Driven-dissipative quantum battery with nonequilibrium reservoirs

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Thank you for informing us of the referee's report on our manuscript (202306-00038v1/Wang). We also extend our gratitude to the referee and editors for their comments and helpful suggestions. We have carefully revised our manuscript by incorporating these comments.

Given the extensive modifications in both the physical content and the English expressions, we have not labeled all individual changes. Instead, we provide a version of the revised manuscript highlighting the major changes in red, and we hope this does not cause any inconvenience to the editors or referees during the second-round review. Please see the detailed point by point answers to the questions raised by the referees below.


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Report of the First Referee--202306-00038v1/Wang

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Comment：

0. 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.

Reply：

We thank the referee for the excellent summarization of our work. Following the comments and suggestions from both referees, we have made significant modifications throughout the paper, including both the physical content and the English expressions.

The point-to-point reply to all comments is provided as follows:


Comment:

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.

Reply:

We thank the referee for the insightful comments. It is true that the model we study consists of two identical qubits, with one of them being continuously driven. This system can be considered as a minimal model for a quantum battery for the following reasons:

(1). In the charging process, a charger interacts with an external energy source and transfers energy to another part, i.e., the battery. This is precisely the process studied in our current manuscript.

(2). In our revised manuscript, we consider both qubits immersed in fermionic reservoirs. This setup closely resembles the behavior of a real ``battery'' system, even in classical contexts.

(3). We have investigated the charging efficiency and power in our quantum battery setup. The charging process is considered complete when the interaction between the two qubits is turned off at a specified moment, $\tau$. At this point, the charging energy, $E(\tau)$, is stored in the battery qubit. While ergotropy quantifies the energy that can be extracted from the battery, the question of how to use the battery to supply energy to a third party is beyond the scope of our current study. In this sense, our work focuses on the performance of the quantum battery setup rather than its practical usage. Therefore, we believe our study remains an integral part of research on quantum battery preparation and design.

Comment:

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. 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, $E(\tau)/\tau$.

Reply:

We thank the referee for the reminder. In the revised version, we have added a discussion on the charging power, $E(\tau)/\tau$. We find that, similar to the efficiency, the compensation mechanism also plays a significant role in enhancing the charging power of our quantum battery setup.

Additionally, as suggested by the referee, we have adopted the notation $R(\tau)$ to denote the efficiency and $P(\tau)$ for the charging power in the revised manuscript.


Comment:

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.

Reply:

It is true that the Redfield master equation cannot guarantee the positivity of the density matrix, as we have investigated in our previous work [Phys. Rev. A {\bf 99}, 042320 (2019)]. However, this issue arises only within a very small parameter regime. Fortunately, in our current work, we find that the Redfield master equation works well across the entire considered parameter regime.

In the revised manuscript, we have included a comment on this point in the conclusion section.

Comment:

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.

Reply:

We thank the referee for the kind reminder. Regarding Fig.~2(a) in the revised manuscript, we have also investigated the charging efficiency as a function of the detuning $\Delta/\lambda$, aiming to explore the connection between entanglement and efficiency.

Unfortunately, we can not specify the exact role of entanglement in determining the charging efficiency. By comparing Fig.~2(a) in the revised manuscript and the efficiency (not shown here), we observe that the line shapes and the positions of the maxima for entanglement versus detuning $\Delta/\lambda$ and efficiency versus detuning $\Delta/\lambda$ are different.

As a result, we have not included this additional result in the revised manuscript. Instead, we treat entanglement and charging efficiency as two separate sections.

Comment:

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?

Reply:

First, in the revised version, we have discarded the discussion on bosonic reservoirs, as real battery systems are typically connected to fermionic reservoirs. Consequently, $c_k$ and $b_k$ in the revised manuscript are now treated as fermionic operators.

Second, the chemical potential difference between the two reservoirs drives particle exchange until chemical equilibrium is reached. Thus, in our setup, when the two reservoirs are in a nonequilibrium state, they exchange fermions through the charger-battery coupled system.


Comment:

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".

Reply:

We apologize for our carelessness. In the revised version, we have made every effort to improve our English expression.

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Report of the Second Referee -- 202306-00038v1/Wang

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Comment:

0. 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.

Reply:

We thank the referee for the positive comments on our work. In particular, we appreciate the referee's recognition of our approach as ``stick to a more realistic steady-state charging protocol'' and the acknowledgment of our ``promising results about how nonequilibrium features can enhance the efficiency in a less traditional nonresonant driving scheme.''

We will address all the comments one by one and revise our manuscript accordingly.

Comment:

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.

Reply:

First, it is true that the Hamiltonian is rotated with
$U_1 = \exp\left[i\omega_d\left(\frac{\sigma_z^{(1)} + \sigma_z^{(2)}}{2}\right)t + i\omega_d\sum_k(b_k^{\dagger} b_k + c_k^{\dagger} c_k)t\right].$
In the revised version, we have not only explicitly presented this transformation but also provided the initial Hamiltonian before the rotation.

Second, we have corrected the expression below Eq.~(25) in the original manuscript, which corresponds to Eq.~(17) in the revised version. Here, the density matrix describes the charger-battery system, with the operators for the reservoirs traced out. Consequently, the reservoir part does not contribute and does not need to be included in this expression.

