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Distinct Markovian embeddings are different unravelings of the same Gaussian bath self-energy, connected by Bogoliubov transformations.

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2026-08-02 21:01 UTC pith:B6KRF7KK

load-bearing objection A useful unifying framework that maps HEOM and Lindblad-pseudomode onto different Green's-function unravelings, but the key boundary-condition step from path integral to operator equations is asserted, not shown. the 2 major comments →

arxiv 2602.21430 v1 pith:B6KRF7KK submitted 2026-02-24 quant-ph

Markovian Embeddings of Non-Markovian Open System Dynamics

Meng Xu , J. T. Stockburger , J. Ankerhold This is my paper
classification quant-ph
keywords open quantum systemsnon-Markovian dynamicsMarkovian embeddingHierarchical Equations of MotionLindblad pseudomodeGaussian unravelinginfluence functionalBogoliubov transformation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that the many distinct Markovian equations used to simulate non-Markovian open quantum systems are not separate methods but different unravelings of the same Gaussian bath self-energy. The key move is to factor the self-energy into U†GV, with G the Green's function of a local differential operator; this turns the time-nonlocal influence functional into a time-local path integral over auxiliary fields. Different choices of G — Keldysh, block-diagonal, or retarded-only — generate the Lindblad-pseudomode equations, HEOM-type hierarchies, and related time-local generators. A sympathetic reader would care because this gives one transparent foundation for a widely used computational toolbox and explains puzzling features such as why HEOM auxiliary modes start in vacuum, while pseudomodes may start in thermal states.

Core claim

On the paper's own terms, the discovery is that the influence functional of a Gaussian bath can be dissolved into a deterministic time-local dynamics by writing the bath self-energy as a left–right product Σ(t)=U†G(t)V, where G obeys K(i∂t)G=δ. The Gaussian integral identity then converts the time-nonlocal phase into a path integral with a local action in auxiliary fields, and the inverse kernel K(i∂t) becomes the generator of damped auxiliary modes. The physical consequences follow from which Green's function structure is chosen: a triangular Keldysh structure leads to a Lindblad equation with the auxiliary mode in a reference thermal state; a block-diagonal structure leads to vacuum-projec

What carries the argument

The central object is the left–right factorization ansatz Σ(t)=U†G(t)V, where Σ is the Gaussian bath self-energy matrix and G(t) is the Green's function of a local differential operator K(i∂t), so K(i∂t)G(t)=δ(t). The factorization is not unique: any invertible L,R give an equivalent triplet (U†L^{-1}, LGR^{-1}, RV), and this gauge freedom is what organizes the family of embeddings. The machinery converts the retarded, time-nonlocal influence phase into a time-local Gaussian path integral over auxiliary coherent-state fields; boundary conditions inherited from the inverse of G determine both the preparation of auxiliary initial states (thermal reference state vs vacuum) and how the reduced d

Load-bearing premise

The load-bearing premise is that the bath self-energy can be factored as U†GV with G the Green's function of a local differential operator — exact for multi-exponential correlations, approximate otherwise — and that boundary terms in converting the path integral to operator equations can always be absorbed into initial and terminal auxiliary states.

