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Proposal for an autonomous quantum heat engine
by Miika Rasola, Vasilii Vadimov, Tuomas Uusnäkki, Mikko Möttönen
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Miika Rasola · Vasilii Vadimov |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2502.08359v2 (pdf) |
| Date submitted: | May 3, 2025, 9:18 a.m. |
| Submitted by: | Miika Rasola |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We propose and theoretically analyse a superconducting electric circuit which can be used to experimentally realize an autonomous quantum heat engine. Using a quasiclassical, non-Markovian theoretical model, we demonstrate that coherent microwave power generation can emerge solely from the heat flow through the circuit determined by non-linear circuit quantum electrodynamics. The predicted energy generation rate is sufficiently high for its experimental observation with contemporary techniques, rendering this work a significant step toward the first experimental realization of an autonomous quantum heat engine based on Otto cycles.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-6-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2502.08359v2, delivered 2025-06-12, doi: 10.21468/SciPost.Report.11396
Strengths
- Plausible experimental feasibility
Weaknesses
- Unclear discussion of the quantum vs. classical effects in the system
Report
Technically the work contains a device proposal and the solution of classical equations of motion of a superconducting circuit, driven by noise from resistors. This appears fine in itself, and probably is reasonable model for the proposed system at temperatures high enough. However, I have some comments below which should be clarified.
Although a classical model is not really a "single quantum" heat engine, for realizing the system it's regardless useful to understand the classical behavior. In the manuscript it is also suggested with some justifications that the principle would scale to low temperatures.
As such, the work is probably a useful step toward realizing this type of heat engines on cQED platform, and may be suitable for publication after open questions are clarified.
Requested changes
1 - In Eq. (8) and below symmetrized correlator is used, so in a sense emission and absorption are summed together. If they are different, omega > T considered in the manuscript, one could expect it may have some importance for the heat engine physics studied here.
Implications on the difference to fully classical limit is discussed in Sec. 4.3., mostly stating that the numerical results change quantitatively, and in the introduction noting that linear equations are same in quantum case. Can the final results be understood in terms of quanta emitted from the hot reservoir and absorbed in the cold or in the driven resonator? Do the equations obtained reflect such physics? The exponential dependence in Fig. 4(cd) at low T may be reasonable, but this is found from numerical results so the origin is not unambiguous. I don't find the discussion of the semiclassical approximation very clear in the present version.
2 - The Otto cycle as discussed in Sec. 4.4 is somewhat hard to see and understand from Fig. 8. From the discussion, I would guess what is implied is that (x,y)=(omega_a', <phi^2>) forms a loop with nonzero interior. This could be contrasted to proposals [45] and others, after converting between field amplitude and photon counts.
3 - The negative dissipation occurs only at nonzero resonator amplitude when there is internal dissipation, unless quality factor is very high. What can be said about the threshold where the resonator B starts to oscillate if it starts at rest?
4 - The results are calculated by time-scale separation arguments. Are these assumptions self-consistent with the results for the parameters used, i.e do the slow equations produce slow dynamics? This is not explicitly discussed in the manuscript, but probably would be useful to note it.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2025-6-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2502.08359v2, delivered 2025-06-12, doi: 10.21468/SciPost.Report.11390
Report
To start with, I miss a more intuitive explanation of the mechanism which drives the engine, i.e. on the origin of the self-sustained oscillations in phi_b. Is there a simple qualitative condition for their appearance? Perhaps a simple schematic figure would help more than the rather involved numerical approach described in the manuscript.
On the other hand, I wonder about the need of such a complex circuit with so many parameters. If this work is motivated by an actual experiméntal device this should be clearly stated. The choice of parameters in table 1 should also be more clearly justified.
Finally, it would be interesting that the authors comment how they plan to approach a complete quantum description of their device.
Recommendation
Ask for minor revision
We would like to thank the reviewers for raising valid points and making keen critical
observations about our work. The remarks helped us in improving the manuscript considerably
over the revision process. The common point of the reviews seems to be the
issues raised with the qualitative explanation and intuitive description of the proposed
device, as well as lacking schematic figures supporting such considerations. These issues
have been addressed in the revised manuscript, along with other points raised by the
reviewers. The most important addition is the schematic figure (Fig. 2) added to the
manuscript, much clarifying the intuitive understanding we want to convey. The point-
by-point response for each of the reviews can be found below. Please consult the revised
manuscript to see the updates mentioned within.
In our manuscript, we originally attempted to avoid extensive phenomenological
discussion since it was presented in a previous work [1] and instead focused more
on a theoretical model which allows one to calculate the output power and the engine
efficiency. However, we agree with the Referee that a qualitative explanation
of QHE operation makes the manuscript more accessible to the readers. Therefore,
we have added a descriptive schematic Figure 2 accompanied by relevant intuitive
description of the device dynamics.
We agree with the Referee that the choice of the circuit was not clearly explained
in the manuscript. Partially, we rely on our previous work [1], cited in the the
manuscript as the source of motivation. However, we also work on the experimental
realization of the proposed quantum heat engine and use our theoretical model
for the design of the actual device and for the analysis of the experimental data.
Therefore, the model cannot be too simplified if we aim for quantitative match
between the theoretical and experimental results. Note that our experiments are
currently on-going and the data is too preliminary to be published at this point.
