Floquet engineering of quantum thermal machines: A gradient-based procedure to optimize their performance

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 optimal-control framework is made available in a public software library

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

The author analyses the use of optimal control methods to improve the finite-time 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 steady-state in terms of a Fourier representation of the driven Linblad master equation. As a concrete example of this approach, the author considers a two-level 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 infinitely-short 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 frequency-space 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.

(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 two-level system?

(iii) The key motivation and advantage stated by the author to use the proposed method in the two-level example considered regards practical feasibility: smooth control functions are more realistic than quick piecewise-constant 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 two-level 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 Floquet-Markov 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 Floquet-engineering and open-quantum-systems 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?

Ask for major revision

1. Clear
2. Well written
3. Interesting

1. The usefulness of the obtained result is not proven in the manuscript.
2. Seems to be a certain inconsistency in the theoretical analysis

An extended referee report is attached below.

Describe in the attached report.

Ask for major revision

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 Born-Markov 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 non-equilibrium 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 Lamb-shift “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 close-to-zero 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 y-axis.
- 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?

Ask for major revision