What would settle it

Take a single Brownian-oscillator bath at strong coupling, fix the same physical initial state, and compare converged reduced dynamics from the Lindblad-pseudomode equation (33) and the HEOM-type equation (42) as the auxiliary basis grows; if they disagree beyond truncation error, the claimed equivalence fails. Alternatively, feed a non-multi-exponential spectral density through the factorization and check whether the time-local equations reproduce the exact influence-functional propagator as K increases.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • HEOM and Lindblad–pseudomode are members of one family; each is a different Green's-function structure for the same Gaussian bath, so results from one can be translated into the other.
  • The auxiliary-mode initial state is a free reference occupation, not the physical bath temperature; simulations may choose it for convenience or stability.
  • The vacuum initialization of HEOM auxiliary density operators is explained: it is the boundary condition of a block-diagonal or retarded unraveling, not an ad hoc choice.
  • Bogoliubov transformations provide a systematic way to redistribute truncation errors: transformed embeddings of the same dynamics can be numerically stable where the standard form diverges.
  • Any bath correlation function approximated as a sum of exponentials can be embedded by assembling U,V,G; the paper supplies explicit recipes for the Brownian-oscillator case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Not directly claimed in the paper, but an immediate design rule: for a given bath, one can choose the Green's-function structure and the reference occupation to optimize a specific simulation platform (e.g., tensor networks or quantum circuits), since all such choices leave the exact reduced dynamics unchanged.
  • The boundary-condition step is asserted rather than fully derived; a rigorous continuum-limit treatment of the discrete time path integral would either close this gap or reveal additional surface terms for non-multi-exponential baths.
  • The equivalence suggests a testable extension: repurpose 'unphysical' pseudomode preparations (negative or high occupation references) as a variance- or error-reduction handle in stochastic simulations of the same open system.
  • Because the framework is linear in the auxiliary-mode path space, it should extend straightforwardly to multiple baths and to fermionic Gaussian environments, where the same factorization should yield hierarchies with fermionic signs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a path-integral framework for Markovian embeddings of non-Markovian open quantum dynamics. Starting from the Feynman–Vernon influence functional, the bath self-energy is factorized as Σ = U† G V with G satisfying a local differential equation. A Gaussian identity recasts the nonlocal influence phase as a local path integral over auxiliary fields, which is then converted into deterministic time-local operator equations. The framework is applied to the Brownian-oscillator spectral density, showing that Keldysh and block-diagonal Green’s function structures reproduce Lindblad–pseudomode and HEOM equations, and that gauge transformations in the factorization correspond to Bogoliubov transformations. The paper also discusses numerical-stability implications of different representations.

Significance. The paper addresses a significant question: clarifying the relationships among major embedding methods such as HEOM and Lindblad–pseudomode, and providing a systematic construction of time-local equations for non-Markovian dynamics. Its strengths are the explicit connections to known formulations (e.g., Eq. (33) recovering Lindblad–pseudomode in the appropriate limit, Eq. (42) recovering HEOM in multiple special cases), which provide strong evidence for correctness. The identification of the factorization gauge freedom with Bogoliubov transformations is conceptually valuable. However, the central equivalence between the exact nonlocal influence functional and the time-local operator equations relies on an asserted boundary-condition calculation that is not supplied. If this gap is filled, the framework would be a useful unifying foundation. As it stands, the paper is a promising but incomplete theoretical contribution.

major comments (2)
  1. [§III, Eqs. (24)–(33), (41)–(42), (50)–(52)] The decisive step from the local path integral (24)–(25) to the operator equations (33), (42), (52) is asserted rather than derived. The text states that initial states and terminal weights are constructed by 'carefully accounting for the boundary conditions' and that the inverse kernel is valid 'up to boundary contributions at the contour endpoints' (Eq. (30)), but no boundary-term calculation is presented. The equivalence of the nonlocal influence phase (8) with the time-local equations over the finite interval [0,t] is the central claim of the paper. The authors should show explicitly how the finite-interval inversion of K(i∂τ) yields the chosen initial auxiliary state (thermal in Eq. (32), vacuum in Eq. (41), etc.) and the final trace/projection operations. Without this, the framework rests on formal manipulation and consistency checks rather than a complete proof.
  2. [§IV and abstract] The claimed relation between different embeddings via Bogoliubov transformations is not demonstrated within this manuscript. The abstract and introduction state that the distinct formulations are related through linear transformations in auxiliary-mode path space, and Sec. IV asserts that the generators in Eqs. (33) (with n_ref = 0) and (42) are connected by a Bogoliubov transformation, citing Refs. [6,43]. If this connection is a central contribution, an explicit transformation should be exhibited, at least for a simple case. Otherwise, the paper should clearly attribute this result to prior work rather than presenting it as a new outcome.
minor comments (5)
  1. [§IIB, Case 1] Typo: 'elevanted' should be 'elevated'.
  2. [§IIIB, Example 5] The phrase 'the equation that identity to Eq. (45)' should read 'the equation that is identical to Eq. (45)'.
  3. [§IIIA and §IIIB] In several places (e.g., Eq. (35) and §IIIB Example 1), the frequency appears as ω but the context indicates ω0. Please make the notation consistent.
  4. [Abstract and §IV] The abstract states that the framework 'enables numerically stable and efficient simulations,' but the paper contains no numerical simulations; the examples are analytical identifications with known equations. Either add a numerical demonstration or soften this claim.
  5. [General] The manuscript relies heavily on the authors' previous work (Refs. [6,43]) for key steps such as the QD-MESS framework and Bogoliubov transformations. The novelty of the present contribution relative to those works should be stated more precisely.

