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On the generality of symmetry breaking and dissipative freezing in quantum trajectories
by Joseph Tindall, Dieter Jaksch, Carlos Sánchez Muñoz
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Submission summary
Authors (as registered SciPost users):  Dieter Jaksch · Joseph Tindall 
Submission information  

Preprint Link:  https://arxiv.org/abs/2204.06585v1 (pdf) 
Date submitted:  20220415 10:26 
Submitted by:  Tindall, Joseph 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
Recently, several studies involving open quantum systems which possess a strong symmetry have observed that every individual trajectory in the Monte Carlo unravelling of the master equation will dynamically select a specific symmetry sector to `freeze' into in the longtime limit. This phenomenon has been termed `dissipative freezing', and in this paper we demonstrate how it is a general consequence of the presence of a strong symmetry in an open system and independent of any microscopic details. Specifically, through several simple, plausible, mathematical arguments, we detail how this phenomenon will happen in most setups with only a few exceptions. Using a number of example systems we illustrate these arguments, highlighting the relationship between the spectral properties of the Liouvillian in offdiagonal symmetry sectors and the time it takes for freezing to occur. In the limiting case that eigenmodes with purely imaginary eigenvalues are manifest in these sectors, freezing fails to occur. Such modes indicate the preservation of information and coherences between symmetry sectors of the system and are associated with phenomena such as nonstationarity and synchronisation. The absence of freezing at the level of a single quantum trajectory provides a simple, computationally efficient way of identifying these traceless modes.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022519 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.06585v1, delivered 20220519, doi: 10.21468/SciPost.Report.5102
Strengths
 Quite accessible introduction to the topic
Weaknesses
1 The paper lacks precision at several instances.
2 Although the statements about dissipative freezing are formulated in a mathematical language, the arguments are rather handwaving and no quantitative aspects of dissipative freezing, i.e. no boundaries, time scales etc. are discussed.
3 To me the paper does not provide much original content compared to previous publications on dissipative freezing. To be candid, what the authors present in Sec. 1 and 2 was the picture that was more or less communicated in previous publications [2426].
Report
The manuscript is quite accessible and explains the phenomenon of dissipative freezing in simple terms. This phenomenon, which describes the timedependent collapse of a wave function into one of the symmetry sectors of the Liouvillian generator of the dynamics was observed recently. Very colloquially, the reason for this phenomenon to occur is the oneway type stochastic evolution of each unravelling, which will "loose" contributions from different symmetry sectors over time but, due to the symmetry, is unable to restore them.
In summary, I think the paper is correct and well written and therefore deserves publication. However, in my personal opinion it lacks the originality and the quantitative precision that would be necessary to publish in SciPost Physics. I therefore think that, after consideration of the comments below, I may be published in SciPost Physics Core instead.
Details to weakness 2:
The authors' general argument for the occurrence of dissipative freezing is based on Eq. (10). However, this implies that all derivatives of this equation go to zero in finite time. I don't see a clear reason why this would be the case. All statements based on the stationary density matrix are asymptotic (i.e. t> infty and N > infty). The order of the two limits is important as well, since if we want to describe the density matrix entries at all times, one has to take N> infty first and then t> infty. Then number of the degrees of freedom, which need to fulfil the constraints grows as fast as the number of constraints. Therefore I am not convinced by the authors' argument.
Even if I was convinced, the lack of quantitative statements about the process persists.
Lack of precision:
 page 4: "D trace unity hermitian steady states", I don't understand what is meant by this expression. There is an infinite number of possible stationary, each one depending on the structure of the initial state. The basis of stationary states is at least Ddimensional.
 page 4: "We emphasize that this unravelling of the GSKL equation is not just a mathematical formalism:..." The authors should be more precise at this point. First of all, one needs to distinguish unravellings with hermitian jump operators from unravellings with nonhermitian jump operators. The latter is per se not a continuous measurement process. Furthermore, even for hermitian jump operators one can easily choose an unravelling that has no measurement counterpart.
 page 5: Definition of dissipative freezing: The authors say the process happens generically but also mention that there are few exceptions. Neither the term generic nor the number of "few" is explained here. Is it clear that the two exceptions which are discussed later in the text are the only exceptions? This doesn't read very convincing.
