Heat current and fluctuations between a dissipative qubit and a monitor under continuous measurement and feedback

Reply to Report #2 by Anonymous (Referee 2)

We have submitted the revised version of our manuscript to arXiv as version 2. It will be published soon.

Report The authors investigate heat current and fluctuations in a problem where a qubit is coupled to reservoirs and is also monitored by continuous measurement and feedback. The problem is timely and the results are interesting. I can recommend the article for publication after the authors address the following points:

We thank Referee 2 for the careful reading and recommendation for the publication for SciPost Phys.

1. Presentation of the model: Several reservoirs are indicated in Fig.1 and in section 2.1. However, nothing is mentioned afterwards about the role of them. Do they have the same temperature? Since the results focus on a single bath, the authors should mention that.

   Throughout this work, we consider multiple reservoirs. The effects of these reservoirs are incorporated into the total emission and absorption rates in the dissipator D_B[rho(t)] (see Eq. (5)). These rates are the sums of the emission and absorption rates at each reservoir, labeled by r, given by Gamma_+=sum_r (pi/2) I_r(Delta)[1+n_r(Delta)] and Gamma_-=sum_r (pi/2) I_r(Delta)n_r(Delta), respectively. Here, I_r(omega) is the spectral density and n_r(omega) is the Bose-Einstein distribution for reservoir r. These definitions are provided below Eq. (5). Therefore, it does not matter if the reservoirs have different temperatures, as long as the total emission and absorption rates remain much smaller than the qubit energy. We have added the comment “As long as this condition is justified, it does not matter if the heat baths have different temperatures” below Eq. (5) in the revised manuscript. A comment “which corresponds to alpha~0.0191 and T/Delta~1.44 for the single heat bath. Note that, for simplicity and without loss of generality, we consider a single heat bath characterized by the temperature T in the following” in the middle of page 5 might be confusing. As mentioned above, the effects of the heat baths are incorporated into the total emission and absorption rates, so the temperature of each reservoir is not uniquely determined for multiple reservoirs. To determine the temperature from the total emission and absorption rates, it is necessary to consider a single reservoir. To avoid restricting the discussion to a single heat beath, we have modified this comment to: “which corresponds to the effective coupling strength alpha_eff~0.0191 and the effective temperature T_eff/Delta~1.44 when the effects of multiple heat baths are represented by an effective single heat bath.”

2. In Fig. 2 the authors describe m and n as monitor and feedback states, respectively. It would be useful if this is also mentioned in Section 3.1.

   We thank Referee 2 for the suggestion. We have added the comment “Thus, the measurement and feedback states are characterized by theta_m and theta_n, respectively” below Eq. (7).

3. Even more important, the results depend on the temperature of the thermal bath. I think that the authors can further stress this point. In Fig.2, no Information is provided on the temperature of the bath.

   As mentioned in our response to Referee 2’s Comment 1, the temperature information is included in the total emission and absorption rates, and the temperatures are not uniquely determined from these rates for the multiple heat baths. We have added the effective coupling strength and the effective temperature, obtained from the total emission and absorption rates, to the caption of Fig. 2.

4. In Section 3.3 they conclude that the temperature of the thermal bath affects the condition of zero current between the qubit and the monitor, but nothing is said about the current between the qubit and the bath. Is the qubit thermalized with this bath or is there heat flowing through qubit between the monitor and the thermal bath?

   At the steady-state limit, the heat exchange between the monitor and the qubit, J=tr_0[H_0 D_M[rho]], and the heat exchange between the heat baths and the qubit, defined as J_B=tr_0[H_0 D_B[rho]], are balanced, J=-J_B. This can be confirmed from the quantum master equation (4) under the steady-state condition, drho(t)/dt=0. Therefore, when no heat flows between the monitor and the qubit, the heat exchange between the heat baths and the qubit also vanishes, J_B=0. In this sense, the qubit is thermalized at the effective temperature T_eff=Delta/ln(Gamma_+/Gamma_-).

5. In section 4 I do not find the explicit definition of J_c(t). In the discussion of the results, it would be nice to find some discussion on how the features of the Poisson noise and backaction noise are related to the behavior of the heat current discussed in fig. 2.