Circularity Check

1 steps flagged

Core Gaussian-unraveling derivation is independent and externally anchored; main circularity concern is a load-bearing Bogoliubov-connection claim supported only by the authors' own QD-MESS papers, plus an asserted (not derived) finite-interval boundary-condition bridge.

specific steps
  1. self citation load bearing [Sec. IV (Discussion), third paragraph; cf. Sec. III.B Example 1, Eq. (43)]
    "For example, the generators in Eqs. (33) (with nβ = 0) and (42) are connected via a Bogoliubov transformation [6, 43]."

    One of the paper's stated contributions — that HEOM and Lindblad–pseudomode embeddings 'are related through linear transformations in the auxiliary-mode path space, corresponding precisely to Bogoliubov transformations' — is supported by asserting that the transformation B (Eq. 43) maps the pure-state generator (42) onto the Keldysh generator (37), with both the transformation and its action cited to the same authors' own QD-MESS papers ([6], RMP in press; [43], arXiv:2307.16790) and not demonstrated here. As presented, this part of the unification claim reduces to an unverified-in-paper self-citation chain. Severity is limited because B is written out explicitly and the target generator (37) is independently derived in Sec. III.A, so the algebra is checkable; the main 'different unravelin

full rationale

The derivation chain is: influence phase Φ (Eq. 8) → left–right factorization Σ = U†GV (Eq. 12, explicitly an ansatz exact only for multi-exponential C(t), with the approximation acknowledged) → Gaussian identity (Eqs. 24–25) → operator equations (Eqs. 32–33, 41–42). The inputs (Gaussian self-energy, external Gaussian identity [62,63], stated factorization) do not contain the target results (HEOM, Lindblad–pseudomode), so there is no self-definitional reduction. The recovered equations are not presented as free predictions: the examples fix free gauge parameters (δ, λ, n_ref) and match independently established benchmarks (Lindblad–pseudomode [26–28]; HEOM [34,65,68]; stabilized HEOM [42,65]), so the framework is validated by external consistency rather than predicting new physics from fitted values — no fitted-input-called-prediction infraction. No uniqueness theorem is imported, and the gauge freedom (Eq. 14) is explicitly non-unique. Two genuine weaknesses are weighed. (1) The bridge from the finite-interval path integral (24) to the operator formulations is asserted, not derived: K(i∂_t) is given only 'up to boundary contributions at the contour endpoints' (Eq. 30), the initial states are introduced as 'assuming a factorized initial state ... thermal state' (Eq. 32), and the check is deferred ('can be verified by tracing out the auxiliary fields, recovering the superoperator form of the influence functional [51]'). This is a real completeness gap in the central equivalence, but it is anchored by the external benchmarks the framework reproduces, so it is a rigor issue rather than circularity. (2) The Bogoliubov-connection claim in Sec. IV rests on the authors' own QD-MESS preprints [6,43]; the transformation is displayed and checkable, limiting severity. Overall, the central claim retains independent content; score 3.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

The central claim is a re-expression of a known influence functional through auxiliary Gaussian fields; it introduces gauge parameters δ, λ, n_ref and assumes a local Green's-function factorization. No physical entities are posited; auxiliary modes are computational devices. The heavy use of the authors' earlier QD-MESS work [6,43] means part of the framework is imported rather than re-derived here.

