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

arXiv:2504.08068 · Physical Review Research

Testing bath correlation functions for open quantum dynamics simulations

Masaaki Tokieda

Published Nov 18, 2025• 1 citation
arXivHTMLPDF

Abstract

Accurate simulations of thermalization in open quantum systems require a reliable representation of the bath correlation function (BCF). Numerical approaches, such as the hierarchical equations of motion and the pseudomode method, inherently approximate the BCF using a finite set of functions, which can impact simulation accuracy. In this work, we propose a practical and rigorous testing framework to assess the validity of approximate BCFs in open quantum dynamics simulations. Our approach employs a harmonic oscillator system, where the computed dynamics can be benchmarked against known exact solutions. To enable practical testing, we make two key methodological advancements. First, we develop numerical techniques to efficiently evaluate these exact solutions across a wide range of BCFs, ensuring broad applicability. Second, we introduce a moment-based state representation that significantly simplifies computations by exploiting the Gaussian nature of the system. Applications to a two-spin system and a transmon-resonator system demonstrate that our testing procedure provides error estimates that capture the qualitative trends observed in thermalization simulations. Using this methodology, we assess the performance of recently proposed BCF construction methods, highlighting both their strengths and a notable challenge posed by sub-Ohmic spectral densities at finite temperatures.

DOI
10.1103/bv19-dtb1

Related papers

High accuracy exponential decomposition of bath correlation functions for arbitrary and structured spectral densities: Emerging methodologies and new approaches

H. Takahashi, Samuel L. Rudge, Christoph Kaspar +2 more

This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.

The Journal of Chemical Physics•May 22, 2024•45 citations

Exact open quantum system dynamics: Optimal frequency vs time representation of bath correlations

Richard Hartmann, Michael Werther, Frank Großmann +1 more

Two different numerically exact methods for open quantum system dynamics, the hierarchy of pure states (HOPS) method, and the multi-Davydov-Ansatz are discussed. We focus on the suitability of the underlying representations of bath correlations. While in the HOPS case the correct description of the bath correlation function (BCF) in the time domain is decisive, it turns out that a windowed Fourier transform of the BCF is an appropriate indicator of the quality of the discretization in the multi-Davydov-Ansatz. For the spin-boson model with sub-Ohmic spectral density considered here, a discretization of the spectral density based on an exponential distribution, used previously, turns out to be most favorable.

The Journal of Chemical Physics•Jun 17, 2019•29 citations

Pseudomodes and the corresponding transformation of the temperature-dependent bath correlation function

David W. Schönleber, Alexander Croy, Alexander Eisfeld

In open-system approaches with non-Markovian environments, the process of inserting an individual mode (denoted as ``pseudomode'') into the bath or extracting it from the bath is widely employed. This procedure, however, is typically performed on basis of the spectral density and does not incorporate temperature. Here, we show how the (temperature-dependent) bath correlation function (BCF) transforms in such a process. We present analytic formulas for the transformed BCF and numerically study the differences between factorizing initial state and global thermal (correlated) initial state of mode and bath, respectively. We find that in the regime of strong coupling of the mode to both system and bath, the differences in the BCFs give rise to pronounced differences in the dynamics of the system.

arXiv:1504.00139•Physical Review A•May 14, 2015•14 citations

Exact Open Quantum System Dynamics Using the Hierarchy of Pure States (HOPS)

Richard Hartmann, Walter T. Strunz

We show that the general and numerically exact Hierarchy of Pure States method (HOPS) is very well applicable to calculate the reduced dynamics of an open quantum system. In particular, we focus on environments with a sub-Ohmic spectral density (SD) resulting in an algebraic decay of the bath correlation function (BCF). The universal applicability of HOPS, reaching from weak to strong coupling for zero and nonzero temperature, is demonstrated by solving the spin-boson model for which we find perfect agreement with other methods, each one suitable for a special regime of parameters. The challenges arising in the strong coupling regime are not only reflected in the computational effort needed for the HOPS method to converge but also in the necessity for an importance sampling mechanism, accounted for by the nonlinear variant of HOPS. In order to include nonzero-temperature effects in the strong coupling regime we found that it is highly favorable for the HOPS method to use the zero-temperature BCF and include temperature via a stochastic Hermitian contribution to the system Hamiltonian.

