A purified input-output pseudomode model for structured open quantum systems

Lingzhen Guo; Mauro Cirio; Neill Lambert; Pengfei Liang; Si Luo

REVIEW 3 major objections 5 minor 3 cited by

A purified input-output pseudomode model for structured open quantum systems

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper builds purified pseudomode models — auxiliary modes kept in pure states by analytic continuation — that reproduce exactly both the reduced dynamics of an open quantum system and the input-output statistics of its bosonic bath…

desk verdict A credible new purification trick for pseudomodes that extends them to bath input-output, but the exactness proof is entirely in a missing supplement and the main text overstates 'numerically exact'. read the letter →

arxiv 2412.04264 v1 pith:Y7MQMNWT submitted 2024-12-05 quant-ph

Pengfei Liang , Neill Lambert , Si Luo , Lingzhen Guo , Mauro Cirio This is my paper · ORCID

classification quant-ph MSC 81S2281V80

keywords openquantumsystemspseudomodetheorynon-Markoviandynamicsinput-outputhierarchicalequationsofmotionnon-Gaussianbathstatescavityelectrodynamicsanalyticalcontinuation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims a way to extend pseudomode methods from simulating the reduced dynamics of an open quantum system to also simulating the bath's own input-output properties, including cases where the bath starts in a non-Gaussian state. The move is to analytically continue each pseudomode's frequency and decay rate into the complex plane and take the large-frequency limit, which splits every bath correlation into forward- and backward-time pieces that can each be represented by a pure auxiliary mode. The resulting purified pseudomode model reproduces the full influence functional of the original continuum bath, so the reduced system density matrix and the bath observables both come out of one Lindblad-like simulation. A numeric demonstration on a coupled-cavity waveguide shows multi-photon transfer with large time delays, an emission spectrum with Fano interference, and a single-photon bath initial state. If the construction holds in general, it provides a non-perturbative input-output theory for strongly coupled, long-memory open quantum systems.

What carries the argument

The load-bearing object is the purified pseudomode: an auxiliary bosonic mode whose correlation is the positive- or negative-time half of a bath correlation, reached by the analytic-continuation path $\Omega \to \Omega \pm ia$, $\Gamma \to \Gamma + a$ in the $a \to \infty$ limit. Its defining property is one-sided action — the mode operators appear on only one side of the density matrix in the master equation — which keeps the ancilla pure and lets the whole model evolve as a state vector. The machinery works because each exponential in a bath correlation is mapped one-to-one onto such a mode, and because the same mapping turns the bath field operators into pseudomode superoperators (for instance $\Phi^l_{\mathrm{PPM},\alpha+}[\cdot] = \lambda''_s d_{\alpha+}[\cdot]$) that allow direct computation of input and output observables. The resulting master equation is equivalent to the free-pole HEOM in the Hermitian-coupling case, which the paper proves in the Supplemental Material.

What would settle it

Simulate a strongly coupled two-emitter waveguide with a spectral density that has a slowly decaying power-law tail, using a fixed small number of exponential terms in the purified pseudomode model; compare the predicted emission spectrum and emitter population against a direct tensor-network simulation of the original continuum model. If the two disagree by more than the declared numerical tolerance and the disagreement does not shrink as more exponentials are added, the 'numerically exact' claim fails. A second check is to compute $\rho_{\mathrm{eff}}$ for increasing finite values of $a$ and verify that the trace equality and bath observables converge before $a$ becomes large enough to cause numerical stiffness.

Watch

Extended reading notes

Core claim

The central claim is that for a system linearly coupled to a bosonic bath, whenever each bath correlation can be written as a finite sum of exponentials, there exists an exact effective model made of 'purified' pseudomodes — auxiliary bosonic modes $d_\pm$ with complex frequencies $\pm\Omega - i\Gamma$ — whose pure-state dynamics gives $\rho_S(t) = \operatorname{Tr}_\pm[\rho_{\mathrm{eff}}(t)]$ and also produces the correlation matrices $\rho^a_S(t)$ that encode generic bath field correlations. The purification is a limit of the standard pseudomode continuation: sending $\Omega \to \Omega \pm ia$ and $\Gamma \to \Gamma + a$ with $a \to \infty$ turns the mode's correlation into $C^\pm_{\mathrm{PPM}}(t) = C(t)\Theta(\pm t)$, so the positive- and negative-time branches of every correlation are carried by separate modes. Because $d_\pm$ and $d^\dagger_\pm$ act on only one side of the density matrix in the generators $\mathcal{L}_\pm$, the ancillas remain pure, making the simulation cheaper and tensor-network friendly. The same replacement rule constructs pseudomode field superoperators, so bath input-output statistics are read out from the same master equation without separate HEOM layers.

Load-bearing premise

The construction is exact only if the limit $a \to \infty$ can be moved freely past the time-ordered integrals and the trace, a step the main text defers to the Supplemental Material, and only if the bath correlations are accurately represented by the chosen exponential decomposition.