free parameters (5)
  • δ
    Gauge/unraveling parameter in U,V factorizations (Eqs. 29, 40); changes representation but not the physical dynamics; chosen by hand in examples (e.g., δ=1, δ=√2, δ=√(2c1)).
  • λ
    Gauge/unraveling parameter paired with δ; chosen by hand in examples (e.g., λ=1, λ=√2, λ=-δ).
  • n_ref = 0 or n_β
    Gaussian reference occupation of the auxiliary mode; sets the Keldysh fluctuation level and the initial auxiliary state; chosen per representation, not fixed by the physical bath temperature.
  • Rank K of Green's-function expansion
    Number of exponential terms retained in the correlation-function decomposition; controls accuracy of the factorization Eq. (12) for non-multi-exponential baths.
  • Matrix gauge transformations L,R
    Free invertible transformations in Eq. (14) that leave the self-energy product invariant; used to move between representations.
axioms (6)
  • standard math Gaussian identity for complex coherent-state path integrals
    Used to convert the bilinear influence phase into a time-local action; taken from Refs. [62,63].
  • domain assumption Noise is complex Gaussian, fully characterized by two-point correlation C(t)
    The entire embedding is based on Gaussian statistics; non-Gaussian baths would need higher-order cumulants.
  • domain assumption Fluctuation-dissipation relation Sβ(ω)=J(ω)/(1−e^{-βω}) and finite bandwidth
    Used to set the correlation structure and physical temperature; standard for thermal harmonic baths.
  • ad hoc to paper Self-energy factorization Σ=U†GV with G satisfying a local differential equation
    Ansatz Eqs. (12)-(13) is not derived; exact for multi-exponential correlations, approximate otherwise. The central claim depends on this.
  • domain assumption Initial state factorizes and auxiliary modes start in thermal/vacuum state
    Eqs. (32), (41), (51) assume ρ(0)=ρs(0)⊗ρm(0); if the physical bath is correlated with the system initially, the embedding fails.
  • domain assumption Inverse kernel K(i∂t) is local up to boundary terms
    The step from G(t) to the differential operator K(i∂t) with boundary conditions is assumed; the paper says 'up to boundary contributions at contour endpoints' but does not derive the cancellation.
invented entities (1)
  • Auxiliary coherent-state fields Ψ=[ϕ+,ϕ−], etc. no independent evidence
    purpose: Unravel the Gaussian influence functional into a time-local action; these are computational degrees of freedom, not physical entities.
    No falsifiable handle outside the paper; they are gauge/unraveling artifacts whose traces/projections recover the physical reduced density operator.

pith-pipeline@v1.3.0-alltime-deepseek · 15883 in / 11981 out tokens · 93882 ms · 2026-08-02T21:01:38.732577+00:00 · methodology

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read the original abstract

Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

80 extracted references · 6 linked inside Pith · cited by 1 Pith paper

  1. [1]

    This con- struction yields the HEOM structure [30, 45, 69–74]

    The resulting equation, when represented in the normalized Fock basis defined by ˆa|n⟩= √n|n−1⟩, generates aux- iliary density operators ˆρm,n(t) =⟨m|ˆρ(t)|n⟩. This con- struction yields the HEOM structure [30, 45, 69–74]. 7 Example 4.–For the symmetric choicec 1 =c 2 =c 0 andλ=−δ, the correlation function simplifies to C(t≥0) =c 0 e−γt cosζt .(44) The au...

  2. [2]

    C. W. Gardiner and P. Zoller,Quantum noise: a hand- book of Markovian and non-Markovian quantum stochas- tic methods with applications to quantum optics, 3rd ed. (Springer, Berlin, 2004)

    2004

  3. [3]

    Weiss,Quantum dissipative systems, 4th ed

    U. Weiss,Quantum dissipative systems, 4th ed. (World Scientific, New Jersey, 2012)

    2012

  4. [4]

    H. P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, New York, 2002)

    2002

  5. [5]

    de Vega and D

    I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys.89, 015001 (2017)

    2017

  6. [6]

    very recently developed a unified framework termed Quantum Dissipation in Minimally Extended State Space (QD-MESS). This framework serves a dual purpose: By demonstrating how these various non-Markovian ap- proaches emerge from the QD-MESS formulation, it pro- vides a unified picture of non-Markovian approaches with extended state space and thus provides ...

