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Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

by Berislav Buca, Cameron Booker, Dieter Jaksch

Submission summary

As Contributors: Berislav Buca
Arxiv Link: (pdf)
Date submitted: 2021-05-03 19:48
Submitted by: Buca, Berislav
Submitted to: SciPost Physics
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Experiment
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


Synchronization is a phenomenon when interacting particles lock their motion into the same phase and frequency. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We give such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles). We show these eigenmodes must be quantum coherent and give an exact analytical solution for \emph{all} such dynamics in terms of a dynamical symmetry algebra. Using our theory we study both stable synchronization and metastable synchronization. Moreover, we give compact algebraic criteria that may be used to prove \emph{absence} of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2103.01808v2 on 3 May 2021

Reports on this Submission

Anonymous Report 1 on 2021-6-1 (Invited Report)


1 - Elegant and illuminating theory of quantum synchronisation using the Liouvillian formalism including perturbative results for metastable synchronisation, linking many results of the field together.
2 - Thorough introduction into the field of synchronisation, strong overview of both the field's history as well as recent results in the literature.
3 - Manuscript provides several examples for theoretical results provided, good experimental relevance.
4 - Extensive appendix provides detailed proofs of all theorems and corollaries including additional examples which provide further context to the main results.


(see Report for details)
1 - Problematic clarity in several places due to, e.g., use of highly specialised terminology.
2 - Introduction and discussion of Liouvillian formalism is flawed.
3 - Structure of manuscript not streamlined optimally in some places.


--- General Remarks ---

This manuscript provides a novel theory of quantum synchronisation using the Liouvillian formalism. By associating the emergence of synchronisation with imaginary eigenvalues of the Liouvillian and symmetries of the dynamical generator, the authors are able to prove general results which are applicable to a wide variety of physical systems. In particular the importance of the evolution's unitality is interesting and noteworthy. The authors both embed their findings in a detailed discussion of the field's history as well as illustrate their main results with several experimentally relevant examples.

Unfortunately, the text suffers from a very high level of specialised language. Furthermore several technical aspects are often not explained in a clear manner. While the results themselves are intriguing and certainly warrant publication, it would be highly desirable if the manuscript could be made more accessible and if the discussion of some key concepts could be corrected and/or clarified. In the following I will provide further details on a section-by-section basis.

--- Section 1 ---

- I very much appreciate the detailed overview of the field, starting from classical considerations and moving to the quantum domain. However, reading through pages 4-6 I became a little lost as to the relevance of many of the discussed topics to the subject of the paper - quantum synchronisation. For example the listing of classes of synchronicity are quickly supplanted with the definitions from Sec. 2. It might be helpful to keep the introduction slightly more streamlined to the paper's central theme and remind the reader of the implication for the quantum domain regularly and clearly.

- On page 8 the sentence "Indeed, even when the stationary state can be found analytically diagonalizing it, as is required to find its support, is still an open problem." needs to be entirely reworked.

--- Section 2 ---

- There is a brief mention on page 9 regarding "alternative cases" which refers to synchronised signals that differ by a phase or a scale factor. This discussion should be expanded to make clear in how far and in which places the results of this work need to be altered to account for them.

- The remark on page 10 regarding the fact that all definitions are understood as equalities up to exponentially small terms in time is very important and could be emphasised more (e.g. by giving it its own paragraph). I also feel like calling it "common-place in physics" is overstating how customary it actually is.

- The Lindblad formalism is not well introduced and explained on page 11. First of all it is not true that all smooth one-parameter families of completely-positive and trace-preserving (CPTP) maps have Lindblad form. This can be seen by considering a general, potentially non-Markovian evolution which originates from partial tracing a unitary evolution of a larger system (which is still a convolution of continuous operations). It is the semigroup property that is crucial which is unfortunately not mentioned at all when refering to the original work by Lindblad. As such the Lindblad formalism is not the most general form of a quantum evolution (even though it is correct that as a point-to-point map for a single point in time, CPTP maps are general). This misconception is present in several parts of the paper which is particularly puzzling since in Sec. 5.1 the authors very explicitly make use of a non-Markovian environment themselves. While I do agree that the Lindblad formalism is a sensible framework for this study, its generality should not be overstated and in particular the role of non-Markovian environments should be discussed since they are explicitly exploited in the manuscript after all. Finally it should be mentioned that the Lindblad formalism can also be employed for strong system-environment coupling and not only in the weak-coupling limit as the authors claim.