Editorial extensions

If this is right

Reduced system dynamics and bath observables are obtained from one simulation, so input-output theory gains a non-perturbative form valid for strong coupling and large time delays.
Non-Gaussian bath preparations, such as single- or multi-photon environmental states, are handled by the same purified model through extra field superoperators.
The equivalence with free-pole HEOM means technical advances on either side (mode optimization, truncation, tensor-network compression) transfer to the other.
Because the ancillary modes remain pure, the method lowers the local Hilbert-space cost of tensor-network simulations of open quantum systems.
Any bosonic bath whose correlation functions admit an exponential decomposition falls within the method's scope.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural stress test is fermionic baths: the one-sided action idea might survive the sign structure of fermion correlations, but the Wick decomposition and superoperator signs would need a separate derivation.
The accuracy of the method hinges on how well a finite set of exponentials fits the bath spectral density; a systematic, a priori error bound on truncation would be a direct extension the paper does not provide.
The $a \to \infty$ limit is taken before the time-ordered integrals; checking finite-$a$ convergence numerically for a non-Gaussian initial state would give an accessible verification of the purified model's exactness.
Since the method reads out bath observables through the same ancillas that mediate the system-bath interaction, it could be adapted to compute heat currents or entanglement between bath partitions, not just photon emission spectra.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper earns a real look. It constructs a 'purified pseudomode' model by analytically continuing pseudomode parameters along a→∞, so that the auxiliary modes stay pure, and then uses it to compute bath observables and input-output quantities for non-Gaussian environmental states—something standard pseudomode and HEOM methods don't directly give. The numerics are also respectable: the small-delay limit matches a Lindblad benchmark, and the emission spectrum reproduces the experimental Fano-interference data of Yu et al. If the method works, it's a useful tool for waveguide QED and quantum-network simulations, not a revolution.

What's genuinely new: the purification construction itself and its use for bath input-output. The paper also honestly notes that for system dynamics the method reduces to the free-pole HEOM of Xu et al. (PRL 129, 230601) — that connection is a strength, not a weakness.

Now the soft spots, in proportion. The central identity ρ_S = Tr_±[ρ_eff] is asserted in the main text and the proof is deferred to a supplement that isn't in front of us. More specifically, the a→∞ limit is taken inside time-ordered integrals and a trace, and for finite a the effective generator is non-Hermitian. The non-Hermiticity alone is fine—unphysical modes are the whole point here—but the limit interchange is genuinely nontrivial and the main text does not justify it. The free-pole HEOM connection makes me think the identity very likely holds, but 'very likely' is not the same as 'numerically exact'.

Second, the method requires an exponential decomposition of every correlation function, and the main text gives no error analysis or truncation criterion. For a methods paper that claims exactness, that's a real gap, though a fillable one.

Third, no code or data is shipped, so the benchmarks can't be independently reproduced without reimplementation.

The stress-test note worries that the asserted limit interchange invalidates the paper; I'd push back only on the severity. The finite-a non-Hermiticity is not itself a flaw—standard pseudomodes are unphysical by design. The flaw is the missing proof, and the right verdict is conditional, not dismissal.

Who should read this: anyone working on non-Markovian open quantum systems, input-output theory, or structured baths. Send it to referees—but ask them to verify the supplement and to request a convergence analysis for the exponential decomposition.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a 'purified pseudomode' construction for open quantum systems, in which auxiliary bosonic modes with complex frequencies are obtained by analytically continuing the parameters of standard pseudomodes. The central claim is that the reduced system dynamics and, more generally, bath input-output correlations -- captured by the auxiliary quantities rho^a_S(t) in Eq. (3) -- are exactly reproduced by a Lindblad-like equation, Eq. (8), for purified modes d_+ and d_-, so that rho_S(t) = Tr_+-[rho_eff(t)]. The method is claimed to work also for non-Gaussian environmental initial states and to establish a connection between pseudomode theory and HEOM. The paper demonstrates the approach on a two-cavity waveguide QED model, computing emitter and cavity dynamics, non-Gaussian bath-initial-state dynamics, and emission spectra.

Significance. If the exactness of Eq. (8) is established, the paper would give a practically useful bridge between pseudomode theory and HEOM, while extending the accessible observables from the reduced system state to bath input-output properties beyond the Markovian regime. The numerical examples are concrete and physically relevant; the comparison with the small-delay Lindblad benchmark in Fig. 3(b) and with the experimental Fano spectrum in Ref. [91] provides meaningful checks. The main limitation is that the central exactness statement is deferred to the Supplemental Material [97], and the exponential-decomposition assumption is not quantified in the main text, so the 'numerically exact' status of the method cannot yet be assessed from the submitted manuscript alone.

major comments (3)

