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Thermodynamics of quantumjump trajectories of open quantum systems subject to stochastic resetting
by Gabriele Perfetto, Federico Carollo, Igor Lesanovsky
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Authors (as registered SciPost users):  Gabriele Perfetto 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.05078v2 (pdf) 
Date submitted:  20211223 11:03 
Submitted by:  Perfetto, Gabriele 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is nonMarkovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantumjumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantumjump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven twolevel and an intermittent threelevel system. Our findings show that stochastic resetting may be exploited as a tool to tailor the statistics of the quantumjump trajectories and the dynamical phases of open quantum systems.
Current status:
Reports on this Submission
Anonymous Report 3 on 2022218 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.05078v2, delivered 20220218, doi: 10.21468/SciPost.Report.4461
Strengths
1 Very clear, especially for experts
2Potentially very useful approach for simulating nonMarkovian dynamics
Weaknesses
1Most of the results are formal
2 Examples seem a bit too limited
3 Motivation for stochastic reset not clear
Report
The authors show how the stochastic resetting dynamics of [39,40] can be used to simulate nonMarkovian quantum dynamics. They likewise derive formal results for the quantum jumps statistics. They apply their method to two examples, a two and threelevel atom system.
I think that the paper is interesting and represents an important contribution to the theory of open quantum systems. I especially like the part of simulating nonMarkovian dynamics.
Requested changes
1 The authors should explain better why their results on the quantum jump statistics are not merely formal and can be used to solve difficult problems.
2 In the introduction the authors introduce stochastic resets with a references. It would be nice for the reader to have more on the motivation for this process: for example, why stochastic, instead of deterministic reset? Why reset at all? Is this something that gives interesting physics and can be simulated or does it model something realistic?
3 Can the authors give a bit more elaborate example than a 3level system? Or explain why the problem cannot be treated with more elementary approaches
Report 2 by Krishnendu Sengupta on 202227 (Invited Report)
 Cite as: Krishnendu Sengupta, Report on arXiv:2112.05078v2, delivered 20220207, doi: 10.21468/SciPost.Report.4333
Report
The paper discusses a theoretical Lindblad operator based approach to the problem of stochastic resetting. The paper provides the detailed derivation of a Lindblad based formalism for generic reset problem. Their analysis constitutes a generalization of the Poissonian resetting problem which is reproduced as a special case. However, the illustrative examples provided in the paper are for somewhat simple systems ( two and three level optics problems). A much more nontrivial example which, in my opinion, could be worked out within their approach would be the reset dynamics of 1D Ising model in a transverse field. Furthermore, the analysis section of the paper is a bit dense; it would help an average reader if it can be rewritten in a slightly more pedagogical manner. For example, the authors do not quite explain how they get Eqs 5 and 6 from Eq 4. I did look up the refs they provide, and it may take quite a bit of time for the reader to figure this out. So I would suggest a bit of explanatory sentences/formulae connecting these equations even if they have been done before (may be put in the appendix, if the authors feel that it would disturb the flow).
Requested changes
1. Include a slightly more complicated example to demonstrate the power of the method.
2. Make the presentation slightly more accessible.
Author: Gabriele Perfetto on 20220530 [id 2542]
(in reply to Report 2 by Krishnendu Sengupta on 20220207)
We thank the Referee for the careful reading of our manuscript. In the following, we address their report. We have highlighted in blue the most important changes we did in the text while revising it.
The referee writes:
However, the illustrative examples provided in the paper are for somewhat simple systems ( two and three level optics problems). A much more nontrivial example which, in my opinion, could be worked out within their approach would be the reset dynamics of 1D Ising model in a transverse field. [...] 1. Include a slightly more complicated example to demonstrate the power of the method.
