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A time-dependent regularization of the Redfield equation
by Antonio D'Abbruzzo, Vasco Cavina, Vittorio Giovannetti
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Submission summary
| Authors (as registered SciPost users): | Antonio D'Abbruzzo |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202211_00019v2 (pdf) |
| Date submitted: | 2023-05-16 16:01 |
| Submitted by: | D'Abbruzzo, Antonio |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We introduce a new regularization of the Redfield equation based on a replacement of the Kossakowski matrix with its closest positive semidefinite neighbor. Unlike most of the existing approaches, this procedure is capable of retaining the time dependence of the Kossakowski matrix, leading to a completely positive divisible quantum process. Using the dynamics of an exactly-solvable three-level open system as a reference, we show that our approach performs better during the transient evolution, if compared to other approaches like the partial secular master equation or the universal Lindblad equation. To make the comparison between different regularization schemes independent from the initial states, we introduce a new quantitative approach based on the Choi-Jamiołkowski isomorphism.
List of changes
- Comments about the weak-coupling regime were added below Eqs. (5) and (34).
- In Sec. 3.2, "2-norm" was replaced by "spectral norm", and definitions of the involved norms were added.
- A citation to Hartmann and Strunz, Phys. Rev. A 101, 012103 (2020) was added (now Ref. [23]).
- Colors and styles of the plots were changed to improve readability.
- In numerical calculations, the value of the coupling constant was lowered.
- A discussion about the role of the spectral width was added in Sec. 4.2, and Fig. 3 was updated appropriately.
- A link to the RHS non-Markovianity measure was added below Eq. (25), with appropriate references.
- Typos were corrected.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2023-6-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202211_00019v2, delivered 2023-06-12, doi: 10.21468/SciPost.Report.7336
Report
The authors have addressed my requests in a convincing way. They have not performed further numerical studies following Hartmann and Strunz as I was suggesting, but I understand there is already a lot of material in the current version of the paper. Moreover, I really like the new figure 3, which partially investigates the trade-off between Redfield and regularised master equation.
I would like to recommend the paper for publication after the following two issues are discussed:
i) I kind of disagree with the authors' assertion about the weak-coupling limit being based on $\tau_\epsilon\ll \tau_S$ only (memory time of the environment being much smaller than the system relaxation time). This is the necessary condition for performing the "first Markov" approximation. However, all the derivations of the master equations I am aware of make use of a perturbative treatment that requires that the system-environment interaction is a perturbation of the total Hamiltonian. That is, the system-bath coupling is much smaller than the system energy ($\gamma \ll \omega_{1,2}$). This is discussed, for instance, in the standard textbook by Breuer & Petruccione. A different derivation based on Nakajima-Zwanzig expansion can also be found in the textbook by Rivas and Helga on open quantum systems. Note that this condition does not immediately imply $\tau_\epsilon \ll \tau_S$, and vice versa. For instance, very high temperatures and cutoff frequencies in the Caldeira-Leggett model (see Section 3.6 of Breuer and Petruccione) can lead to very small $\tau_\epsilon$ even if the coupling constant is large. So, if the authors are not basing their weak-coupling analysis on some different arguments that I currently missing, I believe they should modify the justification of the weak coupling limit in the text. I guess they do not need to modify the values in the numerical experiments, because the weak coupling limit is already kind of satisfied. To be fair, 0.3 in figure 2 is not much smaller than 1, but it is still acceptable in the present context (figure 3 is totally fine).
ii) The authors have correctly replied that right now they cannot say anything about the steady state of the regularised equation and they have made some interesting connections with the mean-force Gibbs state. I would really appreciate if they added these comments to the manuscript, because I believe they can be of interest for a broader audience.
Author: Antonio D'Abbruzzo on 2023-07-13 [id 3807]
(in reply to Report 1 on 2023-06-12)We thank the referee for the additional comments. Below we describe how the two remarks affected the new version of the manuscript.
We modified our description of the weak-coupling limit by adopting the following milder formulation: the inverse coupling constant provides the longest timescale of the problem. We hope this is sufficient to rule out further concerns about the applicability region of the master equations discussed here. A citation to Mozgunov and Lidar, Quantum 4, 227 (2020) was added below Eq. (5), where the question is addressed more thoroughly.
We thank the referee for finding the connection with the mean force Gibbs state interesting. As requested, we added a couple of sentences in the Conclusions section to briefly describe what was said during our correspondence. A citation to Lee and Yao, Phys. Rev. E 106, 054145 (2022) was therefore added.