[Purification of zero-temperature pseudomodes, Eqs. (7)-(8)] The identity rho_S(t) = Tr_+-[rho_eff(t)] is the load-bearing claim of the paper, but it is not proved in the main text. The derivation requires interchanging the limit a -> infinity with the time-ordered integrals in the influence functional (1) and with the trace over the extended Hilbert space. For finite a, the effective generator contains complex mode frequencies (Omega +/- ia), so the equation is not a standard Lindblad master equation and the limit is a singular analytic continuation; no assumptions, regularity conditions, or error bounds are given. The reference to the Supplemental Material [97] is not sufficient for a self-contained paper, and without a proof of this limit interchange the 'numerically exact' status of the method is not established. Please include the proof, or at least a precise statement of the conditions under which the identity holds and a convergence argument, in the main text.
[Purified pseudomode models, Eq. (4)] The construction begins 'whenever an exponential decomposition of each ... correlation is available', and the numerical simulations necessarily truncate this decomposition to a finite number of modes. The manuscript does not state how many exponential terms are used, how the coefficients (w_s, Omega_s, Gamma_s) are determined for the waveguide correlation functions, or what the convergence criterion is. Since generic spectral densities are not finite sums of exponentials, a finite fit introduces an approximation; quantifying this error is essential for the claimed 'numerically exact' description.
[Numerical implementation, Fig. 3] The only independent quantitative benchmark is the small-delay Lindblad equation (black dashed lines in Fig. 3(b)), valid for kappa_{c,n} t_d << 1. The large-delay results (x_d = 1500 lambda_0) and the non-Gaussian bath initial state in Fig. 3(d) are not cross-checked against another method, and the experimental comparison in Fig. 3(e) is qualitative. To support the central claim in the deep non-Markovian regime, please provide convergence tests (e.g., dependence on the number of exponential terms and on the local Hilbert-space dimension) and, if possible, an independent benchmark for at least one large-delay case.

minor comments (5)

[System dynamics and environmental correlations, Eq. (3) vs. Eq. (6)] The symbol a is used both as a subset index in Eq. (3) and as the analytic-continuation energy parameter in Eq. (6); this double use is confusing and should be changed.
[Eq. (7)] The notation Theta(+-t) should be spelled out: C+_PPM corresponds to C(t)Theta(t) and C-_PPM to C(t)Theta(-t), with the value at t = 0 irrelevant for the time-ordered integrals.
[Purified pseudomode models, HEOM connection] The claimed equivalence of Eq. (9) with the free-pole HEOM of Ref. [46] is stated in one sentence and deferred to the Supplemental Material; a short sketch of the mapping in the main text would make the claimed connection verifiable.
[Fig. 3(e) caption] The caption attributes the small oscillation to finite spectral resolution; a brief explanation of the numerical resolution parameter and its effect on the spectrum would help the reader separate artifacts from physical features.
[General] The manuscript does not include a code or data availability statement; given the numerical character of the paper, such a statement would strengthen reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the pseudomode parameters are fitted to the bath correlations by construction in the standard exact-embedding sense, and the physical predictions are checked against independent Lindblad and experimental results.

full rationale

The central identity ρ_S(t)=Tr_±[ρ_eff(t)] (Eq. 8) is presented as a result proved in the Supplemental Material, not as a quantity that is fitted and then renamed a prediction. The mode frequencies Ω−iΓ and coupling strengths λ′_s, λ′′_s are indeed chosen so that the effective pseudomode correlations reproduce the originally specified exponential-decomposed bath correlations via Eqs. (7) and (11); this is the usual construction of an exact effective model, not a hidden fit, because the predicted quantities (emitter populations, cavity occupations, emission spectra) are non-trivial functionals of those correlations. The numerical results are validated against an independent Lindblad master equation in Fig. 3(b) and against the experimental Fano-resonance spectrum in Ref. [91] in Fig. 3(e). The self-citation [94] introduces the ρ^a_S formalism for bath input-output, but the relevant expressions are stated explicitly in Eqs. (2)-(3) and the purified-pseudomode claim does not rest solely on that citation; it is therefore a minor, non-load-bearing self-citation rather than a circular step. The main weakness is a correctness risk, not circularity: the a→∞ limit in Eq. (7) is interchanged with the time-ordered integrals and the trace, and the rigorous justification is deferred to the Supplemental Material [97], which is not included in the manuscript. If that limit interchange fails, the 'numerically exact' status of the method is not established, but this is an unverified mathematical step, not a reduction of the prediction to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The method's central object is an exact effective model, so its honesty rests on the decomposition and limit assumptions rather than on fitted physical constants.

free parameters (2)

exponential decomposition coefficients (w_s, Omega_s, Gamma_s) = not reported in main text
Every bath correlation in Eq. (4) is assumed to be a sum of exponentials; these parameters set the pseudomode frequencies and couplings. The accuracy and truncation of this decomposition are not quantified, so 'numerically exact' depends on this representation.
Hilbert-space truncation for pseudomodes = not specified
The bosonic local dimensions must be truncated in numerical simulation, and no convergence details are provided in the main text.

