Driven-dissipative quantum battery with nonequilibrium reservoirs

1. The authors extend their analysis beyond the scope of a Lindblad master equation by employing the Redfield master equation.

1. The manuscript's presentation quality and use of English are inadequate.
2. The analysis within the manuscript lacks depth, necessitating further investigation.
3. The manuscript does not convincingly explain why the described setup qualifies as a "battery."

This manuscript presents an investigation into an open quantum battery system, consisting of a qubit coupled to a driven qubit (referred to as the charger) and in contact with two thermal reservoirs, both fermionic and bosonic. The study utilizes both Lindblad and Redfield master equations, exploring phenomena such as bistability.

In my opinion, this manuscript fails to do an appropriate analysis due to several important shortcomings.  I find this manuscript to be poorly written and I have found important typos. I believe this article could be published in SciPost Physics only after a serious and major revision.

Here are the main points that need addressing:
1. I am not persuaded that the authors are studying something that could be called a "battery". The authors study the steady state of an out-of-equilibrium problem and report the charging efficiency. It is not clear, however, how this system, is continuously driven by the baths and the classical field, could operate as a battery, storing energy for a long time and providing it to a third system to do something useful.

2. The manuscript defines efficiency as the ratio between ergotropy and average energy. think it would be much better to show also the average energy, as the efficiency alone is not indicative of the energy injected in the battery. As an example, if the battery has a very low amount of energy, but the battery's state is pure, the efficiency would be one (100%). Still, in this case, the battery would be of no use, since the amount of stored energy is very low. Hence, I think that one cannot focus only on the efficiency. As a minor remark, the authors use P(\tau) to denote the efficiency. In the field, this usually denotes the average power, E(\tau)/\tau.
3.  The results obtained with a Lindblad master equation are compared with the ones obtained with a Redfield master equation,  which is claimed to be more accurate. While this is often true, the Lindblad master equation has the property of being Completely Positive and Trace-Preserving (CPTP).  Briefly, this means that all the states produced by the Lindblad master equation are physical. The same does not hold for the Redfield, which is not guaranteed to be CPTP and can yield unphysical results.  The manuscript should address whether the steady states obtained via the Redfield equation are physical and comment on the implications of using non-CPTP dynamics.
4.  The manuscript suggests a link between efficiency and entanglement. This observation appears to be more coincidental and specific to the setup rather than indicative of a deeper principle. The relevance of this finding should be critically examined.
5.  In Eq. 3, the bath's degrees of freedom are given by c_k and b_k.  Are these fermionic and bosonic operators?
The authors also state that "In this paper, we further couple the charger and battery with two independent reservoirs that can exchange energy (for boson reservoirs) or particles (for fermion reservoirs) with the system". Can the authors say more about this particle exchange?
6. In general, the article is written in poor English. There are also a few typos.
The manuscript starts with:
"The stage is yours. Write your article here. The bulk of the paper should be clearly divided into sections with short descriptive titles, including an introduction and a conclusion." which clearly should be removed!
Below,  I guess that "Floquent" should be "Floquet".

In this work the authors study several improvements over the standard scenario of a qubit quantum battery charged by a driven-dissipative qubit ancilla. In particular, they add a different noise on the battery and they treat the whole model using a microscopically sound approach. They derive the corresponding Redfield equation at weak coupling and numerically solve it to study the behavior of the efficiency in various parameter regimes. I really liked how they sticked to a more realistic steady-state charging protocol (instead of the classical switch on/off), and their promising results about how nonequilibrium features can enhance the efficiency in a less traditional nonresonant driving scheme. However, before expressing my final recommendation I would like the authors to address the following remarks.

General remarks

1. It is not entirely clear from the presentation how the rotating frame transformation at frequency $\omega_d$ interacts with the rotating wave approximation. Specifically, from Eqs. (1)-(2) I can guess that the starting universe Hamiltonian has been rotated with something like $\exp[i\omega_d( \sigma_z^{(1)} + \sigma_z^{(2)} )t/2 + i\omega_d \sum_k ( b_k^\dagger b_k + c_k^\dagger c_k )t]$, but to me this is not a completely trivial fact and should be specified. This is perhaps partially mentioned below Eq. (25), but a factor $\omega_d/2$ is missing there, and the reservoir part obviously do not appear. If this is the correct transformation, then I believe the expression Eq. (3) for the system-reservoir coupling $V$ is not the rotated one, since counter-rotating terms should present a factor $\exp(2i\omega_d t)$. This does not impact the results of the paper, since the rotating wave approximation is performed from Sec. 2.1 onward: as a consequence, I suggest to remove counter-rotating terms from Eq. (3) and stick with the rotating-wave form throughout, unless I missed some use of counter-rotating terms in the rest of the paper.