 Eq. (15) can't be correct. If P_1 and P_2 project onto orthogonal subspaces then P_1HP_1=P_2HP_2 implies that the Hamiltonian must be zero in both subspaces. If P_1psi>=psi> then P_2psi>=0. I guess what the authors mean is that in both subspaces the Hamiltonian has the same matrix structure (modulo a unitary transformation since I can always choose a different subsystem basis).
Report 1 by Marco Cattaneo on 2022516 (Invited Report)
 Cite as: Marco Cattaneo, Report on arXiv:2204.06585v1, delivered 20220516, doi: 10.21468/SciPost.Report.5084
Strengths
1 It introduces a satisfactory explanation of why dissipative freezing occurs.
2 It presents some nice examples that corroborate this explanation.
Weaknesses
1 The clarity of some concepts and sentences is occasionally not so high.
Report
The authors present a number of arguments for the emergence of dissipative freezing in quantum trajectories. Dissipative freezing is the fact that, at the level of a single quantum trajectory, the evolved state is quite often found at infinite time in a single subspace only of a strong symmetry. They also show how and why this phenomenon is broken.
Overall, I think the paper is very good and the explanation for the emergence of dissipative freezing is convincing and interesting, especially for the argument on longproducts of matrices. The examples are also really nice and pedagogical.
I must say, however, that sometimes I struggled to follow the text and I believe some clarifications are necessary. For instance, I feel the presentation is too discursive in some sections, while some more formulas wouldn't hurt.
I would like the authors to address the following comments:
1 I do not exactly understand where the time derivatives in Eq.(10) (for m=1,2,...) come from. Anyway, isn't the equation for m=0 sufficient to make the authors' point? If nothing weird happens and the time derivative commutes with the limit, we are just saying that the time derivatives at infinite time of some functions that are asymptotically zero (for t > infty) are zero. Isn't this kind of trivial? So, aren't the time derivatives redundant here?
2 Right after eq. (10), the authors say "at least one of coefficients in the product... will always be zero". For alpha \neq alpha', aren't all the coefficients zero?
3 I have a curiosity about the definition of "traceless nondecaying eigenmode" of the Liouvillian. Apparently, according to the authors this means that there is some nondecaying coherence between different symmetry eigenspaces. Isn't it possible to have a traceless nondecaying eigenmode within a single symmetry subspace (oscillating coherences with the same symmetry value)?
4 The meaning of the last sentences in Section 2.3 "Recovering ... Steady State" is a bit obscure. Given an open quantum system, why shall we be allowed to add symmetry subspaces? Please clarify this or provide some examples.
5 Please provide a more clear definition of "growth rate" in the section about Longproducts of matrices. A formula would help the reader to better grasp this concept.
6 How is the ergodic time average defined in the references [41,42]? Is there some kind of time window in the average, or is it 1/t int_0^t for t that goes to infinity? Please specify this.
7 I miss something in the discussion of the result in Ref. [41]. How is it connected to Eq. (10)? Is it because ergodicity for a single trajectory would exclude the possibility for this trajectory to reach a state in a single symmetry subspace only? What if the trajectory passes ergodically through all the different states in different symmetry sectors, and the time average yields the ensemble average? Maybe this is not really clear, but please, clarify this point through a more extensive discussion.
8 It is really difficult to follow the last sentences of the subsection on ergodicity. Please, try to express with some formulas the final sentence starting with "If the probability of freezing into...". I believe it's much easier for the reader to understand your message if there are some clear and readable equations to look at.
9 In section 2.4 two different kinds of exception are discussed. Am I wrong or the first kind is related to the absence of dissipative freezing despite all the steady states can be written as in Eq. (8) (at least in almost all scenarios), while the second one is due to the failure of Eq.(8)? I think highlighting this difference could be good.
10 Please introduce the "freezetime" in Section 3.1 through a proper equation.
11 Same as for the gap Delta in the following example. Write down in a visible equation what you are talking about.
Finally a few typos: pag. 2 last paragraph, "allows us (to) identify"; the caption of Fig. 2, "unormalised"; the name of the first author of Ref. [42] is misspelled.
Requested changes
Please consider the above points.
Author: Joseph Tindall on 20220711 [id 2649]
(in reply to Report 1 by Marco Cattaneo on 20220516)
1) The timederivatives were in place to emphasize that the expression is `steady’ i.e. that the ensemble average of coefficients is 0 and unchanging in the longtime limit. The referee is correct, however, that this is superfluous and $m=0$ is sufficient because if the limit $t \rightarrow \infty$ converges asymptotically for $m=0$ then its derivative must vanish.
