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Dissipative flow equations

by Lorenzo Rosso, Fernando Iemini, Marco Schirò, Leonardo Mazza

Submission summary

As Contributors: Leonardo Mazza
Arxiv Link: (pdf)
Date submitted: 2020-10-08 09:11
Submitted by: Mazza, Leonardo
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We generalize the theory of flow equations to open quantum systems focusing on Lindblad master equations. We introduce and discuss three different generators of the flow that transform a linear non-Hermitian operator into a diagonal one. We first test our dissipative flow equations on a generic matrix and on a physical problem with a driven-dissipative single fermionic mode. We then move to problems with many fermionic modes and discuss the interplay between coherent (disordered) dynamics and localized losses. Our method can also be applied to non-Hermitian Hamiltonians.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

Dear Editor,

We would like to thank the referee for his/her careful reading of our manuscript and for his/her useful report on it. We append below our answers to the points raised by the referee and the relevant changes we made in the new version of the manuscript.

                                    The authors (L. Rosso, F. Iemini, M. Schirò and L. Mazza)


  1. In the presentation of the different generators η , the authors rely a few times on a perturbative treatment involving some parameter ξ. In order to make a clearer connection to concrete physical problems, the authors may want to give a concrete physical interpretation of ξ in the case of one or two simple examples.

Answer: In general, any situation where a term of the master equation is multiplied by a small prefactor is amenable to this description. For instance, the authors Li, Petruccione and Koch of Scientific Reports 4, 4887 (2014) discuss within this framework a ring of spins with strong dissipation and a perturbatively small spin-spin coherent coupling. The same authors discuss in Phys. Rev. X 6, 021037 (2016) an open Jaynes-Cumming lattice where the spin-photon coupling is small and treated perturbatively. Appropriate referencing to this material and information has been included in the text.

  1. In the first example of section 4, I find that the explanation of the procedure to obtain the results plotted in Fig. 1-4 could be clearer.
  2. What could correspond to the non-Hermitian matrix A (a Liouvillian or a non-Hermitian Hamiltonian ?) ?
  3. What 'flow equations' presented above are concretely solved ? Some references to the equations could be useful.
  4. the 'flow step' in Section 4 is denoted by d and by dℓ in Section 6.2. The expression dℓ is probably better.

Answer: - The use of flow equations is not limited to physics and it has a more general interest, since it just proposes a method to put in diagonal form a matrix (flow equations diagonalizing Hermitian operators are known since the XIX century). Our work has obvious applications in physics, but it could just be seen as a mathematical tool for diagonalizing non-Hermitian matrices; this is what we do in the first example. One could of course interpret the matrix A as a non-Hermitian Hamiltonian, but we discourage this because it’s just a randomly generated matrix and it probably does not satisfy physical constraints.

  • The different figures 1, 2 and 3 show the solution of the flow equations using the three different generators. Explicit reference to the generators is made in the captions to Eq. 17, 22 and 29. The new version of the text details more explicitly what we did.

  • We have modified the draft accordingly.

  • The authors focus mainly on the diagonalization of the Liouvillian in the four different examples. I would suggest the authors provide (e.g. in the second example) the time-evolution and/or steady state of the density matrix that correspond to the parametrization (42) of the Liouvillian, in order to provide a slightly more physical picture.

Answer: We have added a discussion of how to derive steady-state properties for the dissipation properties in the second example.

  1. While Lindblad master equations to describe open quantum system dynamics are common and relevant in many situations, it is sometimes necessary to go beyond the standard Born-Markov approximations to describe non-Markovian effects (e.g. in the solid-state). Would it be possible to generalize the flow equations to these situations (e.g. non-Markovian master equations (time non-local master equations) or more simply Redfield master equations ( time-local but non-positive master equations (see e.g. [1]) ? The authors may want to provide comments on that, as it could provide potentially more interest in the development of the method.

Answer: We thank the referee for this interesting suggestion. At the moment we do not think that retardation effects can be efficiently taken into account because the flow equations are a technique to obtain the eigenvalues and eigenvectors of the generator of the dynamics. This information is useful when the generator of the dynamics is local in time and time-independent. On the other hand, the Redfield equation pointed out by the reviewer seems to be within the possibilities of our method because the generator of the dynamics is a linear operator that is local in time and time independent. We have briefly mentioned this possibility in the conclusions. In general, however, one can always reintroduce the explicit description of the modes of the bath and treat the problem with the standard flow equations for the Hamiltonian, as it was already done in the ‘90s (see the cited literature in the article).

List of changes

List of changes:

• Subsection 3.3: Sentence added: “This situation is not uncommon … and treated perturbatively”.

• Section 4: Paragraph added: “We study the dissipative … is also presented”.

• Subsection 5.2.1 “Steady state properties” added.

• Conclusions: Sentence added: “Although … Redfield master equation”.

Submission & Refereeing History

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Resubmission 2007.12044v2 on 8 October 2020

Reports on this Submission

Report 1 by François Damanet on 2020-10-9 Invited Report


I thank the authors for their detailed response and the changes made. I recognise that the authors have made some effort to improve the weaknesses pointed out in the previous report.
I thus recommend the manuscript for publication in SciPost Physics.

Requested changes


  • validity: good
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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