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arXiv:2312.13233 · Nature Communications

Unified framework for open quantum dynamics with memory

Felix Ivander, Lachlan P. Lindoy, Joonho Lee

Published Sep 15, 2024• 18 citations
arXivHTMLPDF

Abstract

Abstract The dynamics of quantum systems coupled to baths are typically studied using the Nakajima-Zwanzig memory kernel ( $${{{\bf{{{{\mathcal{K}}}}}}}}$$ 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 $${{{\bf{{{{\mathcal{K}}}}}}}}$$ 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.

DOI
10.1038/s41467-024-52081-3

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

Felix Ivander, Lachlan P. Lindoy, Joonho Lee

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.

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The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.

arXiv:2112.05595•Condensed Matter Physics•Jan 1, 2022•4 citations
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.
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.
We develop a data-driven framework for identifying non-Markovian equations of motion for open quantum systems, demonstrated here for qubit-environment dynamics. Starting from the Nakajima-Zwanzig formalism, we vectorize the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel. The learning task is then formulated as a constrained optimization problem over the admissible operator space, where correlation functions are approximated by rational functions using Pade approximants. We establish well-posedness of the learning problem, ensuring existence of minimizers. To assess performance, we construct synthetic data sets from representative test problems of increasing complexity: (i) exactly solvable pure dephasing, with correlation functions expressed in terms of special functions, (ii) a damped Jaynes-Cummings model with an analytic coherence kernel, (iii) a transverse Born model with frequency-resolved bath integrals and population-coherence coupling, and (iv) a non-rotating-wave quantum Rabi model whose memory kernel has no closed form. Numerical experiments demonstrate that Pade captures nontrivial temporal structures such as oscillatory memory, algebraic tails, and phase-sensitive coherence transfer, and that the learned models generalize across ensembles of physically admissible initial states. We perform a parametrization-invariant sensitivity analysis and show that the trajectories are insensitive to the unrecoverable parts of the kernel, so the learned models stay predictive despite severe ill-conditioning in kernel recovery. These results together illustrate that data-driven rational approximation provides an effective route to identifying non-Markovian kernels of practical relevance in quantum technologies.
The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath ϱB(t)\varrho_{B}(t)ϱB​(t) and the system ϱS(t)\varrho_{S}(t)ϱS​(t) are written down. A formal solution of the equation for ϱB(t)\varrho_{B}(t)ϱB​(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in ϱS(t)\varrho_{S}(t)ϱS​(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ϱB(t)\varrho_{B}(t)ϱB​(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
We present a general and practical theoretical framework to investigate how energy is dissipated in open quantum system dynamics. This is performed by quantifying the contributions of individual bath components to the overall dissipation of the system. The framework is based on the Nakajima-Zwanzig projection operator technique, which allows us to express the rate of energy dissipation into a specific bath degree of freedom by using traces of operator products. The approach captures system-bath interactions to all orders, but is based on second-order perturbation theory on the off-diagonal subsystem's couplings and a Markovian description of the bath. The usefulness of our theory is demonstrated by applying it to various models of open quantum systems involving harmonic oscillators or spin baths and connecting the outcomes to existing results such as our previously reported formula derived for locally coupled harmonic baths [Kim and Franco, J. Chem. Phys. 154, 084109 (2021)]. We also prove that the dissipation calculated by our theory rigorously satisfies thermodynamic principles such as energy conservation and detailed balance. Overall, the strategy can be used to develop the theory and simulation of dissipation pathways to interpret and engineer the dynamics of open quantum systems.
We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.