- The use of "trivial Jordan form" on page 12 seems overly convoluted and I only was able to understand the meaning by referencing the appendix. In fact, "(block-)diagonalisable" would be a nomenclature that is more easily accessible to a wide audience I think.

--- Section 3 ---

- I was not quite able to understand in how far Theorem 1 neccessarily implicates a non-equilibrium steady-state. If I consider a Liouvillian that is trivial, e.g. the zero operator, then clearly all states are eigenstates with eigenvalue zero which would fall under the conditions for the theorem. However no state is a non-equilibrium steady state in this case since expectation values of all observables are constant in time because no actual evolution takes place. I seem to be missing an argument as to the sufficiency of the theorem in this and related special cases.

- On page 14 the sentence "In order to have \omega \neq 0 we can see by taking the trace of equation (15) that A must not be unitary but additionally must be traceless.". To my understanding of the previous discussion A still can be unitary but does not have to be. As such I suspect in this sentence it should be "A need not be unitary" instead of "A must not be unitary".

- The discussion at the end of Sec. 3.1 as to the commensurability was not quite clear to me. The authors claim there is a danger in having "too many incommensurable purely imaginary eigenvalues" and state that "it should be understood that considerations regarding commensurability must be made in addition to the results provided.". I do not quite understand the scope to which these statements should be interpreted since the phrasing used here is very vague.

--- Section 4 ---

- While I do like the general discussion in Secs. 4.1 - 4.3 it took me several re-reads to properly place them into the big picture of the section. It might be better to start with the general statements in 4.3 and Theorem 3 (to my understanding this theorem should capture all other cases since \omega = 0 is not excluded from its scope) and move from there to the more specialised cases of 4.1 and 4.2 and see how they emerge from Theorem 3.

- On page 17 and 18 I have trouble to understand the use of the nomenclature "explicity breaking the exchange symmetry". If I understand it well this corresponds to a vanishing anticommutator of the Liouvillian with the operator of exchange symmetry. While this does indeed break the exchange symmetry, so would any scenario in which the commutator does not vanish and one might even argue that a vanishing anticommutator is a kind of very specific breaking of the symmetry that preserves some structure.

--- Section 5 ---

- As mentioned above, the fact that a non-Markovian bath is employed for anti-synchronisation seems very important to me and should not be properly put into the bigger scope of the whole paper.

- The connection between decoherence-free subspaces and synchronisation mentioned at the bottom of page 20 is quite intriguing and might warrant further discussion and/or future study.

- On page 21 the vanishing anticommutator of A with P_{2,3} is called "antisymmetry" which seems to me more appropriate than the "explicit symmetry breaking" nomenclature used in Sec. 4.

- The discussion regarding a general framework of quantum synchronization of the model in Sec. 5.2. was very hard to follow. It also uses several concepts that might not be too familiar to most readers with a background in physics (e.g. the Cartan subalgebra).

--- Section 6 ---

- This section feels somewhat out of place. It might be a good idea to move Theorem 4 and the corresponding discussion to the end of Section 3 where all general results and theorems are collected and discussed.

--- Final Remarks ---

The science presented in the manuscript is intriguing and original. However, there are some flaws that still need to be amended before publication, most notably the clarity of the presentation. Once the authors address these weaknesses in a revision, I would recommend this paper for publication in SciPost Physics.

Requested changes

(see Report for details)
1 - Improve clarity and accessibility of manuscript.
2 - Amend discussion of Liouvillian formalism by, e.g., mentioning the role of non-Markovianity in quantum evolutions, in particular with respect to its direct application in Sec. 5.1.

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

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