    Pith/arXiv arXiv 2026

  7. [7]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium:non-Markovian dynamics in open quantum systems, Rev. Mod. Phys.88, 021002 (2016)

    2016

  8. [8]

    Egger, L

    R. Egger, L. M¨ uhlbacher, and C. H. Mak, Path-integral monte carlo simulations without the sign problem: Mul- tilevel blocking approach for effective actions, Phys. Rev. E61, 5961 (2000)

    2000

  9. [9]

    M. Xu, V. Vadimov, J. T. Stockburger, and J. Ankerhold, Simulating Non-Markovian Dynamics in Open Quantum Systems, Rev. Mod. Phys. 10.1103/w3nw-hbjc (2026), in press

    doi:10.1103/w3nw-hbjc 2026

  10. [10]

    Kobayashi, H

    F. Kobayashi, H. Manabe, G. A. White, T. Farrelly, K. Modi, and T. M. Stace, Tensor-network decoders for process tensor descriptions of non-markovian noise, arXiv:2412.13739 (2024)

    Pith/arXiv arXiv 2024

  11. [11]

    Strathearn, P

    A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-Markovian quantum dynamics us- ing time-evolving matrix product operators, Nat. Com- mun.9, 3322 (2018)

    2018

  12. [12]

    M¨ uhlbacher, J

    L. M¨ uhlbacher, J. Ankerhold, and A. Komnik, Nonequi- librium dynamics of correlated electron transfer in molec- ular chains, Phys. Rev. Lett.95, 220404 (2005)

    2005

  13. [13]

    Makri and D

    N. Makri and D. E. Makarov, Tensor propagator for itera- tive quantum time evolution of reduced density matrices. i. theory, J. Chem. Phys.102, 4600 (1995)

    1995

  14. [14]

    Wang and M

    H. Wang and M. Thoss, From coherent motion to local- ization: Ii. dynamics of the spin-boson model with sub- ohmic spectral density at zero temperature, Chem. Phys. 370, 78 (2010)

    2010

  15. [15]

    Bose and P

    A. Bose and P. L. Walters, A multisite decomposition of the tensor network path integrals, J. Chem. Phys.156, 024101 (2022)

    2022

  16. [16]

    Wang and M

    H. Wang and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent Hartree theory, J. Chem. Phys.119, 1289 (2003)

    2003

  17. [17]

    A. W. Chin, ´A. Rivas, S. F. Huelga, and M. B. Plenio, Ex- act mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal poly- nomials, J. Math. Phys.51, 092109 (2010)

    2010

  18. [18]

    L. P. Lindoy, B. Kloss, and D. R. Reichman, Time evo- lution of ML-MCTDH wavefunctions. II. Application of the projector splitting integrator, J. Chem. Phys.155, 174108 (2021)

    2021

  19. [19]

    Bulla, H.-J

    R. Bulla, H.-J. Lee, N.-H. Tong, and M. Vojta, Numer- ical renormalization group for quantum impurities in a bosonic bath, Phys. Rev. B71, 045122 (2005)

    2005

  20. [20]

    Zeng, W.-Q

    J. Zeng, W.-Q. Xie, and Y. Zhao, Variational approach to entangled non-hermitian open systems, J. Chem. Theory Comput.21, 3857 (2025)

    2025

  21. [21]

    J. Ren, W. Li, T. Jiang, Y. Wang, and Z. Shuai, Time- dependent density matrix renormalization group method for quantum dynamics in complex systems, Wiley Inter- discip. Rev. Comput. Mol. Sci.12, e1614 (2022)

    2022

  22. [22]

    Zhao, The hierarchy of Davydov’s Ans¨ atze: From guesswork to numerically “exact” many-body wave func- tions, J

    Y. Zhao, The hierarchy of Davydov’s Ans¨ atze: From guesswork to numerically “exact” many-body wave func- tions, J. Chem. Phys.158, 080901 (2023)

    2023

  23. [23]

    Borrelli and M

    R. Borrelli and M. F. Gelin, Simulation of quantum dy- namics of excitonic systems at finite temperature: an ef- ficient method based on thermo field dynamics, Sci. Rep. 7, 1 (2017)

    2017

  24. [24]

    Yan and Y

    Y. Yan and Y. Zhao, Multiple davydov ans¨ atze as solu- tions to lindblad master equations, J. Chem. Phys.163, 234108 (2025)

    2025

  25. [25]

    Yu, Non-markovian quantum trajectories versus mas- ter equations: Finite-temperature heat bath, Phys

    T. Yu, Non-markovian quantum trajectories versus mas- ter equations: Finite-temperature heat bath, Phys. Rev. A69, 062107 (2004)

    2004

  26. [26]