assumptions (4)

standard math Gaussian statistics of the bath, enabling Wick's theorem
Used after Eq. (2) to express multi-time correlations as products of two-point functions; relies on the bath being Gaussian.
domain assumption Initial system-bath state is a product state rho(0)=rho_S ⊗ rho_B with Gaussian rho_B
Stated before Eq. (1); limits applicability to initially uncorrelated Gaussian baths, with non-Gaussian inputs encoded via field superoperators.
domain assumption All relevant correlation functions admit an exponential decomposition
Required for Eq. (4) and the construction of pseudomodes; not all spectral densities admit an exact finite exponential representation.
ad hoc to paper The analytic-continuation limit a -> infinity can be interchanged with the dynamics and trace
Eqs. (6)-(8) rely on taking a -> infinity inside the evolution; the validity of this limit is deferred to the Supplemental Material.

invented entities (1)

purified pseudomodes d±
purpose: Auxiliary bosonic modes with complex frequencies ±Omega - iGamma that reproduce the positive/negative-time branches of bath correlations and remain in a pure state
Mathematical ancillas introduced to reduce Hilbert space and compute bath observables; no physical reality or experimental signature is claimed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A purified input-output pseudomode model for structured open quantum systems." pith.science (2026). https://pith.science/paper/Y7MQMNWT

@misc{pith2026241204264,
  author       = {Pith},
  title        = {Pith review of: A purified input-output pseudomode model for structured open quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7MQMNWT}},
  note         = {Machine review of arXiv:2412.04264}
}

read the original abstract

A full understanding of open quantum systems requires the characterization of both system and environmental properties. However, the complexity of the environmental statistics in the presence of strong system-bath hybridization and long memory effects usually prevents effective non-perturbative methods from going beyond the analysis of the reduced system dynamics. Here we present a model consisting of purified auxiliary bosonic modes to describe, alongside properties of the system, the dynamics of environmental observables for bosonic baths prepared in non-Gaussian initial states. We numerically exemplify this method by simulating non-Markovian multi-photon transfer processes on a coupled cavity waveguide system in the large time delay regime.

Figures

Figures reproduced from arXiv: 2412.04264 by the authors.

**Figure 2.** FIG. 2. Analytical continuation of the frequency Ω and decay [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Schematics of the coupled-cavity QED model. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

WaveguideQED.jl: An Efficient Framework for Simulating Non-Markovian Waveguide Quantum Electrodynamics
quant-ph 2024-12 accept novelty 6.0 of 10

WaveguideQED.jl is an efficient, open-source, time-bin simulator for waveguide QED with up to two photons, including full photonic states and non-Markovian feedback.
Markovian Embeddings of Non-Markovian Open System Dynamics
quant-ph 2026-02 conditional novelty 5.0 of 10

Different Markovian embedding schemes for non-Markovian open quantum systems are shown to be different unravelings of the same Gaussian bath self-energy, related by Bogoliubov transformations.
Tensor network methods for non-perturbative dynamics of open quantum systems
quant-ph 2026-08 accept novelty 3.0 of 10

A review that organizes six numerically exact tensor-network methods for open quantum systems into a shared formalism, compares their convergence behavior, and lists software implementations.

Reference graph

Works this paper leans on

97 extracted references · 63 canonical work pages · cited by 3 Pith papers

[91]

Y. Yu, A. M. Delgoffe, A. Miranda, A. Lyasota, B. Dwir, A. Rudra, and E. Kapon, Remote excitation between quantum emitters mediated by an optical fano resonance, Optica 8, 1605 (2021)

work page 2021
[97]

The supplementary material contains necessary technical details to complement the derivations and statements in the main text

work page
[1]

Breuer and F

H.-P. Breuer and F. Petruccione, The Theory Of Open Quantum Systems (Oxford University Press, Oxford, 2002)

2002
[2]

Gardiner and P

C. Gardiner and P. Zoller, Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic meth- ods with applications to quantum optics(Springer Science, 2004)

2004
[3]

S.-H. Wei, B. Jing, X.-Y. Zhang, J.-Y. Liao, C.-Z. Yuan, B.-Y. Fan, C. Lyu, D.-L. Zhou, Y. Wang, G.-W. Deng, H.- Z. Song, D. Oblak, G.-C. Guo, and Q. Zhou, Towards real- world quantum networks: A review, Laser & Photonics Reviews 16, 2100219 (2022)

2022
[4]

A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakin- skiy, and A. N. Poddubny, Waveguide quantum electro- dynamics: Collective radiance and photon-photon corre- lations, Rev. Mod. Phys. 95, 015002 (2023)

2023
[5]

Ac ´ ın, I

A. Ac ´ ın, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, The quantum technologies roadmap: a european commu- nity view, New Journal of Physics 20, 080201 (2018)

2018
[6]

Frisk Kockum, A

A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics 1, 19 (2019)

2019

Show all 97 references

[7]

Forn-D ´ ıaz, L

P. Forn-D ´ ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019)

2019
[8]

Frisk Kockum, P

A. Frisk Kockum, P. Delsing, and G. Johansson, Designing frequency-dependent relaxation rates and lamb shifts for a giant artificial atom, Phys. Rev. A 90, 013837 (2014)