arXiv:1710.08268•Journal of Chemical Theory and Computation•Oct 10, 2017•73 citations

Non-Markovian dynamics in nonstationary Gaussian baths: a hierarchy of pure states approach

Vladislav Sukharnikov, Stasis Chuchurka, Frank Schlawin

Building on the standard hierarchy of pure states (HOPS) approach, we construct a generalized formulation suitable for open quantum systems interacting with nonstationary Gaussian baths, potentially extending its applicability to nonequilibrium baths. This is achieved by extending the conventional exponential decomposition of a bath correlation function (BCF) for nonstationary cases. Using our formulation of HOPS, we derive the corresponding hierarchy of master equations and, when each term in the BCF expansion can be associated with an independent physical bath, we show how the formalism connects to the well-known pseudomode representation. We demonstrate the method's performance on two examples of nonstationary squeezed reservoirs generated via uniform squeezing and degenerate parametric amplification in a one-sided cavity. Benchmarking against the hierarchy of master equations shows that HOPS is more efficient under hierarchy truncation. The pseudomode representation is shown to be more efficient in the strongly non-Markovian regime. Our results highlight HOPS as a versatile and powerful tool for simulating open quantum systems in nonstationary baths, with potential applications ranging from squeezed light-matter interactions to driven quantum materials and dissipative phase transitions.

quant-pharXiv:2511.01358•Physical Review Research•Nov 4, 2025•0 citations

Quasi-Lindblad pseudomode theory for open quantum systems

Gunhee Park, Zhen Huang, Yuanran Zhu +3 more

We introduce a new framework to study the dynamics of open quantum systems with linearly coupled Gaussian baths. Our approach replaces the continuous bath with an auxiliary discrete set of pseudomodes with dissipative dynamics, but we further relax the complete positivity requirement in the Lindblad master equation and formulate a quasi-Lindblad pseudomode theory. We show that this quasi-Lindblad pseudomode formulation directly leads to a representation of the bath correlation function in terms of a complex weighted sum of complex exponentials, an expansion that is known to be rapidly convergent in practice and thus leads to a compact set of pseudomodes. The pseudomode representation is not unique and can differ by a gauge choice. When the global dynamics can be simulated exactly, the system dynamics is unique and independent of the specific pseudomode representation. However, the gauge choice may affect the stability of the global dynamics, and we provide an analysis of why and when the global dynamics can retain stability despite losing positivity. We showcase the performance of this formulation across various spectral densities in both bosonic and fermionic problems, finding significant improvements over conventional pseudomode formulations.