    Tamascelli, A

    D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Nonperturbative treatment of non-Markovian dynamics of open quantum systems, Phys. Rev. Lett.120, 030402 10 (2018)

    2018

  27. [27]

    M. F. Gelin, A. Velardo, and R. Borrelli, Efficient quan- tum dynamics simulations of complex molecular sys- tems: A unified treatment of dynamic and static disorder, J. Chem. Phys.155, 134102 (2021)

    2021

  28. [28]

    de Vega and M.-C

    I. de Vega and M.-C. Ba˜ nuls, Thermofield-based chain- mapping approach for open quantum systems, Phys. Rev. A92, 052116 (2015)

    2015

  29. [29]

    Lentrodt, O

    D. Lentrodt, O. Diekmann, C. H. Keitel, S. Rotter, and J. Evers, Certifying multimode light-matter interaction in lossy resonators, Phys. Rev. Lett.130, 263602 (2023)

    2023

  30. [30]

    Pleasance, B

    G. Pleasance, B. M. Garraway, and F. Petruccione, Gen- eralized theory of pseudomodes for exact descriptions of non-Markovian quantum processes, Phys. Rev. Research 2, 043058 (2020)

    2020

  31. [31]

    S. Luo, N. Lambert, P. Liang, and M. Cirio, Quantum- classical decomposition of gaussian quantum environ- ments: A stochastic pseudomode model, PRX Quantum 4, 030316 (2023)

    2023

  32. [32]

    Huang, G

    Z. Huang, G. Park, G. K. Chan, and L. Lin, Coupled lindblad pseudomode theory for simulating open quan- tum systems, arXiv:2506.10308 (2025)

    arXiv 2025

  33. [33]

    Liang, N

    P. Liang, N. Lambert, S. Luo, L. Guo, and M. Cirio, A purified input-output pseudomode model for structured open quantum systems, arXiv:2412.04264 (2024)

    Pith/arXiv arXiv 2024

  34. [34]

    G. Park, Z. Huang, Y. Zhu, C. Yang, G. K.-L. Chan, and L. Lin, Quasi-lindblad pseudomode theory for open quantum systems, Phys. Rev. B110, 195148 (2024)

    2024

  35. [35]

    Tanimura, Stochastic Liouville, Langevin, Fokker- Planck, and master equation approaches to quantum dis- sipative systems, J

    Y. Tanimura, Stochastic Liouville, Langevin, Fokker- Planck, and master equation approaches to quantum dis- sipative systems, J. Phys. Soc. Jpn.75, 082001 (2006)

    2006

  36. [36]

    Alford, L

    W. Alford, L. P. Bettmann, and G. T. Landi, Subtleties in the pseudomodes formalism, arXiv:2509.16377 (2025)

    Pith/arXiv arXiv 2025

  37. [37]

    Tanimura and R

    Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath, J. Phys. Soc. Jpn.58, 101 (1989)

    1989

  38. [38]

    Yan, Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach, J

    Y. Yan, Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach, J. Chem. Phys.140, 054105 (2014)

    2014

  39. [39]

    Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of mo- tion (HEOM), J

    Y. Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of mo- tion (HEOM), J. Chem. Phys.153, 020901 (2020)

    2020

  40. [40]

    Lambert, S

    N. Lambert, S. Ahmed, M. Cirio, and F. Nori, Mod- elling the ultra-strongly coupled spin-boson model with unphysical modes, Nat. Commun.10, 3721 (2019)

    2019

  41. [41]

    Chen and I

    X. Chen and I. Franco, Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in struc- tured thermal environments, J. Chem. Phys.163, 104109 (2025)

    2025

  42. [42]

    Y. Ke, R. Borrelli, and M. Thoss, Hierarchical equations of motion approach to hybrid fermionic and bosonic en- vironments: Matrix product state formulation in twin space, J. Chem. Phys.156, 194102 (2022)

    2022

  43. [43]

    Ke, Tree tensor network state approach for solving hierarchical equations of motion, J

    Y. Ke, Tree tensor network state approach for solving hierarchical equations of motion, J. Chem. Phys.158, 211102 (2023)

    2023

  44. [44]