2014
[9]

A. F. Kockum, G. Johansson, and F. Nori, Decoherence- free interaction between giant atoms in waveguide quan- tum electrodynamics, Phys. Rev. Lett. 120, 140404 (2018)

2018
[10]

Andersson, B

G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, Non-exponential decay of a giant artificial atom, Nature Physics 15, 1123 (2019)

2019
[11]

L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, Oscillating bound states for a giant atom, Phys. Rev. Res. 2, 043014 (2020)

2020
[12]

C. A. Gonz´ alez-Guti´ errez, J. Rom´ an-Roche, and D. Zueco, Distant emitters in ultrastrong waveguide qed: Ground- state properties and non-markovian dynamics, Phys. Rev. A 104, 053701 (2021)

2021
[13]

de Vega, U

I. de Vega, U. Schollw¨ ock, and F. A. Wolf, How to dis- cretize a quantum bath for real-time evolution, Phys. Rev. B 92, 155126 (2015)

2015
[14]

W. Li, J. Ren, and Z. Shuai, Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evo- lution schemes, The Journal of Chemical Physics 152, 024127 (2020)

2020
[15]

J. Ren, W. Li, T. Jiang, Y. Wang, and Z. Shuai, Time- dependent density matrix renormalization group method for quantum dynamics in complex systems, WIREs Com- putational Molecular Science 12, e1614 (2022)

2022
[16]

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

1985
[17]

Martinazzo, B

R. Martinazzo, B. Vacchini, K. H. Hughes, and I. Burghardt, Communication: Universal Markovian re- duction of Brownian particle dynamics, J. Chem. Phys. 134, 011101 (2011)

2011
[18]

Iles-Smith, N

J. Iles-Smith, N. Lambert, and A. Nazir, Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems, Phys. Rev. A 90, 032114 (2014)

2014
[19]

M. P. Woods, R. Groux, A. W. Chin, S. F. Huelga, and M. B. Plenio, Mappings of open quantum systems onto chain representations and Markovian embeddings, J. Math. Phys. 55, 032101 (2014). 6

2014
[20]

Strasberg, G

P. Strasberg, G. Schaller, T. L. Schmidt, and M. Esposito, Fermionic reaction coordinates and their application to an autonomous Maxwell demon in the strong-coupling regime, Phys. Rev. B 97, 205405 (2018)

2018
[21]

Wertnik, A

M. Wertnik, A. Chin, F. Nori, and N. Lambert, Op- timizing co-operative multi-environment dynamics in a dark-state-enhanced photosynthetic heat engine, J. Chem. Phys. 149, 084112 (2018)

2018
[22]

A. W. Chin, A. Rivas, S. F. Huelga, and M. B. Ple- nio, Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials, J. Math. Phys. 51, 092109 (2010)

2010
[23]

Prior, A

J. Prior, A. W. Chin, S. F. Huelga, and M. B. Plenio, Efficient simulation of strong system-environment inter- actions, Phys. Rev. Lett. 105, 050404 (2010)

2010
[24]

Tamascelli, A

D. Tamascelli, A. Smirne, J. Lim, S. F. Huelga, and M. B. Plenio, Efficient simulation of finite-temperature open quantum systems, Phys. Rev. Lett. 123, 090402 (2019)

2019
[25]

N¨ ußeler, I

A. N¨ ußeler, I. Dhand, S. F. Huelga, and M. B. Plenio, Efficient simulation of open quantum systems coupled to a fermionic bath, Phys. Rev. B 101, 155134 (2020)

2020
[26]

Holstein, Studies of polaron motion: Part I

T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Annals of Physics 8, 325 – 342 (1959)

1959
[27]

Holstein, Studies of polaron motion: Part II

T. Holstein, Studies of polaron motion: Part II. The “small” polaron, Annals of Physics 8, 343 – 389 (1959)

1959
[28]

Jackson and R

B. Jackson and R. Silbey, On the calculation of transfer rates between impurity states in solids, The Journal of Chemical Physics 78, 4193–4196 (1983)

1983
[29]

Silbey and R

R. Silbey and R. A. Harris, Variational calculation of the dynamics of a two level system interacting with a bath, J. Chem. Phys. 80, 2615 (1984)

1984
[30]

R. A. Harris and R. Silbey, Variational calculation of the tunneling system interacting with a heat bath. II. Dynamics of an asymmetric tunneling system, J. Chem. Phys. 83, 1069 (1985)

1985
[31]

W¨ urger, Strong-coupling theory for the spin-phonon model, Phys

A. W¨ urger, Strong-coupling theory for the spin-phonon model, Phys. Rev. B 57, 347 (1998)

1998
[32]

Wilson-Rae and A

I. Wilson-Rae and A. Imamo˘ glu, Quantum dot cavity- QED in the presence of strong electron-phonon interac- tions, Phys. Rev. B 65, 235311 (2002)

2002
[33]