2. I do not understand why the constants $K_\pm$ are introduced below Eq. (10), since they are identical to the $G_\pm$ constants. Since $U^{-1} = U^\dagger$, why not use the same notations of Eq. (9)? Moreover, it should be true that $\omega_\pm/G_\pm = K_\pm/2M$, but it is not immediately clear to me that such a relation holds.

3. Eq. (14) shows the general form of the Redfield equation in interaction picture, while Eq. (15) shows its rielaborated form in Schroedinger picture, but the same symbol $\rho$ for the system state is used: I suggest to differentiate the notation to avoid confusion. Moreover, Eq. (14) is written with an interaction $V$ that is $\omega_d$-rotated, but it is not clear if Eq. (15) is rotated as well or not. I guess it is still in the rotated frame, but since in Sec. 2.2 a transformation that inverts the $\omega_d$ rotation is introduced, some clarifications can avoid misunderstanding. I would also add a citation to Breuer-Petruccione just above Eq. (14) and explicitly write that the Redfield equation is obtained under a weak-coupling assumption.

4. A similar notation problem happens in Eq. (28), where the symbol $H_B$ appears to refer to the battery Hamiltonian, even though the same symbol is used in Eq. (2) for the reservoir Hamiltonian. Moreover, it is stated that $H_B = \omega |e\rangle \langle e|$, but with the previous notations it should be $\omega_2/2 (|e \rangle \langle e| - |g \rangle \langle g|)$. I believe this energy shift does not influence the results, but uniformization can make the paper more clear.

5. Below Eq. (37) it is stated that the concurrence is used to measure the stead-state entanglement between charger and battery. I would briefly mention the expression you used to calculate it, just to make the paper more self-contained.

6. The Conclusion section beautifully summarizes the obtained results, but my feeling is that the same cannot be said about the abstract. The one submitted to arXiv is written considerably better. In both versions, however, it is not clear that the "compensation mechanism" the authors mention is related to having temperature or chemical potential difference: in my view, the efficiency enhancement that nonequilibrium features bring to the nonresonant driving scheme is the most important result of the paper and should be emphasized more in the abstract.

Follow-up questions

7. It is interesting to see how assuming resonant driving opens up a nontrivial steady-state space. In Lindblad theory, the closure of the Liouvillian gap is often associated with dissipative phase transitions (see Ref. [58]). I guess some subtleties should be taken into consideration when looking at the Redfield equation, but do you think it is possible to interpret the bistability you obtain using arguments from DPTs and symmetry breaking? A related question is why the steady-state concurrence in Fig. 3 appears to be so discontinuous as a function of $\Delta$.

8. In Fig. 7b you show how the efficiency at equilibrium with resonant driving drops for a certain value of $\mu$ that is not dependent on $T$. Do you have an interpretation for why this happens and/or why it happens at that specific value of $\mu$?

9. It is a known fact in the theory of open quantum systems that the Redfield equation is not positive, meaning that it could lead to the prediction of meaningless negative probabilities for measurement outcomes. Did you have problems in this sense, or your values for $\alpha_{1,2}$ were sufficiently low to not encounter issues?

Writing issues

10. There is a nonnegligible number of grammatical and orthographic issues. For example: "nonequilibrium" is misspelled several times; "eigen states" and "eigen values" instead of "eigenstates" and "eigenvalues"; other statements are ill-formed, especially in the Introduction (e.g., "the quantum battery which is subject to the open system"). There is also a piece of the SciPost template the beginning of the Introduction. I suggest to perform a complete writing recheck.

11. Related to the previous point, I noticed typos in some mathematical expressions. Below Eq. (4), it should be $\delta(\omega - \omega_{bk})$ instead of $\delta(\omega - \omega_1)$ and $\delta(\omega - \omega_{ck})$ instead of $\delta(\omega - \omega_2)$. In Eq. (8), $E_3$ is repeated twice, whereas $E_4$ should appear instead in second occurrences. Below Eq. (25), $\omega_d/2$ should appear in $U_1(\tau)$ I believe.

12. Few suggestions about the figures: in Fig. 2 the label $\Re[\Lambda/\lambda]$ is used to indicate the Liouvillian gap but in the text the notation $\Lambda = \Re[\lambda_1]$ is used; in Fig. 3 maybe it should be specified in the caption that the tomography is performed at $\Delta = 0$; in Fig. 4 the majority of the information is concentrated around $\Delta = 0$ but the plot is too zoomed out to appreciate the chemical potential differences; Figs. 9-11 are maybe too small.