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Solution of master equations by fermionicduality: Timedependent charge and heat currents through an interacting quantum dot proximized by a superconductor
by Lara C. Ortmanns, Maarten R. Wegewijs, Janine Splettstoesser
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Authors (as registered SciPost users):  Maarten Wegewijs 
Submission information  

Preprint Link:  https://arxiv.org/abs/2210.04973v1 (pdf) 
Date submitted:  20221012 17:53 
Submitted by:  Wegewijs, Maarten 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We analyze the timedependent solution of master equations by exploiting fermionic duality, a dissipative symmetry applicable to a large class of open systems describing quantum transport. Whereas previous studies mostly exploited duality relations after partially solving the evolution equations, we here systematically exploit the invariance under the fermionic duality mapping from the very beginning when setting up these equations. Moreover, we extend the resulting simplifications  so far applied to the local state evolution to nonlocal observables such as transport currents. We showcase the exploitation of fermionic duality for a quantum dot with strong interaction  covering both the repulsive and attractive case  proximized by contact with a largegap superconductor which is weakly probed by charge and heat currents into a wideband normalmetal electrode. We derive the complete timedependent analytical solution of this problem involving nonequilibrium Cooper pair transport, Andreev bound states and strong interaction. Additionally exploiting detailed balance we show that even for this relatively complex problem the evolution towards the stationary state can be understood analytically in terms of the stationary state of the system itself via its relation to the stationary state of a dual system with inverted Coulomb interaction, superconducting pairing and applied voltages.
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Reports on this Submission
Anonymous Report 2 on 20221125 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2210.04973v1, delivered 20221125, doi: 10.21468/SciPost.Report.6194
Strengths
1 It presents a careful description of the method, including all details
2 If presents a powerful and innovative method for the computation of transient transport in quantum dot devices
Weaknesses
1 (maybe contradictory with strength #1): the manuscript is very long
2 the introduction is difficult to read for non experts.
Report
The manuscript "Solution of master equations by fermionicduality: Timedependent charge and heat currents through an interacting quantum dot proximized by a superconductor" presents a detailed analysis of the transient dynamics in a device composed of a proximized singleĺevel quantum dot (by being coupled to a infinitegap superconductor) tunnel coupled to a normal metal contact. This is neither an unexplored configuration neither a new problem (relevant references are cited), however the authors introduce a powerful method that considerably simplifies its solution, adding important insights on the relevance of fermionic symmetries. The method is also not totally new :the authors have published a number of references using it in simple (simpler) configurations, but it is shown here to be useful for more complex situations, at the same time including new and interesting results in the form of constraints of the dual quantities and the universal and simple relation between their stationary expected values. The text is "in average" clearly written, with all the details of the method carefully exposed. For these reasons, I consider it meets the acceptance criteria to be published in SciPost.
Below, I include a list of more detailed comments/suggestions that may be considered:
 The results are kept on the formal level, with numerical results being referred to a "manuscript in preparation" by the same authors (ref. [18]). Being unnecessary to make the manuscript longer, I wonder if it would increase its presentation by plotting some results (e.g., a comparison of the limiting cases discussed in section 7).
 As I said, the manuscript is carefully and "in average" clearly written . With this, I mean that most of it is very clear and easy to follow, especially in what concerns the derivation of the results, what will be very useful for the interested readers. Unfortunately, this is not the case for the introduction: it is written in a very precise way which I'm afraid is only accessible for specialists or readers that have already known about fermionic duality before. For instance, most of it refers to duality, and the advantages of using "dualitydictated observables" without the reader clearly knowing what duality actually is. Other concepts like the difference between "transition" and "transport rates", or "a shifted version of the transition rate matrix" are not totally transparent. They become clearer as the text advances, but are confusing in the introduction, I wonder if some of those more technical discussions could be more convenient in the conclusions section than before the concepts are presented. I would encourage the authors to try to make the introduction a bit easier to read for non experts.
 In my opinion, the main motivation of the manuscript is to present the strengths of the dualitybased treatment of the transient dynamics of quantum dot systems. In that case, it would be useful to know better about its limitations. For instance, a sequential tunneling rate equation is used. Can similar arguments apply to higher order expansions (cotunneling), other master equations (Lindblad, Redfield...)? A wideband limit seems to have been assumed. Is the duality affected if tunneling rates depend on energy?
 Rates $\gamma_C$ and $\gamma_S$ turn out to have a physical interpretation but they are introduced as an ansatz in eq. (29). I wonder whether this treatment can be generalized to (even) more complex configurations where additional rates may be relevant and revealed this way, or does one need to have a previous intuition/knowledge of which they are?
Minor things:
 page 10, first paragraph of section 5: is is said that rate variables "can occur by their own" what does this mean?
 footnote 6: there is a ·the" too much ("...but our choice ensures...")
 below eq. (43), doesn't one need to transform $\gamma_s\rightarrow\gamma_s$ as well?
Anonymous Report 1 on 20221114 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2210.04973v1, delivered 20221114, doi: 10.21468/SciPost.Report.6128
Report
In the manuscript, the Authors demonstrate how to construct the fully analytical solution to an interacting quantum transport problem guided by the fermionic duality symmetry. The latter is a symmetry of fermionic open quantum systems previously explored by (part of) the Authors, see Ref.[8,9] of the manuscript, which includes the PT symmetry of the (Markovian) Lindblad dynamics as a special case and is valid in the presence of strong coupling/nonMarkovianity.