Jang, Y.-C

S. Jang, Y.-C. Cheng, D. R. Reichman, and J. D. Eaves, Theory of coherent resonance energy transfer, The Journal of Chemical Physics 129, 101104 (2008)

2008
[34]

Jang, Theory of coherent resonance energy transfer for coherent initial condition., The Journal of Chemical Physics 131, 164101 (2009)

S. Jang, Theory of coherent resonance energy transfer for coherent initial condition., The Journal of Chemical Physics 131, 164101 (2009)

2009
[35]

Nazir, Correlation-dependent coherent to incoherent transitions in resonant energy transfer dynamics, Phys

A. Nazir, Correlation-dependent coherent to incoherent transitions in resonant energy transfer dynamics, Phys. Rev. Lett. 103, 146404 (2009)

2009
[36]

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
[37]

Tanimura, Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath, Phys

Y. Tanimura, Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath, Phys. Rev. A 41, 6676 (1990)

1990
[38]

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
[39]

Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J

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

2006
[40]

Ishizaki and G

A. Ishizaki and G. R. Fleming, Unified treatment of quan- tum coherent and incoherent hopping dynamics in elec- tronic energy transfer: Reduced hierarchy equation ap- proach, J. Chem. Phys. 130, 234111 (2009)

2009
[41]

A. G. Dijkstra and Y. Tanimura, Non-Markovian entangle- ment dynamics in the presence of system-bath coherence, Phys. Rev. Lett. 104, 250401 (2010)

2010
[42]

Z. Li, N. Tong, X. Zheng, D. Hou, J. Wei, J. Hu, and Y. Yan, Hierarchical Liouville-space approach for accu- rate and universal characterization of quantum impurity systems, Phys. Rev. Lett. 109, 266403 (2012)

2012
[43]

J. Ma, Z. Sun, X. Wang, and F. Nori, Entanglement dynamics of two qubits in a common bath, Phys. Rev. A 85, 062323 (2012)

2012
[44]

Tanimura, Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities, J

Y. Tanimura, Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities, J. Chem. Phys. 141, 044114 (2014)

2014
[45]

Song and Q

L. Song and Q. Shi, Hierarchical equations of motion method applied to nonequilibrium heat transport in model molecular junctions: Transient heat current and high- order moments of the current operator, Phys. Rev. B 95, 064308 (2017)

2017
[46]

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

2022
[47]

X. Dan, M. Xu, J. T. Stockburger, J. Ankerhold, and Q. Shi, Efficient low-temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model, Phys. Rev. B 107, 195429 (2023)

2023
[48]

M. Xu, V. Vadimov, M. Krug, J. T. Stockburger, and J. Ankerhold, A universal framework for quantum dissipa- tion: Minimally extended state space and exact time-local dynamics (2023), arXiv:2307.16790 [quant-ph]

2023 arXiv
[49]

Tanimura, Numerically “exact” approach to open quan- tum dynamics: The hierarchical equations of motion (HEOM), J

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

2020
[50]

Nakamura and Y

K. Nakamura and Y. Tanimura, Optical response of laser-driven charge-transfer complex described by Hol- stein–Hubbard model coupled to heat baths: Hierarchical equations of motion approach, J. Chem. Phys.155, 064106 (2021)

2021
[51]

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
[52]

Zhang, R.-X

H.-D. Zhang, R.-X. Xu, X. Zheng, and Y. Yan, Nonpertur- bative spin–boson and spin–spin dynamics and nonlinear Fano interferences: A unified dissipaton theory based study, The Journal of Chemical Physics 142, 024112 (2015)

2015
[53]

Y. Yan, J. Jin, R.-X. Xu, and X. Zheng, Dissipation equation of motion approach to open quantum systems, Front. Phys. 11, 110306 (2016)

2016
[54]

R.-X. Xu, Y. Liu, H.-D. Zhang, and Y. Yan, Theory of quantum dissipation in a class of non-Gaussian environ- ments, Chin. J. Chem. Phys. 30, 395 (2017)

2017
[55]

R.-X. Xu, Y. Liu, H.-D. Zhang, and Y. Yan, Theories of quantum dissipation and nonlinear coupling bath descrip- tors, Chin. J. Chem. Phys. 148, 114103 (2018)

2018
[56]

Wang and Y

Y. Wang and Y. Yan, Quantum mechanics of open sys- tems: Dissipaton theories, The Journal of Chemical Physics 157, 170901 (2022). 7

2022
[57]

Su, Z.-H

Y. Su, Z.-H. Chen, Y. Wang, X. Zheng, R.-X. Xu, and Y. Yan, Extended dissipaton equation of motion for elec- tronic open quantum systems: Application to the Kondo impurity model, The Journal of Chemical Physics 159, 024113 (2023)

2023
[58]

X. Li, Y. Su, Z.-H. Chen, Y. Wang, R.-X. Xu, X. Zheng, and Y. Yan, Dissipatons as generalized Brownian particles for open quantum systems: Dissipaton-embedded quan- tum master equation, The Journal of Chemical Physics 158, 214110 (2023)

2023
[59]

