SciPost Submission Page
Floquet engineering of quantum thermal machines: A gradientbased procedure to optimize their performance
by Alberto Castro
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Alberto Castro 
Submission information  

Preprint Link:  https://arxiv.org/abs/2405.09126v1 (pdf) 
Code repository:  https://qocttools.readthedocs.io/en/stable/ 
Date submitted:  20240516 09:56 
Submitted by:  Castro, Alberto 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
A procedure to find optimal regimes for quantum thermal engines (QTMs) is described and demonstrated. The QTMs are modelled as the periodicallydriven nonequilibrium steady states of open quantum systems, whose dynamics is approximated in this work with Markovian master equations. The action of the external agent, and the couplings to the heat reservoirs can be modulated with control functions, and it is the timedependent shape of these control functions the object of optimisation. Those functions can be freely parameterised, which permits to constrain the solutions according to experimental or physical requirements.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
1  High quality of the presentation
2  The development of efficient control strategies for open quantum systems is a timely topic
3  Simulation code to use the proposed optimalcontrol framework is made available in a public software library
Weaknesses
1  Both the novelty and the applicability of the optimal control method proposed is not well substantiated
2  The example considered and the related constraints imposed remain abstract, limiting their relevance for realistic applications
3  Some technical aspects of the example model used are unclear
Report
The author analyses the use of optimal control methods to improve the finitetime performance of a quantum heat engine. The author considers a model of a heat engine as a periodically driven system coupled to Markovian heat baths. The author formulates the optimal control problem of maximizing a target functional of the nonequilibrbium steadystate in terms of a Fourier representation of the driven Linblad master equation. As a concrete example of this approach, the author considers a twolevel heat engine that was previously introduced and analysed by other authors. The author compares the ideal optimal performance, obtained in these earlier works by means of infinitelyshort piecewise constant controls, with the results obtained through optimal control with smooth control functions in a fixed finite time, including a penalty for the use of high frequencies. The impact of this frequency penalty is analysed, showing how it limits the performance.
The paper is well written, the quality of the presentation is high and the results are presented in a clear way. However, while there are some interesting aspects in the work (for instance, regarding the frequencyspace formulation of the optimal control problem), I do not see enough novel results to justify publication in SciPost Phys. I specify my doubts in relation to the requested changes below.
Requested changes
(i) methodologically, the optimal control methodology follows standard procedures. The approach is in particular very close to the one used by the same author in recent works (Refs 24, 27), such that it is hard to consider it as a core novel contribution of this work. Can the author argue more strongly for the novelty of the approach proposed?
(ii) the applicability of the optimization procedure to more general problems is not much discussed: how computationally hard it is to go beyond the optimization of a single function and of a twolevel system?
(iii) The key motivation and advantage stated by the author to use the proposed method in the twolevel example considered regards practical feasibility: smooth control functions are more realistic than quick piecewiseconstant ones. However, no connection to potential experimental setups is given, which would provide a more concrete context and realistic constraints. What could be, for instance, realistic frequency cutoffs and driving periods T in a potential implementation?
(iv) some aspects of the open twolevel model used are unclear to me: in how far is it justified to continuously modulate the system with multiple harmonics, while not changing the description of the dissipative parts? [e.g., by switching to FloquetMarkov master equations, using the nomenclature of Phys Rep. 304 229—354 (1998), in the system's Floquet basis].
Minor points:
 I appreciate the effort of the author to place the work in a broader context, but I also find the introduction at times distracting: the discussion of the actual contributions of this paper is, in my opinion, rather compressed as compared to the background.
 concerning Floquet engineering and optimal control, the author writes, in the introduction, "recently, this author and collaborators have coupled this concept with OCT". However, the use of optimal control in the context of Floquet engineering dates quite back [I can think, e.g., of PRL 113, 010501 (2014)]. Various subsequent works also used similar approaches for optimized Floquet engineering [e.g., PRX 13, 031008 (2023), PRL 126, 250504 (2021)].
 in the same paragraph, the author states "[...] it can be termed as Floquet engineering of QHT". While, on the one hand, this wording might not hurt, it also seems to imply that the joint use of Floquetengineering and openquantumsystems theory in the context of quantum heat engines is a novel aspect of this work. If so, can the author clarify what aspects are novel in this respect?
Recommendation
Ask for major revision
Strengths
1. Clear
2. Well written
3. Interesting
Weaknesses
1. The usefulness of the obtained result is not proven in the manuscript.
2. Seems to be a certain inconsistency in the theoretical analysis
Report
An extended referee report is attached below.
Requested changes
Describe in the attached report.
Recommendation
Ask for major revision
Author: Alberto Castro on 20240910 [id 4751]
(in reply to Report 2 on 20240630)(reply in attachment)
Attachment:
Report
The paper’s main focus is to find driving protocols (i.e. time dependent functions which correspond to driven coherent and incoherent evolution) that optimise a particular cost function associated with the systems performance as a thermodynamic engine. The reduced system state (after tracing out the thermal baths or environments) is described by a standard Lindblad master equation i.e. it assumes the BornMarkov approximation.
In particular, the current work focussed solely on the power output over a cycle as the key quantity to be optimised. It is assumed that all driving protocols are periodic, which leads the system to relax to a nonequilibrium steady state. The intermediate timescale where some transient effects play a role is not investigated.
My main critique of the paper, is that there does not seem to be a clear novel improvement over previous works e.g. Refs. 16 and 17. The author must make clear why their contribution is not an incremental improvement over previous papers.
I would not recommend the paper in it’s current form for publication. For publication, the author would need to thoroughly address both this main point and my additional comments listed below.
 The introduction is well structured, but does have some strange and redundant phrasing at times. It could be improved upon.
 Fig. 1 is never referenced in the main text that I can see.
 It is not clear what are the main quantum effects being exploited. Does the coherence of the qubit play an important role? What results would one get from an analogous classical master equation?
 Some notation I find a bit awkward such as with f(t) and f_k(t). Maybe the use of vectors would be useful here. Similarly with u and u^{(k)}.
 The author notes that the Lambshift “can be ignored in the weak coupling limit”. Recent discussion here [arXiv:2305.08941] suggests that this is a somewhat subtle point. Some more discussion on this is probably warranted.
 The notation \rho is used to denote both the general solution to the master equation and the NESS. I would suggest using a subscript for the NESS for clarity.
 There has been recent work e.g. [Quantum 5, 590 (2021) and Sci. Adv.8, eadd0828 (2022)] which show how the usual Lindblad master equation must be modified to ensure thermodynamic consistency when the system is driven.  Since the current work is focussed on quantifying thermodynamics performance, should this modified master equation be employed?
 The main focus has been on optimising output power. Does this have any tradeoff relations with the efficiency (even from the numerics)? Would it make sense to use a joint cost function which combines the two?
 It is claimed that the Lindblad operators can be modulated in time. This is rather unusual. It would be nice if some physical insight into how this is done in practice was included.
 “…which in a realistic setup cannot be taken to arbitrarily closetozero values”. It would be useful to state how small this can really be made for the setups of interest.
 Panel (b) of Figs. 2 and 3 need units for the yaxis.
 For the parameterisation of the control function, would it not be simpler to just use a Fourier series (as in Eq. A2) and just bound the sum of the absolute values of the amplitudes to cap the amplitude?
Recommendation
Ask for major revision
Author: Alberto Castro on 20240910 [id 4750]
(in reply to Report 1 on 20240606)(reply in attachment)
Author: Alberto Castro on 20240910 [id 4752]
(in reply to Report 5 on 20240701)(reply in attachment)
Attachment:
responsereport5.pdf