   We thank you for your feedback. We have added the definition of the conditional heat current J_c(t) as Eq. (16), instead of the conditional heat change Q_c(t). As Referee 2 pointed out, the standard Poisson noise S, for example, in an electronic nanowire, is related to the electronic current I as S=2eI. In our work, we found the Poisson noise has a similar expression (see Eq. (21)). However, since it is described solely by the quantum jump process, this Poisson noise is not related to the heat current in the same way as the standard Poisson noise. This suggests that continuous quantum measurement and feedback induce the characteristic statistics, and the resulting fluctuations allow access to the quantum jump, which is indistinguishable from the backaction of no detection at the level of the heat current. We have revised the discussion at the end of Section 4.1.

Recommendation Ask for major revision

We thank Referee 2 again for the recommendation for publication in SciPost Physics. We have responded to all the comments and revised our manuscript in accordance with most of the suggestions.

Reply to Report #1 by Anonymous (Referee 1)

We have submitted the revised version of our manuscript to arXiv as version 2. It will be published soon.

Strengths - Addresses a timely topic of interest in quantum thermodynamics. - Goes beyond a description in terms of average quantities by considering the heat current fluctuations. Weaknesses - Only the steady-state configuration is addressed, while a previous study by the authors on a similar system also considered the transient dynamics.

We thank Referee 1 for recognizing the strength of our manuscript. As Referee 2 also reported, our work is timely topic in quantum thermodynamics. Additionally, we spotlighted the heat current fluctuations due to quantum measurement. This work focuses on the steady-state heat current because the statistics arising from quantum measurement is the main topic. However, when considering the backaction noise S_1, the transient dynamics is accounted for through the excess heat Q_ex, which was also introduced in our previous work [42] to discuss the transient dynamics. We have added a comment regarding this below Eq. (24). In the first part of our manuscript, we derived the maximum and minimum values of the steady-state heat current. As our previous work [42], heat current can exceed these steady-state bounds in the transient regime, reaching +(-)gamma Delta as the maximum and minimum values. The heat current achieves the maximum (minimum) value when |m>=|g (e)>, |n>=|e (g)>, and the qubit is in a pure state rho=|g (e)><g (e)|. Therefore, the maximum or minimum heat current occurs only when the qubit is prepared in a pure state. Subsequently, the heat current relaxes to the steady-state value, as our previous work [42]. We have added the discussion in the revised version at the end of Section 3.1.

Report The authors study a dissipative qubit in the presence of continuous measurement and feedback by a monitoring system, extending a previous study by the same authors (Ref. [42]), where only the measurement process was included. They employ a master equation approach to obtain the steady-state heat exchange between qubit and monitor. They assume that the qubit is coupled to a collection of bosonic baths, that are treated as an Ohmic environment. The analysis is complemented by a quantum trajectory analysis allowing them to study the stochastic evolution of the monitored qubit. The key results are: - the simultaneous presence of measurement and feedback allows to selectively heat or cool the qubit; - the measurement-induced backaction produces fluctuations in the heat current that deviate from the Poissonian limit, and the system is often found in the sub-Poissonian regime. I believe this work is of good quality and deserves publication. However, I am not confident that publication in SciPost Physics is appropriate, as the journal's acceptance criteria are not met in my opinion. Specifically, the authors indicate that their work provides a novel and synergetic link between different research areas, but I fail to see that this is the case. To my understanding, the only connection mentioned by the authors with a different research field (other than quantum thermodynamics/energetics) is with electronic transport. This connection is in my opinion rather faint: it is indeed based on the analysis of the Fano factor - which is also used in electronic transport - but I do not think that this fact by itself constitutes a novel and synergetic link between different research areas. Instead, I believe this works meets the acceptance criteria of SciPost Physics Core and therefore recommend publication in that journal.

We appreciate the thorough review and the comment about the publication; “I believe this work is of good quality and deserves publication”. Referee 2 pointed out a poor presentation of the connection with different research fields. The current noise has provided new perspectives in mesoscopic transport beyond the mean current, as mentioned “The noise is the signal” by Rolf Landauer. On the other hand, while quantum measurement can induce the current noise, its property remains largely unexplored. In this work, we made a first attempt to investigate the differences between the electronic current in a low-transmission nanowire and the heat current induced by continuous measurement. Our result shows that the statistics induced by continuous measurement follows sub-Poissonian. This arises from the fact that the probability of a quantum jump depends on the qubit state, making the quantum jump not completely random. Similarly, for the electronic current, the sub-Poissonian statistics emerge when electrons do not flow completely randomly, as influenced by the Pauli exclusion principle and the Coulomb interactions. Consequently, we identified a link between mesoscopic physics (correlated electrons) and quantum measurement from the perspective of the current noise. We have added the discussion in Section 4.3. We also stress the feasibility of our setup. Recently, heat current has been measured precisely in superconducting circuits using methods developed in mesoscopic physics. By monitoring the temperature of the heat bath, we can access quantum heat directly. A comment has been added to the summary.

