Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Advanced Quantum Technologies

Variational Approach to Non‐Markovian Dynamics and Two‐Time Correlations Using Pseudomodes

Yiying Yan, Yang Zhao

Published Oct 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT Markovian pseudomode models are shown to accurately describe non‐Markovian and non‐perturbative reduced dynamics and multi‐time correlation functions under certain constraints. However, it is challenging to solve the Lindblad master equation for such models when a number of pseudomodes are involved. A variational approach based on multiple Davydov states is used to solve the master equation of pseudomode models for the multi‐mode Jaynes–Cummings (JC) model, and the spin‐boson system across four distinct configurations, enabling computation of the two‐time correlation function that incorporates bath phonon occupation dynamics. In parallel, calculations of the unitary evolution of full Hamiltonians are chosen as benchmark references. Benchmark comparisons demonstrate that the variational approach for Markovian pseudomode models successfully predicts both the single‐time and two‐time expectations within the multi‐mode JC model. In contrast, in the spin‐boson model, the variational simulations for the pseudomode system and the full Hamiltonian yield inconsistent bath occupation numbers in a non‐perturbative strong coupling regime despite that these two kinds of simulations predict almost identical reduced dynamics, resulting from a fitting artefact in the pseudomode model. Our findings offer insights into the applicability of the non‐unitary variational approach on the simulation of two‐time correlation functions within the pseudomode framework.

DOI
10.1002/qute.70491

Related papers

Accurate Two-Time Correlation from a Pseudomode-Based Variational Approach with the Multiple Davydov D1 Density Operator

Yiying Yan, Zhiguo LÜ, Yuguo Su +2 more

The pseudomode framework provides an efficient description of open quantum systems in non-Markovian and nonperturbative coupling regimes by utilizing simple Markovian Lindblad master equations that incorporate a set of damped bosonic pseudomodes. Developing an efficient numerical scheme to solve these equations accurately is highly desirable. In this work, we propose a modified multiple Davydov D1 density operator ansatz and combine it with the Dirac-Frenkel time-dependent variational principle to simulate the dynamics governed by these master equations. We apply this variational method to the spin-boson model, both with and without a periodic external driving field, and compute both single- and two-time expectation values in non-Markovian, strong-coupling regimes. The accuracy of our variational approach is validated by comparing its predictions with results obtained from the small matrix path integral method. We demonstrate that the variational and path-integral approaches are in excellent agreement for both single- and two-time expectations across nondriven and driven cases, spanning both low- and high-temperature regimes. Our results establish the reliability and broad applicability of the pseudomode-based variational approach for calculating two-time correlation functions in open quantum systems.

Journal of Chemical Theory and Computation•Oct 16, 2026•0 citations

Pseudomode description of general open quantum system dynamics: non-perturbative master equation for the spin-boson model

Graeme Pleasance, Francesco Petruccione

We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.

arXiv:2108.05755•Aug 12, 2021•2 citations

Non-Hermitian pseudomodes for strongly coupled open quantum systems: Unravelings, correlations, and thermodynamics

Paul Menczel, Ken Funo, Mauro Cirio +2 more

The pseudomode framework provides an exact description of the dynamics of an open quantum system coupled to a non-Markovian environment. Using this framework, the influence of the environment on the system is studied in an equivalent model, where the open system is coupled to a finite number of unphysical pseudomodes that follow a time-local master equation. Building on the insight that this master equation does not need to conserve the hermiticity of the pseudomode state, we here ask for the most general conditions on the master equation that guarantee the correct reproduction of the system's original dynamics. We demonstrate that our generalized approach decreases the number of pseudomodes that are required to model, for example, underdamped environments at finite temperature. We also provide an unraveling of the master equation into quantum jump trajectories of non-Hermitian states, which further facilitates the utilization of the pseudomode technique for numerical calculations by enabling the use of easily parallelizable Monte Carlo simulations. Finally, we show that pseudomodes, despite their unphysical nature, provide a natural picture in which physical processes, such as the creation of system-bath correlations or the exchange of heat, can be studied. Hence, our results pave the way for future investigations of the system-environment interaction leading to a better understanding of open quantum systems far from the Markovian weak-coupling limit. Published by the American Physical Society 2024

arXiv:2401.11830•Physical Review Research•Sep 3, 2024•38 citations

Multiple Davydov Ansätze as solutions to Lindblad master equations

Yiying Yan, Yang Zhao

Lindblad master equations describing driven quantum systems coupled to multiple bosonic modes are central to modeling cavity quantum electrodynamics and pseudomode models. In this work, we employ the density-operator-based Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D2Ansatz to provide optimal solutions for Lindblad master equations in both multimode and driven scenarios. Our approach is benchmarked against numerically "exact" methods in two representative models. The first describes a driven qubit subject to spontaneous emission and coupled to a lossy cavity, while the second corresponds to the pseudomode Lindblad master equation derived from the driven spin-boson model, incorporating seven discrete pseudomodes. The proposed variational approach achieves excellent agreement with numerically "exact" results. Furthermore, we analyze the accuracy of the variational solutions using an error metric based on the Frobenius norm, confirming their reliability. Overall, the present method offers an accurate and computationally efficient framework for simulating Lindblad master equations in complex open quantum systems.

