Thermodynamics of autonomous optical Bloch equations

1. The presented thermodynamic framework opens a new pathway to characterize the thermodynamics of quantum devices and has clear potential for multi-pronged follow-up work. The manuscript thereby meets the acceptance criteria for the journal and I am happy to recommend publication.

2. The manuscript provides detailed derivation and analysis, connecting to existing frameworks in a comprehensive way.

3. The developed framework seems to be very general.

1. It is not always clear which of the results are general and which are restricted to the setup of OBE.

2. In the conclusions, the author state that they "answer a long-lasting question in quantum thermodynamics: how can we measure heat-like and work-like quantities in the quantum realm, knowing that measuring a quantum system alters its thermodynamic balance because of measurement back-action?" I don't agree with this. The long-lasting discussion is to my understanding about fluctuations of work. The average of work, and how it is measured, is typically not debated since it can be obtained from averaging over many measurements, each of which with vanishing backaction. The framework provided by the authors only addresses average values and it remains unclear how work fluctuations would be defined in that framework.

3. The manuscript shares ideas with Ref. [59], which provides a thermodynamic framework for a slightly different setting (systems embedded in cavities). This does not become apparent from the manuscript.

In this manuscript, the authors develop a new thermodynamic framework for an atom coupled to a one-dimensional waveguide. In contrast to the established framework, the light leaving the atom carries work, which leads to less dissipation and a tighter second law of thermodynamics. This framework is promising for describing cascaded quantum systems, where the light leaving a quantum system is used to drive another quantum system. While the analysis is restricted to an atom coupled to a one-dimensional waveguide, the results seem to be easily generalizable to more complicated setups. Given the relevance of light-matter coupled systems, and the potential that comes with a detailed thermodynamic understanding, I believe that the formalism developed here will enable many interesting follow-up studies. I therefore recommend publication.

I have a number of suggestions, questions, and comments that I would like the authors to address:

1. I think Eq. (7) is incorrect. Conventionally, the work is defined through the time-derivative of the Hamiltonian in the lab frame. The Hamiltonian in Eq. (7) is in a rotating frame. In particular, for $\omega_L=\omega_0$, the work in Eq. (7) vanishes, which is not what happens conventionally. Related to this, the authors sometimes call the regime $\omega_L=\omega_0$ the "absence of a drive" (see for instance Sec. 3.4). I think this is misleading since there still is a drive, it just has the same frequency as the atom.

2. Below Eq. (11), the authors state that they only consider right-moving modes (see also the sum in Eq. (11)). Why is this physically justified? Could the atom not emit into left-moving modes and thereby result in reflection of light?

3. In Eq. (21), the right-hand side depends on $n$. Does this imply that the Rabi-frequency depends on time? In Eq. (2) it seems to be time independent.

4. In Eq. (26) and (27), two "photon flows" are introduced. Are these indeed physically distinct flows? Eq. (25) seems to imply that both the photons carrying the heat and the photons carrying the work are exchanged with the same field. In other words is the "photon flow exchanged with the thermal component of the input field" a meaningful physical concept?

5. I am a bit confused about the classical limit of the drive defined on page 12. Should not $\gamma\rightarrow 0$ to obtain the classical limit? Such that the heat current vanishes but work remains finite because $\langle b_{in}\rangle\rightarrow\infty$ such that $\langle b_{in}\rangle \sqrt{\gamma} = const.$ (also Ref. [20] seems to require small $\gamma$).

6. I find the different states introduced for the field to be somewhat confusing. In particular, there is $\eta_n$ which below Eq. (18) seems to act on the full Hilbert space of the field. In appendix H however, $\eta_n$ seems to only act on a single mode of the field.

7. In Eq. (52), the initial displacement of the field, which defines the Rabi drive, seems not to be included in the reference state, since it is not included in $\eta_n^\beta$ (as defined below Eq. (18)). This strikes me as odd and it seems to differ from the results of Ref. [59], could the authors comment on this? Also, what about the modes that have not yet interacted with the atoms, are they still displaced in (52)?

8. The caption of Fig. 3 refers to insets, which I assume have been promoted to panels.

9. I did not understand the physical relevance of the "saturation parameter" in Eq. (54). Could the authors give a definition?

10. In Eq. (69), should there not be a contribution from ${\rm Tr}{\sum_nV_n^{(an)}(t)\partial_t\rho(t)}$?

In addition, I have two questions that the authors may choose to address:

a) The authors consider the energy stored in the average field as work. This raises the question if energy stored in the field differently is necessary heat. Did the authors consider the ergotropy of the outgoing field and how it is related to their results?

b) Since Eq. (33) is equivalent to the OBEs, the physics must remain the same. Is the concept of "self-drive" then a physical concept or just another way of viewing the same physics?

1. I would like the authors to address my comments 1.-10. in the report.

2. I think it would be helpful for the reader if the connections between the present manuscript and Ref. [59] are mademore transparent. Given the dates these works appeared on the arXiv, there is no issue about novelty of the present manuscript but I think pointing out this connection makes it easier for a reader to navigate the literature.

3. I think the authors should soften their claim on "answering a long-lasting question in quantum thermodynamics" in the conclusions, since I believe that this long-lasting question pertains to work fluctuations which are not actually addressed in the present manuscript.

4. Maybe the authors could further clarify which of their results take over to other systems coupled to light and which are restricted to the OBE setting.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1) The paper deals with a fundamental issue of a the Thermodynamics of optical Bloch equations. The Bloch equation is the template of the quantum Master equation. Originally guessed by Bloch it has since been derived from first principles. 2) The paper addresses specifically an autonomous derivation meaning the the drive is also a quantum system. 3) The authors show the connection to a collision model one of the rigorous approaches to derive the Master equation. 4) Definitions of heat and work are obtained connecting to thermodynamics. 5) the derivation is clear.

1) The authors overlook closely related work on an autonomous driven master equation:

Dann, et. al. "Quantum thermo-dynamical construction for driven open quantum systems." Quantum 5 (2021): 590.

Dann, et al. "Unification of the first law of quantum thermodynamics." New Journal of Physics 25, no. 4 (2023): 043019

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