Requested changes

1. There is a slightly misleading statement about the Poissonian noise in electronic transport (in the introduction and below equation 19). Noninteracting electrons in a nanowire exhibit Poissonian noise in the weak transmission limit only. Otherwise, the noise obeys a well-known T(1−T) behavior (where T is the nanowire transmission), while the current is just proportional to T. Consequently, there can be deviation from the Poissonian noise limit. I suggest that the authors correct the statements by explicitly indicating that the Poissonian regime in electric transport is fulfilled at weak transmission.

   As Referee 1 pointed out, the explanation of Poisson noise in electronic transport was insufficient. We have revised it in the introduction and in the discussion below Eq. (21).

2. The following sentence in the introduction is unclear. "By considering continuous quantum measurement and feedback, we can address the steady-state properties of the heat exchange, it’s a steady-state heat current, when accounting for dissipation, which is inevitable but ubiquitous in nature.". I would suggest to reformulate it.

   We have revised the sentence to make it clear as follows; “Most studies on quantum measurement have been conducted in the context of ideal systems without dissipation. It is crucial to take dissipation into account not only because of its ubiquity in nature but also due to its qualitative effects on heat exchange. This consideration enables us to address steady-state properties of heat exchange under continuous quantum measurement and feedback, as measurement and feedback alone induce no heat exchange in the steady-state limit.”

3. The authors state that their model can find application as a measurement/feedback-based quantum refrigerator. Given that they have already calculated the heat fluctuations, could they also provide the efficiency of the cooling process and compare it with standard performance quantifiers, such as the thermodynamic uncertainty relation?

   The cooling of the qubit via quantum measurement and feedback can induce quantum measurement cooling, where energy is extracted from the colder heat bath into the quantum system due to quantum measurement. We confirmed that the quantum measurement cooling occurs by calculating the heat current flowing from the cold heat bath to the qubit for the case of two heat baths, r=c, h. We have added the discussion about the quantum measurement cooling and its efficiency, the coefficient of performance (COP), in the newly added Section 3.4. As Referee 1 pointed out, given the mean and variance of the heat current and the entropy production, we might bound the efficiency of the quantum measurement cooling. However, the variance pertains to the heat current from the cold heat bath, not the heat current between the qubit and the monitor, which is focus of this work. While it would indeed be interesting to discuss the efficiency bound of the quantum measurement cooling using the thermodynamic uncertainty relation, this is beyond the scope of the present work.

4. I would like to encourage the authors to upload the data for the trajectory simulations to an established data repository (e.g., Zenodo), to facilitate the validation and reproducibility of their analysis.

   We have uploaded the numerical data on quantum trajectories used in Figs. 4, 5, and 7, to Zenodo. The DOI is 10.5281/zenodo.14252485.

5. Grammar can be improved. For instance: -page 4: "the steady-state heat current holds the following inequality" → "fulfils/satisfies the following inequality"; -page 5: "the measurement to the ground state and the feedback to the excited state" → "the measurement projects the system in the ground state..."?; -page 5: "the qubit energy is heated by the measurement" → either "the qubit is heated" or "the qubit energy is increased"; -caption of Fig. 2: "the heat current never takes negative" → "never takes negative values"; -page 7: "the correlation function is decomposed of the delta function... and the time non-local correlator" → "is decomposed in a delta function... and a time non-local correlator"; -page 9: "the heat change by the quantum jump and the excess heat are opposite signs" → "... are of opposite signs" or "have opposite signs"; -Page 11: "It shows zero at four points" → "it vanishes at four points" or "it crosses zero at four points" or simply "it is zero at four points".

   We thank Referee 1 for highlighting the grammatical issues. We have addressed all the points raised.

Recommendation Accept in alternative Journal (see Report)

Validity: High Significance: Good Originality: Good Clarity: High Formatting: Excellent Grammar: Reasonable

We have replied to all the comments and continue to believe that our work is suitable for publication in SciPost Physics. In the revised version of our manuscript, we have incorporated discussions and comments in accordance with the suggestions of Referee 1.