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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/2204.06585v3 (pdf) 
Date submitted:  20220804 16:40 
Submitted by:  Tindall, Joseph 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, 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 argue, by presenting several simple mathematical perspectives on the problem, that it is a general consequence of the presence of a strong symmetry in an open system with only a few exceptions. Using a number of example systems we illustrate these arguments, uncovering an explicit 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.
Author comments upon resubmission
We have fixed the two minor errors pointed out by the referee.
On behalf of the authors,
Joseph Tindall
List of changes
1. Eq.(10) has now had the $m \in ...$ part removed.
2. A prime on the second $\beta$ indice has been added to $\rho_{R} ...$ which appears in Section 2.4.
Current status:
Reports on this Submission
Anonymous Report 2 on 202294 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.06585v3, delivered 20220904, doi: 10.21468/SciPost.Report.5636
Strengths
1  Wellstructured and accessible manuscript, suitable for a wide readership.
2 Timely topic
3 Nice set of examples to help understanding and giving concretness for the general results.
Weaknesses
1 Lack of a significant breakthrough on a previouslyidentified and longstanding research discovery (i.e., dissipative freezing)
Report
I read carefully all the versions of the manuscripts and the previous referee's comments. I find the topic interesting and the results worthy of publication and I acknowledge the improvements made by the authors, but I do stick to the opinion of Referee 2 on 2022519 that the manuscript will be more suited for SciPost Physics Core.
Indeed, I do not really see how the present manuscript meets at least one of the "Expectations" criteria of SciPost Physics:
1. There is no "groundbreaking theoretical/experimental/computational discovery", since dissipative freezing was already discovered in a previous work.
2. I do not find that the manuscript "presents a breakthrough on a previouslyidentified and longstanding research stumbling block" (i.e. dissipative freezing). Indeed, even if Ref [24] was only presenting dissipative freezing in a particular case, it was already clear that the presence of a strong symmetry was a key ingredient. In the present manuscript, the authors provide significant mathematical and physical insight into why freezing will emerge, but I would not qualify it as a breakthrough. It is rather an elaboration and a generalisation of a previously found result.
3. I do not find either that this manuscript "opens a new pathway in an existing or a new research direction, with clear potential for multipronged followup work". The manuscript provides a way to identify better whether or not a system undergoes dissipative freezing, but I would not qualify it as a new research direction: I do not find the manuscript impactful and helpful enough for the scientific community for guiding researchers in new research directions. How could one use the results of this paper for this purpose ?
4. I do not find either that it "provides a novel and synergetic link between different research areas". They identify a direct connection between rigorous mathematical results on ergodicity in open systems and dissipative freezing, but again there was already the idea in [24] thet dissipative freezing was an nonergodic phenomenon. Also, the synergy between different fields is unclear.
Hence, while the present manuscript constitutes a significant elaboration on the mechanisms underlying dissipative freezing, I do not think it satisfies the publication criteria of SciPost Physics and how it would open significant research directions for the readership. However, I find that it clearly meets more adequately all the acceptance criteria of SciPost Physics Core:
1. "Address an important (set of) problem(s) in the field using appropriate methods with an abovethenorm degree of originality;"
2. "Detail one or more new research results significantly advancing current knowledge and understanding of the field."
and I would thus recommend it for publications in SciPost Physics Core, except if the authors can revise their manuscript so to provide materials that convince more strongly on how their manuscript meets the acceptance criteria of SciPost Physics.
Requested changes
 I suggest (as Referee 1 on 202289) a typo check.
In the caption of Figure 2, it is said \gamma = 4 \omega. To make clearer the connection with Eq. (1), the number M of jump operators and/or the index j of \gamma should be specified.
 In the example 3.3, the authors work with qudits but below Eq. (29) mentions the "spin3/2" operator. The dimension of the qudits used in the example should be mentioned more clearly.
Anonymous Report 1 on 202289 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.06585v3, delivered 20220809, doi: 10.21468/SciPost.Report.5515
Strengths
1. Well structured and accessible as only minimal knowledge of linear algebra is required.
2. Convincing examples demonstrate dissipative freezing and exceptions to it.
Weaknesses
1. Lacks precision and rigor in its arguments and conclusions
Report
Dissipative freezing corresponds to a recently observed property of the timeevolution of an open system with a strong symmetry to asymptotically project each quantum trajectory into a single subspace of the strong symmetry, thereby spontaneously breaking the latter at the level of individual trajectories.
The manuscript attempts to provide a broader view on this phenomenon that should be accessible to most physicists while also attempting to elevate the discussion previously constrained to specific examples to a more formal level. Unfortunately, the manuscript falls short of its ambitions, ending up as neither fish nor fowl.
The point where this becomes most painfully clear is the box on page 5, which is unsuitable as a mathematical definition but also provides little physical insight as no concise conditions for exceptions are provided.
The level of the analysis of dissipative freezing presented in the manuscript, nevertheless, significantly exceeds the existing literature. I am therefore generally inclined to recommend publication if the following minor points can be clarified/corrected or improved:
Requested changes
1. Content:
In the introduction, I would appreciate a paragraph on the physical relevance of dissipative freezing. It is discussed later, but in my opinion should also be emphasized in the introduction.