The Journal of Chemical Physics•Dec 15, 2025•6 citations

One-to-one correspondence between Hierarchical Equations of Motion and Pseudomodes for Open Quantum System Dynamics

Kai Müller, Walter T. Strunz

We unite two of the most widely used approaches for strongly damped, non-Markovian open quantum dynamics, the Hierarchical Equations of Motion (HEOM) and the pseudomode method by proving two statements: First, every physical bath correlation function (BCF) that can be written as a sum of $N$ exponential terms can be obtained from a physical model with $N$ interacting pseudomodes which are damped in Lindblad form. Second, for every such BCF there exists a non-unitary, linear transformation which mirrors the evolution of the system-pseudomode state onto the HEOM hierarchy, and vice versa. Our proofs are constructive and we give explicit expressions for the mirror transformation as well as for the pseudomode Lindbladian corresponding to a given exponential BCF. This approach also gives insight and provides elegant derivations of the corresponding Hierarchy of stochastic Pure States (HOPS) method and its nearly-unitary version, nuHOPS. Our result opens several avenues for further optimization of non-Markovian open quantum system dynamics methods.

quant-pharXiv:2604.06466•Apr 9, 2026•0 citations

Cavity Modified Dissipative Dynamics of the Two‐Spin Spin‐Boson Model

Ningshuang Zhao, Yiying Yan, Zhiguo Lü +1 more

ABSTRACT We investigate the dynamics of two spins that are simultaneously coupled to a high‐Q cavity and an Ohmic phonon bath using the Dirac–Frenkel time‐dependent variational principle together with two types of the multiple Davydov ansatz. In our framework, the cavity mode can either be treated as part of the bath modes within the multi‐D 2 ansatz or regarded together with the spins as a dressed composite system within the multi‐D 1 ansatz. Although both approaches yield consistent results for the spin, cavity, and phonon observables, the multi‐D 1 ansatz proves to be numerically much more efficient than the multi‐D 2 ansatz. In the regime of relatively weak spin‐bath coupling, we find that the cavity induces intricate beat patterns in the spin population dynamics as well as multipeaked phonon spectra. These cavity‐related features can also be reproduced by a polaron‐like transformed Nakajima–Zwanzig master equation, where the multipeaked phonon spectra arise from transitions between dressed states. By contrast, under relatively strong spin‐bath coupling, both the beat patterns and the multipeaked phonon spectra become suppressed. Furthermore, even with strong spin‐cavity coupling, sufficiently strong spin‐bath interaction leads to spin localization, reminiscent of the single‐spin spin‐boson model. Overall, our results reveal how spin‐bath and spin‐cavity couplings compete to shape the dissipative dynamics.