Z.-H. Chen, Y. Wang, R.-X. Xu, and Y. Yan, Open quan- tum systems with nonlinear environmental backactions: Extended dissipaton theory vs core-system hierarchy con- struction, The Journal of Chemical Physics 158, 074102 (2023)

2023
[60]

Li, S.-X

X. Li, S.-X. Lyu, Y. Wang, R.-X. Xu, X. Zheng, and Y. Yan, Towards quantum simulation of non-Markovian open quantum dynamics: A universal and compact theory (2024), arXiv:2401.17255 [quant-ph]

2024 arXiv
[61]

B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997)

1997
[62]

Arrigoni, M

E. Arrigoni, M. Knap, and W. von der Linden, Nonequilib- rium dynamical mean-field theory: An auxiliary quantum master equation approach, Phys. Rev. Lett. 110, 086403 (2013)

2013
[63]

Dorda, M

A. Dorda, M. Nuss, W. von der Linden, and E. Arrigoni, Auxiliary master equation approach to nonequilibrium correlated impurities, Phys. Rev. B 89, 165105 (2014)

2014
[64]

Titvinidze, A

I. Titvinidze, A. Dorda, W. von der Linden, and E. Ar- rigoni, Transport through a correlated interface: Auxil- iary master equation approach, Phys. Rev. B 92, 245125 (2015)

2015
[65]

Schwarz, M

F. Schwarz, M. Goldstein, A. Dorda, E. Arrigoni, A. We- ichselbaum, and J. von Delft, Lindblad-driven discretized leads for nonequilibrium steady-state transport in quan- tum impurity models: Recovering the continuum limit, Phys. Rev. B 94, 155142 (2016)

2016
[66]

Dorda, M

A. Dorda, M. Sorantin, W. von der Linden, and E. Arrigoni, Optimized auxiliary representation of non- Markovian impurity problems by a Lindblad equation, New Journal of Physics 19, 063005 (2017)

2017
[67]

Mascherpa, A

F. Mascherpa, A. Smirne, S. F. Huelga, and M. B. Plenio, Open systems with error bounds: Spin-boson model with spectral density variations, Phys. Rev. Lett. 118, 100401 (2017)

2017
[68]

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 (2018)

2018
[69]

Lambert, S

N. Lambert, S. Ahmed, M. Cirio, and F. Nori, Modelling the ultra-strongly coupled spin-boson model with unphys- ical modes, Nature Communications 10, 3721 (2019)

2019
[70]

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
[71]

Mascherpa, A

F. Mascherpa, A. Smirne, A. D. Somoza, P. Fern´ andez- Acebal, S. Donadi, D. Tamascelli, S. F. Huelga, and M. B. Plenio, Optimized auxiliary oscillators for the simulation of general open quantum systems, Phys. Rev. A 101, 052108 (2020)

2020
[72]

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
[73]

Albarelli, B

F. Albarelli, B. Vacchini, and A. Smirne, Pseudomode treatment of strong-coupling quantum thermodynamics (2024), arXiv:2407.17886 [quant-ph]

2024 arXiv
[74]

Menczel, K

P. Menczel, K. Funo, M. Cirio, N. Lambert, and F. Nori, Non-Hermitian pseudomodes for strongly coupled open quantum systems: Unravelings, correlations and thermo- dynamics (2024), arXiv:2401.11830

2024 arXiv
[75]

L. K. Zhou, G. R. Jin, and W. Yang, Systematic and efficient pseudomode method to simulate open quantum systems under a bosonic environment, Phys. Rev. A 110, 022221 (2024)

2024
[76]

Medina, F

I. Medina, F. J. Garc ´ ıa-Vidal, A. I. Fern´ andez-Dom ´ ınguez, and J. Feist, Few-mode field quantization of arbitrary electromagnetic spectral densities, Phys. Rev. Lett. 126, 093601 (2021)

2021
[77]

Lednev, F

M. Lednev, F. J. Garc ´ ıa-Vidal, and J. Feist, Lindblad master equation capable of describing hybrid quantum systems in the ultrastrong coupling regime, Phys. Rev. Lett. 132, 106902 (2024)

2024
[78]

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. B 110, 195148 (2024)

2024
[79]

Lambert, T

N. Lambert, T. Raheja, S. Cross, P. Menczel, S. Ahmed, A. Pitchford, D. Burgarth, and F. Nori, Qutip-bofin: A bosonic and fermionic numerical hierarchical-equations- of-motion library with applications in light-harvesting, quantum control, and single-molecule electronics, Phys...