    Takahashi, R

    H. Takahashi, R. Borrelli, M. F. Gelin, and L. Chen, Fi- nite temperature dynamics in a polarized sub-ohmic heat bath: A hierarchical equations of motion-tensor train study, J. Chem. Phys.160, 164106 (2024)

    2024

  45. [45]

    T. Li, Y. Yan, and Q. Shi, A low-temperature quan- tum fokker–planck equation that improves the numerical stability of the hierarchical equations of motion for the brownian oscillator spectral density, J. Chem. Phys.156, 064107 (2022)

    2022

  46. [46]

    M. Xu, V. Vadimov, M. Krug, J. T. Stockburger, and J. Ankerhold, A universal framework for quantum dissi- pation: minimally extended state space and exact time- local dynamics, arXiv:2307.16790 (2023)

    Pith/arXiv arXiv 2023

  47. [47]

    G. W. Ford, J. T. Lewis, and R. F. O’Connell, Quantum langevin equation, Phys. Rev. A37, 4419 (1988)

    1988

  48. [48]

    Y. Su, Y. Wang, and W. Dou, Nonperturbative open quantum dynamics bypass influence functional, arXiv:2503.00297 (2025)

    Pith/arXiv arXiv 2025

  49. [49]

    A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissi- pative two-level system, Rev. Mod. Phys.59, 1 (1987)

    1987

  50. [50]

    R. P. Feynman and F. L. Vernon, The theory of a gen- eral quantum system interacting with a linear dissipative system, Ann. Phys.24, 118 (1963)

    1963

  51. [51]

    Kubo, The fluctuation-dissipation theorem, Rep

    R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys.29, 255 (1966)

    1966

  52. [52]

    Kubo, Generalized cumulant expansion method, J

    R. Kubo, Generalized cumulant expansion method, J. Phys. Soc. Jpn.17, 1100 (1962)

    1962

  53. [53]

    A. Garg, J. N. Onuchic, and V. Ambegaokar, Ef- fect of friction on electron transfer in biomolecules, J. Chem. Phys.83, 4491 (1985)

    1985

  54. [54]

    Aurell, R

    E. Aurell, R. Kawai, and K. Goyal, An operator deriva- tion of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an-harmonic baths, J. Phys. A Math.53, 275303 (2020)

    2020

  55. [55]

    Huang, L

    Z. Huang, L. Lin, G. Park, and Y. Zhu, Unified analysis of non-markovian open quantum systems in gaussian environment using superoperator formalism, arXiv:2411.08741 (2024)

    arXiv 2024

  56. [56]

    Meier and D

    C. Meier and D. J. Tannor, Non-markovian evolution of the density operator in the presence of strong laser fields, J. Chem. Phys.111, 3365 (1999)

    1999

  57. [57]

    Grabert, U

    H. Grabert, U. Weiss, and P. Talkner, Quantum theory of the damped harmonic oscillator, Z. Physik B - Condensed Matter55, 87 (1984)

    1984

  58. [58]

    H¨ anggi and G.-L

    P. H¨ anggi and G.-L. Ingold, Fundamental aspects of quantum Brownian motion, Chaos15, 026105 (2005)

    2005

  59. [59]

    A. D. Somoza, O. Marty, J. Lim, S. F. Huelga, and M. B. Plenio, Dissipation-assisted matrix product factorization, Phys. Rev. Lett.123, 100502 (2019)

    2019

  60. [60]

    Tanaka and Y

    M. Tanaka and Y. Tanimura, Quantum dissipative dy- namics of electron transfer reaction system: Nonpertur- bative hierarchy equations approach, J. Phys. Soc. Jpn. 78, 073802 (2009)

    2009

  61. [61]

    Risken,The Fokker-Planck Equation(Springer- Verlag, Berlin, 1984)

    H. Risken,The Fokker-Planck Equation(Springer- Verlag, Berlin, 1984)

    1984

  62. [62]

    Altland and B

    A. Altland and B. D. Simons,Condensed matter field theory(Cambridge University Press, 2010)

    2010

  63. [63]

    Lepp¨ akangas, N

    J. Lepp¨ akangas, N. Vogt, K. R. Fratus, K. Bark, J. A. Vaitkus, P. Stadler, J.-M. Reiner, S. Zanker, and M. Marthaler, Quantum algorithm for solving open- system dynamics on quantum computers using noise, Phys. Rev. A108, 062424 (2023)

    2023

  64. [64]

    S. Bai, S. Zhang, C. Huang, and Q. Shi, Hierarchical equations of motion for quantum chemical dynamics: Re- cent methodology developments and applications, Acc. Chem. Res.57, 3151 (2024)

    2024

  65. [65]

    demonstrated that transforming these conventional structured HEOM into a representation such as (33) can effectively eliminate this numerical instability. B. Pure-State Path-Integral Representation An alternative and conceptually distinct representa- tion emerges when the self-energy is expressed in a basis that renders the 2×2 Green’s function matrixG(t)...