Advanced Quantum Technologies•Nov 20, 2025•2 citations
ABSTRACT Markovian pseudomode models are shown to accurately describe non‐Markovian and non‐perturbative reduced dynamics and multi‐time correlation functions under certain constraints. However, it is challenging to solve the Lindblad master equation for such models when a number of pseudomodes are involved. A variational approach based on multiple Davydov states is used to solve the master equation of pseudomode models for the multi‐mode Jaynes–Cummings (JC) model, and the spin‐boson system across four distinct configurations, enabling computation of the two‐time correlation function that incorporates bath phonon occupation dynamics. In parallel, calculations of the unitary evolution of full Hamiltonians are chosen as benchmark references. Benchmark comparisons demonstrate that the variational approach for Markovian pseudomode models successfully predicts both the single‐time and two‐time expectations within the multi‐mode JC model. In contrast, in the spin‐boson model, the variational simulations for the pseudomode system and the full Hamiltonian yield inconsistent bath occupation numbers in a non‐perturbative strong coupling regime despite that these two kinds of simulations predict almost identical reduced dynamics, resulting from a fitting artefact in the pseudomode model. Our findings offer insights into the applicability of the non‐unitary variational approach on the simulation of two‐time correlation functions within the pseudomode framework.
The pseudomode framework provides an efficient description of open quantum systems in non-Markovian and nonperturbative coupling regimes by utilizing simple Markovian Lindblad master equations that incorporate a set of damped bosonic pseudomodes. Developing an efficient numerical scheme to solve these equations accurately is highly desirable. In this work, we propose a modified multiple Davydov D1 density operator ansatz and combine it with the Dirac-Frenkel time-dependent variational principle to simulate the dynamics governed by these master equations. We apply this variational method to the spin-boson model, both with and without a periodic external driving field, and compute both single- and two-time expectation values in non-Markovian, strong-coupling regimes. The accuracy of our variational approach is validated by comparing its predictions with results obtained from the small matrix path integral method. We demonstrate that the variational and path-integral approaches are in excellent agreement for both single- and two-time expectations across nondriven and driven cases, spanning both low- and high-temperature regimes. Our results establish the reliability and broad applicability of the pseudomode-based variational approach for calculating two-time correlation functions in open quantum systems.
We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.
The pseudomode framework provides an exact description of the dynamics of an open quantum system coupled to a non-Markovian environment. Using this framework, the influence of the environment on the system is studied in an equivalent model, where the open system is coupled to a finite number of unphysical pseudomodes that follow a time-local master equation. Building on the insight that this master equation does not need to conserve the hermiticity of the pseudomode state, we here ask for the most general conditions on the master equation that guarantee the correct reproduction of the system's original dynamics. We demonstrate that our generalized approach decreases the number of pseudomodes that are required to model, for example, underdamped environments at finite temperature. We also provide an unraveling of the master equation into quantum jump trajectories of non-Hermitian states, which further facilitates the utilization of the pseudomode technique for numerical calculations by enabling the use of easily parallelizable Monte Carlo simulations. Finally, we show that pseudomodes, despite their unphysical nature, provide a natural picture in which physical processes, such as the creation of system-bath correlations or the exchange of heat, can be studied. Hence, our results pave the way for future investigations of the system-environment interaction leading to a better understanding of open quantum systems far from the Markovian weak-coupling limit. Published by the American Physical Society 2024
Lindblad master equations describing driven quantum systems coupled to multiple bosonic modes are central to modeling cavity quantum electrodynamics and pseudomode models. In this work, we employ the density-operator-based Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D2Ansatz to provide optimal solutions for Lindblad master equations in both multimode and driven scenarios. Our approach is benchmarked against numerically "exact" methods in two representative models. The first describes a driven qubit subject to spontaneous emission and coupled to a lossy cavity, while the second corresponds to the pseudomode Lindblad master equation derived from the driven spin-boson model, incorporating seven discrete pseudomodes. The proposed variational approach achieves excellent agreement with numerically "exact" results. Furthermore, we analyze the accuracy of the variational solutions using an error metric based on the Frobenius norm, confirming their reliability. Overall, the present method offers an accurate and computationally efficient framework for simulating Lindblad master equations in complex open quantum systems.
We unite two of the most widely used approaches for strongly damped, non-Markovian open quantum dynamics, the Hierarchical Equations of Motion (HEOM) and the pseudomode method by proving two statements: First, every physical bath correlation function (BCF) that can be written as a sum of NNN exponential terms can be obtained from a physical model with NNN interacting pseudomodes which are damped in Lindblad form. Second, for every such BCF there exists a non-unitary, linear transformation which mirrors the evolution of the system-pseudomode state onto the HEOM hierarchy, and vice versa. Our proofs are constructive and we give explicit expressions for the mirror transformation as well as for the pseudomode Lindbladian corresponding to a given exponential BCF. This approach also gives insight and provides elegant derivations of the corresponding Hierarchy of stochastic Pure States (HOPS) method and its nearly-unitary version, nuHOPS. Our result opens several avenues for further optimization of non-Markovian open quantum system dynamics methods.
ABSTRACT We investigate the dynamics of two spins that are simultaneously coupled to a high‐Q cavity and an Ohmic phonon bath using the Dirac–Frenkel time‐dependent variational principle together with two types of the multiple Davydov ansatz. In our framework, the cavity mode can either be treated as part of the bath modes within the multi‐D 2 ansatz or regarded together with the spins as a dressed composite system within the multi‐D 1 ansatz. Although both approaches yield consistent results for the spin, cavity, and phonon observables, the multi‐D 1 ansatz proves to be numerically much more efficient than the multi‐D 2 ansatz. In the regime of relatively weak spin‐bath coupling, we find that the cavity induces intricate beat patterns in the spin population dynamics as well as multipeaked phonon spectra. These cavity‐related features can also be reproduced by a polaron‐like transformed Nakajima–Zwanzig master equation, where the multipeaked phonon spectra arise from transitions between dressed states. By contrast, under relatively strong spin‐bath coupling, both the beat patterns and the multipeaked phonon spectra become suppressed. Furthermore, even with strong spin‐cavity coupling, sufficiently strong spin‐bath interaction leads to spin localization, reminiscent of the single‐spin spin‐boson model. Overall, our results reveal how spin‐bath and spin‐cavity couplings compete to shape the dissipative dynamics.