2023
[80]

Strathearn, B

A. Strathearn, B. W. Lovett, and P. Kirton, Efficient real- time path integrals for non-markovian spin-boson models, New Journal of Physics 19, 093009 (2017)

2017
[81]

Strathearn, P

A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-markovian quantum dynamics using time-evolving matrix product operators, Nature Commu- nications 9, 3322 (2018)

2018
[82]

M. R. Jørgensen and F. A. Pollock, Exploiting the causal tensor network structure of quantum processes to effi- ciently simulate non-markovian path integrals, Phys. Rev. Lett. 123, 240602 (2019)

2019
[83]

Fowler-Wright, B

P. Fowler-Wright, B. W. Lovett, and J. Keeling, Efficient many-body non-markovian dynamics of organic polari- tons, Phys. Rev. Lett. 129, 173001 (2022)

2022
[84]

G. E. Fux, E. P. Butler, P. R. Eastham, B. W. Lovett, and J. Keeling, Efficient exploration of hamiltonian parameter space for optimal control of non-markovian open quantum systems, Phys. Rev. Lett. 126, 200401 (2021)

2021
[85]

Richter and S

M. Richter and S. Hughes, Enhanced tempo algorithm for quantum path integrals with off-diagonal system-bath coupling: Applications to photonic quantum networks, Phys. Rev. Lett. 128, 167403 (2022)

2022
[86]

Gribben, D

D. Gribben, D. M. Rouse, J. Iles-Smith, A. Strathearn, H. Maguire, P. Kirton, A. Nazir, E. M. Gauger, and B. W. Lovett, Exact dynamics of nonadditive environments in non-markovian open quantum systems, PRX Quantum 3, 010321 (2022)

2022
[87]

Link, H.-H

V. Link, H.-H. Tu, and W. T. Strunz, Open quantum system dynamics from infinite tensor network contraction, Phys. Rev. Lett. 132, 200403 (2024)

2024
[88]

Cygorek, J

M. Cygorek, J. Keeling, B. W. Lovett, and E. M. Gauger, Sublinear scaling in non-markovian open quantum sys- tems simulations, Phys. Rev. X 14, 011010 (2024)

2024
[89]

Hughes, Coupled-cavity qed using planar photonic crystals, Phys

S. Hughes, Coupled-cavity qed using planar photonic crystals, Phys. Rev. Lett. 98, 083603 (2007). 8

2007
[90]

Yao and S

P. Yao and S. Hughes, Macroscopic entanglement and violation of bell’s inequalities between two spatially sepa- rated quantum dots in a planar photonic crystal system, Opt. Express 17, 11505 (2009)

2009
[92]

Aurell, R

E. Aurell, R. Kawai, and K. Goyal, An operator derivation of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an- harmonic baths, Journal of Physics A: Mathematical and Theoretical 53, 275303 (2020)

2020
[93]

Cirio, P.-C

M. Cirio, P.-C. Kuo, Y.-N. Chen, F. Nori, and N. Lambert, Canonical derivation of the fermionic influence superop- erator, Phys. Rev. B 105, 035121 (2022)

2022
[94]

Cirio, P

M. Cirio, P. Liang, and N. Lambert, Input-output hi- erarchical equations of motion (2024), arXiv:2408.12221 [quant-ph]

2024 arXiv
[95]

Cirio, S

M. Cirio, S. Luo, P. Liang, F. Nori, and N. Lambert, Modeling the unphysical pseudomode model with physical ensembles: Simulation, mitigation, and restructuring of non-markovian quantum noise, Phys. Rev. Res. 6, 033083 (2024)

2024
[96]

Lambert, M

N. Lambert, M. Cirio, J. dong Lin, P. Menczel, P. Liang, and F. Nori, Fixing detailed balance in ancilla-based dissipative state engineering (2023), arXiv:2310.12539 [quant-ph]

2023 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [91]

Y. Yu, A. M. Delgoffe, A. Miranda, A. Lyasota, B. Dwir, A. Rudra, and E. Kapon, Remote excitation between quantum emitters mediated by an optical fano resonance, Optica 8, 1605 (2021)

work page 2021

[2] [97]

The supplementary material contains necessary technical details to complement the derivations and statements in the main text

work page

[3] [1]

Breuer and F

H.-P. Breuer and F. Petruccione, The Theory Of Open Quantum Systems (Oxford University Press, Oxford, 2002)

2002

[4] [2]

Gardiner and P

C. Gardiner and P. Zoller, Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic meth- ods with applications to quantum optics(Springer Science, 2004)

2004

[5] [3]

S.-H. Wei, B. Jing, X.-Y. Zhang, J.-Y. Liao, C.-Z. Yuan, B.-Y. Fan, C. Lyu, D.-L. Zhou, Y. Wang, G.-W. Deng, H.- Z. Song, D. Oblak, G.-C. Guo, and Q. Zhou, Towards real- world quantum networks: A review, Laser & Photonics Reviews 16, 2100219 (2022)

2022

[6] [4]

A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakin- skiy, and A. N. Poddubny, Waveguide quantum electro- dynamics: Collective radiance and photon-photon corre- lations, Rev. Mod. Phys. 95, 015002 (2023)

2023

[7] [5]

Ac ´ ın, I

A. Ac ´ ın, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, The quantum technologies roadmap: a european commu- nity view, New Journal of Physics 20, 080201 (2018)

2018

[8] [6]

Frisk Kockum, A

A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics 1, 19 (2019)

2019