2) For a given trajectory (i.e. specific $i$) if freezing has occurred then there is only a single $\alpha$ for which the coefficients $c^{(i)}_{\alpha, \beta}(t)$ are not zero. Hence, for a given $i$, the product of coefficients for two different values of $\alpha$ is zero. We have now made this clear.
3) Yes, we did not intend our definition to imply you cannot have traceless nondecaying modes within a single symmetry subspace. We have renamed these as `intersector traceless modes’ in order to avoid suggesting these are the only possible traceless, nondecaying modes that can appear in a Liouvillian’s spectrum.
4) One is always able to introduce a new symmetry subspace to the system by expanding their Hamiltonian and jump operators via a direct sum. Physically this would correspond to considering an additional, independent open system on top of the existing one.
5) We now provide a mathematical definition of this average growth rate.
6) This time average is defined via an average between times zero and infinity. We have specified this and improved the readability here as the sentence was confusing in its original form.
7) The connection we were implying is that establishing an ergodic theorem with a single trajectory across multiple symmetry sectors is not possible due to the problem being over constrained in analogy with Eq. (10). In the interests of avoiding discursive discussion we have removed the reference to Eq. (10) here.
8) We have introduced explicit formulas for the quantities mentioned at the end of Section 2.3. With the addition of more explicit formulas for the freezetime and intersector spectral gap (see point 10), this should be easier to follow.
9) Yes, the first is a breakdown of freezing despite Eq. (8) holding whilst the second is a scenario in which Eq. (8) is no longer true. Section 2.5 has been reworded to make this clearer.
10 + 11) We have introduced an equation for the freeze time and intersector spectral gap. Our expression for the freezetime now follows from a new section where we derive a lowerbound on the timescale of freezing.
Author: Joseph Tindall on 20220711 [id 2648]
(in reply to Report 2 on 20220519)Response to main criticism:
Originality: We disagree that References [2426] communicated the picture we provide in Section 1 and 2 of our work. All these references considered a single microscopic model and numerically observed dissipative freezing (DF) (with Ref. 24 the only one to provide a mathematical proof and this was specific to the exceptional situation H = L). We instead make no assumptions about the details of the Liouvillians—other than its possession of a strong symmetry—and provide significant mathematical and physical insight into why freezing will emerge. We also identify a direct connection between rigorous mathematical results on ergodicity [4042] in open systems and dissipative freezing – something which Refs. [2426] all missed. We did not intend our work to constitute a rigorous mathematical proof of DF. The aim of this paper is, instead, to provide an insightful, comprehensive, mathematical and physical understanding of its origin, complemented with a range of concrete numerical examples. Alongside the intuition and comprehension it provides, such a perspective has allowed us to develop a number of original results without needing to rely on heavy mathematical artillery: exceptions to DF, a lower bound on the timescale on which DF occurs and a computationally efficient method for identifying traceless nondecaying modes in a Liouvillian. In our revised version of the manuscript we have made these points much clearer.
Precision: Both referees comments have significantly helped us improve the precision of the manuscript and the changes made are listed at the end of the document. These changes include mathematical definitions for certain quantities, rectifying grammatical and mathematical errors, avoiding discursive discussions, providing several elaborations when requested and deriving a rigorous lower bound on the timescale of dissipative freezing – which we demonstrate in our numerics. We believe that these changes have significantly improved the clarity and precision of the manuscript (see list of changes to manuscript).
Responses (to referee comments under `Lack of Precision’):
1) The basis of steady states is at least D dimensional, this is now stated explicitly.
2) We have provided additional discussion and references about the relationship between a stochastic unravelling of the GSKL equation and a continuous measurement process in an open system. We note that nonHermitian jump operators can result in a physical unravelling – for instance via photon counting in the environment.
3) The ‘Traceless’ mode exception completely covers the situation in which the intersector spectral gap vanishes, leading to Eq. (10) not being valid and freezing not occurring. The `Similar Symmetry Subspaces’ exception we detail is the only instance we are aware of where Eq. (10) can hold and the system does not undergo dissipative freezing. It is possible other examples where the structure of the two subspaces is sufficiently similar to prevent freezing exist. We have made this explicit, along with the fact we are, however, unaware of any such examples.
4) We thank the referee for spotting this error—we meant to say the projected Hamiltonian has the same form when focussing on the diagonal block where it is nonzero and have now made this mathematically precise, including the part about the unitary transform (change of basis).