfield theory, I present a collaborative effort to extend its applicability into highly non-Markovian regions via a mode freezing approach. In this method, a portion of bath modes characterized by low frequencies are treated as sources of static disorder and used to calculate modified Redfield dynamics. Application of the method to the spin-boson and FMO complex models indicates that the Redfield+frozen modes scheme consistently produces dynamics that are as good or better than bare Redfield dynamics. Next, we explore GQME approach coupled to the self-consistent solution of the memory kernel, which requires the calculation of auxiliary kernels. Previous implementations of the method had shown impressive boosts in efficiency and, when approximate methods were used to calculate the auxiliary kernels, accuracy over direct calculation of nonequilibrium averages. We show that this method, when formulated from the Mori perspective, is equally applicable to nonequilibrium averages and equilibrium correlation functions. In addition, we examine the dependence of the improvements afforded by the GQME framework on the choice projection operator and kernel closure. In particular, we demonstrate that improvements in efficiency, which rely on short memory lifetimes, are sensitively dependent on the choice of projection operator, and that the choice of kernel closure directly dictates the improvements in accuracy. In addition, we present evidence that indicates that the success of the GQME formalism when the auxiliary kernels are calculated via semi- and quasi-classical methods is largely due to the exact sampling of bath operators at t=0 required by the calculation of specific kernel closures. Next, we provide analytical arguments that delineate when the GQME framework coupled to the self-consistent solution of the memory kernel is likely to provide improvements in efficiency and accuracy. Finally, we present a path integral framework that can efficiently render the partially Wigner-transformed canonical density operator for systems coupled linearly to harmonic baths. This approach permits the direct calculation of any thermodynamic quantity and can be integrated into dynamical schemes like the Ehrenfest, surface hopping, or linearized semi-classical initial value representation methods to calculate equilibrium correlation functions.
The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112, 110401 (2014)] 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. 21, 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. This work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel KN and the continuous-time NZ-QME kernel K(NΔt). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δt → 0. While the 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 K(NΔt) from exact dynamics or obtaining accurate dynamics from exact K(NΔt) warrants further investigation.
We introduce a general procedure for computing higher-order moments of correlation functions in open quantum systems, extending the scope of our recent work on Memory Kernel Coupling Theory (MKCT) [Liu et al., arXiv:2407.01923 (2024)]. This approach is demonstrated for arbitrary system-bath coupling that can be expressed as a polynomial, HSB=V̂(α0+α1q̂+α2q̂2+⋯) , where we show that the recursive commutators of a system operator obey a universal hierarchy. Exploiting this structure, the higher-order moments are obtained by evaluating the expectation values of the system and bath operators separately, with bath expectation values derived from the derivatives of a generating function. We further apply MKCT to compute the dipole autocorrelation function for the spin-boson model with both linear and quadratic coupling, achieving agreement with the hierarchical equations of motion approach. Our findings suggest a promising path toward accurate dynamics for complex open quantum systems.