Our response:
The main result of the manuscript is the establishment of the analytical result, summarized in Eqs.~(50)(55), as a general method to exactly compute the largedeviation statistics in open quantum systems subject to nonPoissonian resetting. This method is general, as we discuss in the manuscript, as it can be applied to any open quantum system whose evolution in the absence of resetting is ruled by the Lindblad equation. This result is important as for nonPoissonian resetting the SCGF cannot be computed by other means, such as the thermodynamics of quantum trajectories formalism of Ref.~[9], since the tilted generator is time dependent (in a non local way in its most natural representation). In addition, we mention, as written also in the reply to the Referee 3, that the characterization of the SCGF in the context of nonPoissonian resetting is a tough and demanding problem even at the numerical level. As a matter of fact, the SCGF $\theta(k)$ encodes the information about the rare tails of the activity distribution in the long time limit $t \to \infty$. These tails are exponentially suppressed as a function of $t$ and therefore a larger and larger numerical sampling of quantum trajectories is needed as $t \to \infty$. It is also interesting to mention that for Markovian quantum systems, trajectories at the tails of the activity distribution can be efficiently simulated by resorting to the quantum Doob tranformation of Ref.~[9,17], which allows to construct a Markovian Lindblad generator which presents as typical trajectories the rare trajectories of the original Lindblad dynamics. In the present context of the generalized nonMarkovian Lindblad equation (25), it is not, however, known how to construct the Doob transformation.
The availability of the analytical result in Eqs.~(50)(55) therefore provides a remarkable prediction for the SCGF, which could not be computed analytically otherwise, and that would require, to the best of our knowledge, a demanding and inefficient sampling for the numerical estimation.
It is fundamental to stress that the aforementioned difficulties apply already to the class of systems we study in Sec.~4, the two and threelevel systems. The latter systems are therefore valuable examples which show "the power of the method'', using the words of the Referee, since their quantum jump statistics cannot be computed for nonPoissonian resetting without the scheme we present. The "power of the method'', in this perspective, relies in capability of providing analytical predictions for the highly challenging task of characterizing the long time rareevent statistics of the activity obtained from a nonMarkovian dynamics. We have improved the presentation of the manuscript by adding a discussion of these points in the Introduction, in Sec.~2,3, at the beginning of Sec.~4 on page 19, and in the conclusion.
The Referee also suggests considering the onedimensional dissipative transversefield Ising model as an additional example. We thank the Referee for this suggestion, which allows us to draw a nice connection with the physics of the threelevel system discussed in the text. The quantumjump statistics of the Ising chain, without stochastic resetting, has been characterized in Ref.~[12] of the manuscript. In this work, it has been shown that the transverse field drives the system across a firstorder (discontinuous) phase transition, where quantum trajectories are active for high values of the transverse field, and inactive for low values of this parameter. For intermediate values, intermittency in the quantum trajectories is observed. This firstorder transition has been predicted in the thermodynamic limit on the basis of meanfield calculations. At the numerical level, via exact diagonalization techniques for $N=7$ spins, the transition clearly turns into a smoothened crossover around $k=0$ in the SCGF $\theta(k)$. This smoothened crossover is analogous to the one we observe in Subsec.~4.2 for the threelevel system. Also in this case, as a matter of fact, quantumjump trajectories are intermittent, as discussed in the text.
We therefore expect, for a finite number $N$ of spins, for the dissipative transversefield Ising chain similar results to the ones we present for the threelevel system. The results for the threelevel system are therefore also paradigmatic for the effect of resetting on possibly manybody quantum systems, with a finite number $N$ of particles, displaying intermittency in the quantumjump trajectories. In the thermodynamic limit $N \to \infty$, where in Ising chain the crossover becomes an actual discontinuous transition, differences from the threelevel intermittent physics are clearly expected. The quantification of this aspect requires a largescale numerical study on its own which goes beyond the scope of the manuscript as one needs to perform an advanced numerical analysis beyond exact diagonalization. We have discussed these aspects in Subsec.~4.2 and in the conclusions. The corresponding parts of the text are highlighted in blue.
The referee writes:
Furthermore, the analysis section of the paper is a bit dense; it would help an average reader if it can be rewritten in a slightly more pedagogical manner. For example, the authors do not quite explain how they get Eqs 5 and 6 from Eq 4. I did look up the refs they provide, and it may take quite a bit of time for the reader to figure this out. So I would suggest a bit of explanatory sentences/formulae connecting these equations even if they have been done before (may be put in the appendix, if the authors feel that it would disturb the flow). [...] 2. Make the presentation slightly more accessible.
Our response:
We thank the Referee for pointing out this issue. The explanation of the steps from Eqs.~(4) to (5) and (6) was, indeed, not clear. We have added an extended discussion of the interpretation of Eq.~(5). This part of the text is right after Eq.~(6) and it is highlighted in blue. We have also therein cited Ref.~[8], which was not cited in the first version of the manuscript at that point. Reference [8], is the first, to our knowledge, work where the decomposition in Eq.~(5) of the Lindblad time evolution operator is reported. The detailed derivation of Eq.~(5) of the manuscript can be found in the second and third pages of Ref.~[8] (Subsec.~II A ``Twolevel system''), with Eq.~(20) therein corresponding to Eq.~(5) of the manuscript.