Third, in the original manuscript, we incorrectly left the interaction Hamiltonian unrotated. As pointed out by the referee, this leads to the presence of counter-rotating terms with a factor of $\exp(2i\omega_d t)$. Following the referee's suggestion, we have now applied the rotating wave approximation. However, we find that this prevents us from obtaining analytical results in the resonantly driven case, and the bistable behavior also disappears. Therefore, we have removed related discussions on analytical results and bistable behaviors in the revised manuscript.

Comment:

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.

Reply:

We are sorry but slightly puzzled by the referee's statement.

First, in our manuscript, we have defined $G_{\pm} = \sqrt{F^{2} + \lambda^{2} \pm \lambda\sqrt{F^{2} + \lambda^{2}}}$ and $K_{\pm} = \sqrt{F^{2} \pm 2\lambda\omega_{\pm}}$, so $K_{\pm} \neq G_{\pm}$.

Second, the transformation $U$ is unitary rather than Hermitian, meaning that $U^{-1} = U^{\dagger}$, not $U^{-1} = U$. Therefore, it is not necessary to satisfy the relation $\omega_\pm / G_{\pm} = K_{\pm} / 2M$.

Third, we have removed this analytical part from the revised manuscript due to the reasons mentioned in our reply to the previous comment.


Comment:

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.

Reply:

First, following the referee's suggestion, we will denote the density matrix in the interaction picture as $\rho^{I}$ in Eq.~(14) and that in the Schroedinger picture as $\rho$ in Eq.~(15). These equations now correspond to Eqs.~(9) and (10) in the revised manuscript.

Second, Eq.~(15) [Eq.~(10) in the revised version] is still formulated in the rotated frame because $|E_i\rangle \,(i=1,2,3,4)$ are the eigenstates of the rotated Hamiltonian.

Third, to clarify, we have cited Breuer-Petruccione's textbook and explicitly pointed out that the approach operates under the weak-coupling assumption.


Comment:

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.

Reply:

We thank the referee for the reminder. In the revised version, we have clarified that $H_B = \omega_2/2 (|e\rangle\langle e| - |g\rangle\langle g|)$ to ensure consistency with the previous definition. Accordingly, the mean charging energy $E_B(\tau)$ is defined as
$E_B(\tau) = {\rm Tr} [H_B \rho_B(\tau)] - {\rm Tr} [H_B \rho_B(0)].$

Comment:

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.


Reply:

We thank the referee for the beneficial suggestion. In the revised version, we have added a new paragraph (the second paragraph in Sec.~3) to briefly introduce the expression of the concurrence. Additionally, we have cited Wootters' work as a reference.


Comment:

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.

Reply:

In the revised version, we have modified the abstract to clarify the meaning of the ``compensation mechanism.'' Meanwhile, following the referee's suggestion, we have emphasized the role of non-resonance in enhancing the performance of the quantum battery, not only in terms of charging efficiency but also in power. Furthermore, we have revised the conclusion section to improve its clarity and presentation.


Comment:

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 $\Delta$.

Reply:

We agree with the referee that bistability is closely related to dissipative phase transitions (DPT) and symmetry breaking. However, as mentioned earlier, our previous submission did not properly account for the rotating frame. After correcting this oversight, we find that bistability no longer appears. Consequently, we are unable to connect our results to DPT, and the steady-state concurrence as a function of $\Delta/\lambda$ becomes continuous, as shown in Fig.~2 of the revised manuscript.


Comment:

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
$\mu$?

Reply:

We acknowledge that we are unable to provide a definitive answer to this comment. However, we observe that this minimum value appears at a position slightly greater than $0$. One possible explanation is that when the chemical potentials of both reservoirs are zero, they cannot exchange particles with the charger-battery system, resulting in low or potentially zero efficiency. The minor deviation might originate from the external driving applied to the charger.

These are speculative interpretations and, therefore, have not been included in the revised manuscript.


Comment:

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?

Reply:

It is true that the Redfield master equation sometimes fails to guarantee the positivity of the density matrix. Fortunately, in our current work, we have verified the positivity across the entire considered parameter regime. One possible reason for this might be the relatively small values of $\alpha_{1(2)}$, as noted by the referee.


Comment:

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.

Reply:

We apologize for the carelessness in our English expressions in the original manuscript. In the revised version, we have thoroughly polished the language to the best of our ability and hope it now meets the standards of SciPost Physics.


Comment:

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.

Reply:

We thank the referee for the kind reminder. In the revised manuscript, we have corrected the above typos.

Comment:

12. Few suggestions about the figures: in Fig. 2 the label $\rm {R}[\Lambda/\lambda]$
is used to indicate the Liouvillian gap but in the text the notation $\Lambda={\rm R}[\lambda_1]$ 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.

Reply:

Due to the disappearance of the bistable behavior, we have removed Figs.~2 and 3 from the revised manuscript. Meanwhile, we have enlarged Figs.~9-11, leaving only one figure, which is now presented as Fig.~4. Additionally, we have standardized all the figures in the manuscript.

We sincerely thank the referee and editor for their helpful comments and suggestions. We have revised the manuscript accordingly and believe that these modifications have significantly improved its quality. We hope that the revised manuscript meets the standards and can be accepted for publication in SciPost Physics.

Sincerely,

Zhihai Wang, Hongwei Yu and Jin Wang