  66. [66]

    L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys.79, 096001 (2016)

    2016

  67. [67]

    Kamenev,Field theory of non-equilibrium systems (Cambridge University Press, 2023)

    A. Kamenev,Field theory of non-equilibrium systems (Cambridge University Press, 2023)

    2023

  68. [68]

    Y. Yan, T. Xing, and Q. Shi, A new method to im- prove the numerical stability of the hierarchical equa- tions of motion for discrete harmonic oscillator modes, J. Chem. Phys.153, 204109 (2020)

    2020

  69. [69]

    I. S. Dunn, R. Tempelaar, and D. R. Reichman, Re- 11 moving instabilities in the hierarchical equations of mo- tion: Exact and approximate projection approaches, J. Chem. Phys.150, 184109 (2019)

    2019

  70. [70]

    Krug and J

    M. Krug and J. Stockburger, On stability issues of the heom method, The European Physical Journal Special Topics , (2023)

    2023

  71. [71]

    H. Liu, L. Zhu, S. Bai, and Q. Shi, Reduced Quantum Dynamics with Arbitrary Bath Spectral Densities: Hi- erarchical Equations of Motion Based on Several Differ- ent Bath Decomposition Schemes., J. Chem. Phys.140, 134106 (2014)

    2014

  72. [72]

    M. Xu, Y. Yan, Q. Shi, J. Ankerhold, and J. T. Stockburger, Taming quantum noise for efficient low temperature simulations of open quantum systems, Phys. Rev. Lett.129, 230601 (2022)

    2022

  73. [73]

    Suess, W

    D. Suess, W. T. Strunz, and A. Eisfeld, Hierarchical equa- tions for open system dynamics in fermionic and bosonic environments, J. Stat. Phys.159, 1408 (2015)

    2015

  74. [74]

    V. Link, K. M¨ uller, R. G. Lena, K. Luoma, F. m. c. Damanet, W. T. Strunz, and A. J. Daley, Non-markovian quantum dynamics in strongly coupled multimode cavi- ties conditioned on continuous measurement, PRX Quan- tum3, 020348 (2022)

    2022

  75. [75]

    Vilkoviskiy and D

    I. Vilkoviskiy and D. A. Abanin, Bound on approximat- ing non-markovian dynamics by tensor networks in the time domain, Phys. Rev. B109, 205126 (2024)

    2024

  76. [76]

    Y. Yan, F. Yang, Y. Liu, and J. Shao, Hierarchical ap- proach based on stochastic decoupling to dissipative sys- tems, Chem. Phys. Lett.395, 216 (2004)

    2004

  77. [77]

    Zhang, N

    C. Zhang, N. Lambert, X.-Q. Li, M. Cirio, and P. Liang, Purified pseudomode model for nonlinear system-bath in- teractions, Phys. Rev. B112, 115131 (2025)

    2025

  78. [78]

    Ishizaki and Y

    A. Ishizaki and Y. Tanimura, Quantum dynamics of sys- tem strongly coupled to low-temperature colored noise bath: Reduced hierarchy equations approach, J. Phys. Soc. Jpn.74, 3131 (2005)

    2005

  79. [79]

    Q. Shi, L. Chen, G. Nan, R. Xu, and Y. Yan, Electron transfer dynamics: Zusman equation versus exact theory, J. Chem. Phys.130, 164518 (2009)

    2009

  80. [80]

    Ikeda and G

    T. Ikeda and G. D. Scholes, Generalization of the hierar- chical equations of motion theory for efficient calculations with arbitrary correlation functions, J. Chem. Phys.152, 204101 (2020)

    2020