Anonymous Report 1 on 202222 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.05078v2, delivered 20220202, doi: 10.21468/SciPost.Report.4307
Strengths
1 clarity
Weaknesses
1 the paper is a bit formal and technical
Report
In this paper, the authors consider Markovian open quantum systems subjected to stochastic resetting.
This means that the dissipative dynamics, usually described by a Linbladian, is now
reset, randomly in time, to some specific fixed state. The study of resetting of stochastic processes
has recently attracted a lot of interest, though mainly for classical systems. It was recently
realized that resetting might also generate interesting behaviours of quantum dynamics.
The present paper presents a study of such an open quantum systems in the presence of
resetting where the times at which resetting occurs may have a nonPoissonian (i.e., nonexponential)
distribution. In this case, the corresponding effective quantum dynamics is
nonMarkovian and it is shown to be described by a "generalized Lindblad equation". Despite
of this, the authors show that it is still possible to compute the large deviation function
associated to the activity (i.e., the number of quantum jumps in a given quantum trajectory)
in the limit of large observation time $t$. This result relies on a renewal approach, which is a key
tool to study resetting dynamics. This large deviation function is then expressed implicitly
and is computed numerically for two and threelevels systems.
The paper is quite well written and presents interesting and original results, which
will may be relevant for the communities working (i) on open quantum systems and
(ii) on resetting dynamics. Therefore, I would like to recommend the publication
of the present paper in SciPost, which I believe is a relevant platform to publish such
results. The authors may however consider the following (minor) comments:
1) The central result of the paper is the computation of the large deviation $\theta(k)$,
given (in an implicit form) in Eq. (55). I think that the authors should indicate
this result  by refereeing to that equation  when they present their main results on p. 4.
This would help the reader, since the paper might appear otherwise a bit formal.
2) Above Eq. (3): "nonHermitean" > "nonHermitian".
3) In Eq. (9): $d_{N1}$ > $dt_{N1}$.
4) "r.h.s." below Eq. (11)  although standard  should be defined.
Author: Gabriele Perfetto on 20220530 [id 2541]
(in reply to Report 1 on 20220202)
We thank the Referee for the supportive report and the careful reading of our manuscript. In the following, we address their report.
The Referee writes:
Therefore, I would like to recommend the publication of the present paper in SciPost, which I believe is a relevant platform to publish such results. The authors may however consider the following (minor) comments: 1) The central result of the paper is the computation of the large deviation $\theta(k)$, given (in an implicit form) in Eq.~(55). I think that the authors should indicate this result  by refereeing to that equation  when they present their main results on p. 4. This would help the reader, since the paper might appear otherwise a bit formal.
Our response:
We thank the Referee for this suggestion, which helped in improving the presentation of the main result of the paper. We have explicitly referenced this result and Eqs.~(50)(55) for the calculation of $\theta(k)$ in the introduction on page 4 of the revised manuscript. We have further referenced Eqs.~(50)(55) at the beginning of Sec.~3 on page 14 of the revised manuscript. The sentences we added are highlighted in blue.
The Referee writes
2) Above Eq.~(3): "nonHermitean" $\to$ "nonHermitian". 3) In Eq.~(9): $\mbox{d}N1 \to \mbox{dt}_{N1}$. 4) "r.h.s." below Eq. (11)  although standard  should be defined.
Our response
We thank the Referee for pointing out these issues and typos in our manuscript. We have corrected all the points 2)4) mentioned by the Referee.
Anonymous on 20220615 [id 2579]
(in reply to Gabriele Perfetto on 20220530 [id 2541])In this revised version, the authors have satisfactorily taken into account my comments. Therefore, I wish to recommend the publication of the present paper in SciPost.
Author: Gabriele Perfetto on 20220530 [id 2543]
(in reply to Report 3 on 20220218)We thank the Referee for the careful reading of our manuscript and for pointing out some fundamental aspects which were not clearly explained in the text. In the following, we address their report. We have highlighted in blue the most important changes we did in the text while revising it.
The referee writes:
Our response: We thank the Referee for pointing out this problem. We believe that the judgment about our results being "merely formal'' was motivated by a lack of clarity of the presentation, as we now explain. "Formal results'', within our understanding, are restatements of already known facts, which do not lead to new scientific predictions. Our results, on the contrary, posses a predictive power as they represent a generically applicable method for the exact calculation of the SCGF associated to the activity statistics for a nonMarkovian quantum dynamics. This is a ``difficult problem'', which cannot be solved by other means, for the following reasons.