Below equation (14) there appears to be a typo as neither $\alpha$ nor $\alpha’$ are defined in this context. I therefore suggest to replace “is largest in $\alpha$ or $\alpha'$” by “is larger in $\alpha_1$ or $\alpha_2$”.
At the beginning of page 10, it doesn’t seem necessary, nor guaranteed by the definition of A that the steady state of each symmetry sector is unique. To be explicit, A might not split the Hilbert space in the maximal number of subspaces.
At the beginning of page 11 I fail to follow the argument
“if $\tilde{p}^{(i)}(\alpha_{1}, t)\tilde{p}^{(i)}(\alpha_{2}, t) \ll 1$ we have that one of $p^{(i)}(\alpha_{1},t)$ or $p^{(i)}(\alpha_{2}, t)$ will be on the order of $\epsilon^{2}$”
In this context it might also help to introduce the symbol $t_f$ for the freezetime.
In section 2.5 $\Delta^{(\alpha_{1}, \alpha_{2})} = 0 \ \forall \alpha_{1} \neq \alpha_{2}$ should read $\Delta^{(\alpha_{1}, \alpha_{2})} \neq 0 \ \forall \alpha_{1} \neq \alpha_{2}$. However, as is explained in the following section on intersector traceless modes, even this statement is not generally valid.
Conversely in equation (26) one should have $\Delta^{(\alpha_{1}, \alpha_{2})} = 0$. On a more general note, this condition doesn’t seem very useful as, on the one hand, it is not being linked to any properties of the Hamiltonian or Lindblad operators. On the other hand, it is also computationally expensive to check. If possible I would appreciate if the authors code provide a statement to which extend equation (26) may be of use.
The argument on page 17 “If the trajectory is left to evolve over a timescale sufficiently longer than that associated with the inverse of $\Delta^{(\alpha_{1}, \alpha_{2})}$, then any signatures of freezing should be apparent. If they are not then it is reasonable to assume freezing will not occur and traceless nondecaying modes are present.” is circular as an evolution to $t\sim 1/\Delta^{(\alpha_{1}, \alpha_{2})}$ excludes the case of traceless nondecaying modes characterized by $\Delta^{(\alpha_{1}, \alpha_{2})}$
2. Grammar/Spelling:
While the general level of grammar is very good, in its current form the manuscript is riddled with punctuation errors. Due to their huge number, I will not provide the detailed location, but rather the immediate context:
 far reaching implications which→ farreaching implications, which
 model at hand an analytical → model at hand, an analytical
 numerical examples, from a range → numerical examples from a rang
 absense → absence
 dissipative freezing we provide a novel → dissipative freezing, we provide a novel
 system symmetries can no longer → system, symmetries can no longer
 also true, if there → also true if there
 Evan’s → Evans or Evans’
 Unravellings which are to higher order → Unravellings which are of a higher order
 symmetry subspace then the phenomenon → symmetry subspace, then the phenomenon
 offdiagonal block we will refer them as → offdiagonal block, we will refer to them as
 states in different subspaces then dissipative → states in different subspaces, then dissipative
 constraint independently → constraint, independently
 of the GSKL equation we can → of the GSKL equation, we can
 this make sense → this makes sense
 jump operators it follows from → jump operators, it follows from
 quantities values → quantities’ values
 the, randomly selected → the randomly selected
 freezing we now → freezing, we now
 our analysis we work → our analysis, we work
 blockdiagonal then the Liouvillian → blockdiagonal, the Liouvillian
 If this is true then → If this is true, then
 simlarity → similarity
 any, other forms of → any other forms of
 respresent → represent
 is protected from the system’s → are protected from the system’s
 examples we → examples, we
 routine we → routine, we
 As our first example we → As our first example, we
 Lindblad operators which are block → Lindblad operators, which are block
 and in this example these → and, in this example, these
 a certain time the total → a certain time, the total
 the longtime limit the probability → the longtime limit, the probability
 identical, aside → identical, aside
 Importantly this→ Importantly, this
 intuition we have built up → intuition we have built
 all commute there → all commute, therefor
 follows the number → follows that the number
 Notably as $\gamma$ → Notable, as $\gamma$
 is traceless modes → are the traceless modes
 for sufficiently long times we can → for sufficiently long times, we can
 diverges then any given → diverges, then any given
 are not then → are not, then
 much more computationally → a much more computationally
 vanishes then the Liouvillian → vanishes, then the Liouvillian
 posseses → possesses
 exceptions would → exceptions, would
 possesion → possession
 system the bound → system, the bound
 and then general dependence of the the freezing→ and then the general dependence of the freezing
 where the $c^{(i)}_{\alpha, \beta}(t)$ are the coefficients of the $i$th trajectory → where $c^{(i)}_{\alpha, \beta}(t)$ is the coefficient of the $i$th trajectory
 to ensemble average → to ensembleaverage
 open systems) the more constraints → open systems), the more constraints
 qubit → qudit
Author: Joseph Tindall on 20220923 [id 2845]
(in reply to Report 1 on 20220809)We are thankful for the thorough reading and comments on the manuscript. We have made a number of changes based on their recommendations.
Author: Joseph Tindall on 20220923 [id 2844]
(in reply to Report 2 on 20220904)We are thankful for the thorough reading and comments on the manuscript. We have made the minor changes requested by the referee.