Thus we limit the scope of the present manuscript to the theoretical analysis. Furthermore,
we would like to point out that the proposed superconducting microwave
circuit is relatively straightforward to fabricate with the current state-of-the-art
methods. The choice of the parameters for the device was simply dictated by experimental
feasibility — the parameters can be realised and the theory shows that
the device can produce enough output power for experimental observation. This
has been more clearly mentioned in the revised manuscript.
Full quantum description of the QHE engine is, in principle, possible using nonlinear
response theory developed in Ref. [2]. This theory allows to describe the dynamics
of the superconducting circuit using a hierarchical equation of motion and evaluate
various correlators of the output field. This approach is technically challenging but
not impossible since HEOM has been proven to be efficient for simulating open
quantum systems in structured environments [3].
[1] Miika Rasola and Mikko Möttönen. Autonomous quantum heat engine based on
non-markovian dynamics of an optomechanical hamiltonian. Scientific Reports,
14(1):9448, Apr 2024.
[2] V. Vadimov, M. Xu, J. T. Stockburger, J. Ankerhold, and M. Möttönen. Nonlinear-
response theory for lossy superconducting quantum circuits. Physical Review Research, 7(1):013317, March 2025.
[3] Meng Xu, J T Stockburger, G Kurizki, and J Ankerhold. Minimal quantum thermal
machine in a bandgap environment: non-markovian features and anti-zeno advantage.
New Journal of Physics, 24(3):035003, March 2022.
Author: Miika Rasola on 2025-08-20 [id 5747]
(in reply to Report 2 on 2025-06-12)General response:
We would like to thank the reviewers for raising valid points and making keen critical
observations about our work. The remarks helped us in improving the manuscript considerably
over the revision process. The common point of the reviews seems to be the
issues raised with the qualitative explanation and intuitive description of the proposed
device, as well as lacking schematic figures supporting such considerations. These issues
have been addressed in the revised manuscript, along with other points raised by the
reviewers. The most important addition is the schematic figure (Fig. 2) added to the
manuscript, much clarifying the intuitive understanding we want to convey. The point-
by-point response for each of the reviews can be found below. Please consult the revised
manuscript to see the updates mentioned within.
In our work, we use quasiclassical Langevin equation derived in Ref. [62] of the
manuscript. The noise source in Eq. (7e) is real-valued, which implies symmetric
form of the correlator in Eq. (8). However, the photon absorption is included via
damping term in Eq. (7e). For example, in case of linear open quantum systems, the
correlation functions obtained by solving quasiclassical Langevin equation can be
proven to be exact. Our system is indeed nonlinear, therefore this formalism is only
approximate. However, we expect it to capture the main features even if \hbar\omega > k_BT
Point-by-point:
1 - Within our quasiclassical formalism, the only quantum feature is the correlation
function of the noise Eq. (8). In the classical limit, \hbar → 0, it reduces to the white Johnson–Nyquist
noise. The fully classical limit discussed in 4.3. is understood in this sense.
To provide more intuitive explanation of the physics of the proposed QHE,
we have added a schematic Fig. 2: the fast resonator A absorbs higher frequency
photons from the hot bath and emits lower frequency photons to the cold bath. The
difference of energy is the produced work which goes to the slow engine mode B.
To connect this cartoon picture with the used formalism, we notice that stochastic
source in the right hand side of Eq. (7e) contains both thermal noise and
quantum zero-point fluctuations (ZPF). If the temperature of hot reservoir is low
k_BT_h ≪ \hbar\omega_h only ZPF part of the noise remain. Therefore, power and efficiency
shown in Fig. 4(cd) is suppressed despite the temperature of cold bath is well below
Th since it is also driven by the ZPF noise. In the photon language, this corresponds
to the vanishing population of the both reservoirs. This picture breaks in classical
limit \hbar → 0 due to the absence of zero-point fluctuations in the noise sources. In
this case, T_c = 0 corresponds to absence of noise and there is always flow of energy
from the hot bath to the cold.
2 - We are grateful to the Referee for this comment. We have changed Fig. 9 of the
manuscript and added the corresponding cycle in the plane (\omega_a', n_a),
where n_a is the mean photon number in the working fluid mode A.
The latter has been estimated as 2E_m/(\hbar\omega_a′), where E_m is the
magnetic energy of the mode A (Eq. (33) of the revised manuscript).
The loop has indeed non-vanishing area and direction of this
loop is such that positive work is produced.
3 - The condition on the quality factor required for the mode B oscillations to start
from zero is even stricter. However, we observe that it is possible in principle: Fig. 4
shows that in case Q = inf we have negative dissipation even at zero amplitude.
To illustrate this better, we have added panels in Figs. 5, 6, and 7 where we show
(i) threshold Q-factor where the resonator B starts to oscillate from zero and (ii)
threshold Q-factor at which self-sustained oscillations are impossible.
4 - Indeed, the fulfilment of this assumption was not explicitly mentioned in the manuscript,
but was only readable from the data presented in Figures. From here we find the
slow dynamics to have a characteristic rate of change less than one per mill of that
of the intermediate dynamics. We have explicitly mentioned this in the updated
manuscript.