First, there are, in general, no analytical methods to accomplish such a task. The most used analytical method, indeed, to compute the SCGF of the activity statistics in open quantum systems is the thermodynamics of quantumjump trajectories formalism of Ref.~[9]. This method relies on the spectral properties of the tilted $\textit{timeindependent}$ Lindbladian generator controlling the rare trajectories statistics of a MarkovianLindblad dynamics. In the case of nonPoissonian resetting, which is the main focus of the paper, the tilted generator is $\textit{timedependent and nonlocal in time}$ in its most natural representation in Eq.~(48). The thermodynamics of quantumjump trajectories method cannot be therefore applied in this case. It is therefore not clear whether it is analytically possible at all to compute the SCGF. The scheme we develop in the manuscript to analytically compute the SCFG, summarized in Eqs.~(50)(55), is to our knowledge the only analytical method to compute the SCGF for the nonMarkovian generator (48) describing open quantum systems subject to nonPoissonian resetting.
Second, from the numerical perspective, the problem of computing the SCGF for nonPoissonian resetting is very challenging. As a matter of fact, the SCGF $\theta(k)$ encodes the information about the rare tails of the activity distribution in the long time limit $t \to \infty$. These tails are exponentially suppressed as a function of $t$ and therefore a larger and larger numerical sampling of quantum trajectories is needed as $t \to \infty$. It is also interesting to mention that for Markovian quantum systems, trajectories at the tails of the activity distribution can be efficiently simulated by resorting to the quantum Doob tranformation of Ref.~[9,17], which allows to construct a Markovian Lindblad generator which presents as typical trajectories the rare trajectories of the original Lindblad dynamics. In the present context of the generalized nonMarkovian Lindblad equation (25), it is not, however, known how to construct the Doob transformation.
In this sense the presence of the analytical result in Eqs.~(50)(55) is a stepchange as it provides an analytical prediction for the SCGF for a nonMarkovian quantum dynamics, which could not be analytically obtained by other means and that would require a challenging and inefficient numerical analysis.
We have improved the presentation of our results by discussing in a more detailed way the aforementioned points in the Introduction, in Sec.~2 and 3 and in the conclusions. The revised parts of the manuscript are highlighted in blue.
The referee writes:
Our response: We thank the Referee for pointing out this point. We, indeed, realized that stochastic resetting was not clearly introduced in the first version of the manuscript. We have added a paragraph in the introduction where stochastic resetting is motivated in order to model many different physical phenomena such as animal foraging for food, biophysics models for proteins binding and searching algorithms. We have then included Refs.~[3952]. We further mentioned that resetting is also widely studied in physics as a paradigmatic dynamics attaining a nonequilibrium stationary state and, possibly, dynamical transitions in the relaxation to such stationary state. We have accordingly added Ref.~[56]. The revised paragraph of the introduction is highlighted in blue.
The referee writes:
Our response: We find this point raised by the Referee very much related to the point 1 discussed above and therefore that it is caused by a lack of clarity in the presentation.
First of all, in the case the Referee is aware of "more elementary approaches'' to compute the SCGF in the framework analyzed in the manuscript, we would be very much interested in a pointer towards the relevant literature. To our knowledge, the main "elementary approach'', to use the words of the Referee, to compute the largedeviation statistics of quantumjumps in MarkovianLindblad open quantum systems is the thermodynamics of quantumjumps (Ref.~[9] of the manuscript). In the case addressed in the manuscript of nonPoissonian resetting, this method cannot be applied because it relies on the spectrum of a timeindependent tilted generator, while the tilted generator we derive for nonPoissonian resetting depends on time (in a nonlocal way in its most natural representation).
The main result of the manuscript is the establishment of the analytical result, summarized in Eqs.~(50)(55), as a general method to exactly compute the largedeviation statistics in open quantum systems subject to nonPoissonian resetting. In Sec.~4, we have chosen the two and threelevel systems just to exemplify our generic findings. We, however, emphasize that already for the two and threelevel systems with nonPoissonian resetting there are to the best of our knowledge no other analytical methods (see the discussion above) to compute exactly the SCGF. Numerical methods in this context are as well very limited, as explained in the reply to point 1.