arXiv:2408.15529•Physical review. B./Physical review. B•Nov 25, 2024•32 citations
Accurate simulations of thermalization in open quantum systems require a reliable representation of the bath correlation function (BCF). Numerical approaches, such as the hierarchical equations of motion and the pseudomode method, inherently approximate the BCF using a finite set of functions, which can impact simulation accuracy. In this work, we propose a practical and rigorous testing framework to assess the validity of approximate BCFs in open quantum dynamics simulations. Our approach employs a harmonic oscillator system, where the computed dynamics can be benchmarked against known exact solutions. To enable practical testing, we make two key methodological advancements. First, we develop numerical techniques to efficiently evaluate these exact solutions across a wide range of BCFs, ensuring broad applicability. Second, we introduce a moment-based state representation that significantly simplifies computations by exploiting the Gaussian nature of the system. Applications to a two-spin system and a transmon-resonator system demonstrate that our testing procedure provides error estimates that capture the qualitative trends observed in thermalization simulations. Using this methodology, we assess the performance of recently proposed BCF construction methods, highlighting both their strengths and a notable challenge posed by sub-Ohmic spectral densities at finite temperatures.
This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.
Two different numerically exact methods for open quantum system dynamics, the hierarchy of pure states (HOPS) method, and the multi-Davydov-Ansatz are discussed. We focus on the suitability of the underlying representations of bath correlations. While in the HOPS case the correct description of the bath correlation function (BCF) in the time domain is decisive, it turns out that a windowed Fourier transform of the BCF is an appropriate indicator of the quality of the discretization in the multi-Davydov-Ansatz. For the spin-boson model with sub-Ohmic spectral density considered here, a discretization of the spectral density based on an exponential distribution, used previously, turns out to be most favorable.
In open-system approaches with non-Markovian environments, the process of inserting an individual mode (denoted as ``pseudomode'') into the bath or extracting it from the bath is widely employed. This procedure, however, is typically performed on basis of the spectral density and does not incorporate temperature. Here, we show how the (temperature-dependent) bath correlation function (BCF) transforms in such a process. We present analytic formulas for the transformed BCF and numerically study the differences between factorizing initial state and global thermal (correlated) initial state of mode and bath, respectively. We find that in the regime of strong coupling of the mode to both system and bath, the differences in the BCFs give rise to pronounced differences in the dynamics of the system.
We show that the general and numerically exact Hierarchy of Pure States method (HOPS) is very well applicable to calculate the reduced dynamics of an open quantum system. In particular, we focus on environments with a sub-Ohmic spectral density (SD) resulting in an algebraic decay of the bath correlation function (BCF). The universal applicability of HOPS, reaching from weak to strong coupling for zero and nonzero temperature, is demonstrated by solving the spin-boson model for which we find perfect agreement with other methods, each one suitable for a special regime of parameters. The challenges arising in the strong coupling regime are not only reflected in the computational effort needed for the HOPS method to converge but also in the necessity for an importance sampling mechanism, accounted for by the nonlinear variant of HOPS. In order to include nonzero-temperature effects in the strong coupling regime we found that it is highly favorable for the HOPS method to use the zero-temperature BCF and include temperature via a stochastic Hermitian contribution to the system Hamiltonian.
Building on the standard hierarchy of pure states (HOPS) approach, we construct a generalized formulation suitable for open quantum systems interacting with nonstationary Gaussian baths, potentially extending its applicability to nonequilibrium baths. This is achieved by extending the conventional exponential decomposition of a bath correlation function (BCF) for nonstationary cases. Using our formulation of HOPS, we derive the corresponding hierarchy of master equations and, when each term in the BCF expansion can be associated with an independent physical bath, we show how the formalism connects to the well-known pseudomode representation. We demonstrate the method's performance on two examples of nonstationary squeezed reservoirs generated via uniform squeezing and degenerate parametric amplification in a one-sided cavity. Benchmarking against the hierarchy of master equations shows that HOPS is more efficient under hierarchy truncation. The pseudomode representation is shown to be more efficient in the strongly non-Markovian regime. Our results highlight HOPS as a versatile and powerful tool for simulating open quantum systems in nonstationary baths, with potential applications ranging from squeezed light-matter interactions to driven quantum materials and dissipative phase transitions.
We introduce a new framework to study the dynamics of open quantum systems with linearly coupled Gaussian baths. Our approach replaces the continuous bath with an auxiliary discrete set of pseudomodes with dissipative dynamics, but we further relax the complete positivity requirement in the Lindblad master equation and formulate a quasi-Lindblad pseudomode theory. We show that this quasi-Lindblad pseudomode formulation directly leads to a representation of the bath correlation function in terms of a complex weighted sum of complex exponentials, an expansion that is known to be rapidly convergent in practice and thus leads to a compact set of pseudomodes. The pseudomode representation is not unique and can differ by a gauge choice. When the global dynamics can be simulated exactly, the system dynamics is unique and independent of the specific pseudomode representation. However, the gauge choice may affect the stability of the global dynamics, and we provide an analysis of why and when the global dynamics can retain stability despite losing positivity. We showcase the performance of this formulation across various spectral densities in both bosonic and fermionic problems, finding significant improvements over conventional pseudomode formulations.