The system used for the demonstration is an interacting spinfull singlelevel quantum dot weakly connected to a wideband metallic lead and proximized by a supeconductor with infinitely large superconducting gap.
By exploiting the fermionic duality of the system, the transient energy and charge currents are expressed in terms of stationary expetaction values of operators such as the dot polarization, and the elements of the rate superoperator expressed in a suitable basis, whose choice is enforced by the duality symmetry.
The approach is useful and interesting, the manuscript very well written, and the exposition clear and detailed. The derivations are sound, no flaws are apparent to me.
For these reasons the manuscript deserves in my opinion publication in : SciPost Physics.
I have a few minor comments
 The general conditions under which the fermionic duality holds could be stated for completeness
 The Authors should comment on the practical limitations of the approach in general and in the specific example considered here (e.g. finite gap, nondegenerate case)
 A quantum dot connected to superconducting leads in the case of nonequilibrium, finite gap, and AC drive is addressed in the recent article J. Siegl 2022, arXiv:2205.13936
Author: Maarten Wegewijs on 20221221 [id 3170]
(in reply to Report 1 on 20221114)
We thank the referee for the positive appraisal of the work, in particular of the full exposition.
Response to the referee's suggestions:
The general conditions under which the fermionic duality holds could be stated for completeness
Revision A brief overview of the class of open systems, which exhibit a fermionic duality, is now given in the introduction Section 1. It now clearly mentions the general form and conditions of the fermionic duality, referring to prior works going beyond the weakcoupling assumption made here. Section 4 now clearly refers to this more general framework.
The Authors should comment on the practical limitations of the approach in general and in the specific example considered here (e.g. finite gap, nondegenerate case)
Response In Section 2 we already mention the limitations of the applicability of our specific discussion.
Revision In the revised conclusion, Section 8, we point out our limited use of fermionic duality, namely only as a tool to simplify the solution and analysis of a given quantum master equation (In our case taken from Sothman et al in Ref. 8). We highlight that for finitegap superconducting reservoirs duality has so far not been exploited to simplify the preliminary task of computing the memory kernel from the underlying microscopic model as it has been by Schulenborg et al in the supplement of PRB 93, 081411(R) (2016).
A quantum dot connected to superconducting leads in the case of nonequilibrium, finite gap, and AC drive is addressed in the recent article J. Siegl 2022, arXiv:2205.13936
Revision We appreciate this interesting reference, it is now cited in the revised introduction, Section 1, and summary, Section 8, where we address the referee's previous question regarding the limitations of the approach.
Author: Maarten Wegewijs on 20221221 [id 3171]
(in reply to Report 2 on 20221125)We thank the referee for careful reading of the manuscript despite its admitted considerable length and the positive appraisal.
Response We have long considered this possibility but have decided against it since any example plot of dynamics requires an initial condition which is experimentally wellmotivated and consistent with the approximations in Section 2. Introducing this requires quite some care and involves novel results which cannot be briefly conveyed. They are presented in the cited follow up work [Ref. 19 of revised manuscript] with the application and numerical analysis of all regimes that our concrete analytical result covers.
Revision Concrete examples the limiting cases which the referee suggests were reported already in the cited works, in particular Schulenborg etal in Phys. Rev. B 93, 081411(R) (2016) and Appl. Phys. Lett. 116, 243103 (2020). After Eq. (89) we now refer more clearly to these works for tangible plots and discussion.
Revision We thank the referee for this useful feedback. We have revised and shortened the introduction with the nonspecialist reader in mind.
Revision We now introduced duality immediately from the beginning and furthermore decided to not introduce the concept of dualitydictated observables in the introduction.
Revision We have clarified the difference between the two types of rates.
Revision In line with the referee's suggestion we have removed this paragraph and merged it with the conclusion Section 8.
Revision These were mentioned but the revised introduction now highlights these more clearly.
Revision For weak tunneling the duality can be extended to energydependent spectral density see Schulenborg, PRB 98, 235405 (2018) where it was applied charge and heat transport with normal conducting reservoirs. We now mention this more prominently in the introduction Section 1.
It is good that the referee brings up this point: The derivation of duality presented in the above work in fact also applies to proximized quantum dot Hamiltonians (1) in our present study and this is easily overlooked. It is highlighted now in Section 1 and 8 (also in response to Referee 1).
Response The duality mapping implies a general formal strategy for constructing invariant quantities from linear combinations of the transition rates (instead of mapping one problem onto the other as in previous works). Some of the invariant quantities will have more tangible physical meaning (e.g. as decay rates i.e. eigenvalues of the rate matrix) whereas others characterize the various asymmetries between the various transition rates. This works in general but the details may vary with the complexity of the system.
For weak coupling it can be applied separately to each electrode that independently adds a term to the master equation rate matrix. For normal electrodes this amounts to collecting the symmetric and antisymmetric part of fermifunctions which is easy as illustrated in Schulenborg B 93, 081411(R) (2016).
In the present paper the duality invariant rates were constructed to also cover the transport equations. As a most general ansatz, we chose them to be (anti) symmetric with respect to inversion of the particle index ($\eta$) or the Andreev state index ($\tau$) or both. The particle index $\eta$ can be expected to play a role in these kernel rate combinations, as it naturally occurs in the total charge current. The other Andreev state index $\tau$, however, also occurs in the current kernel rates, when we explicitly account for a transient displacement current into the quantum dot. Only in the stationary limit, this displacement current vanishes due to charge conservation. Therefore we expect the Andreev state index $\tau$ also plays a role.
Finally, we note that we here shown how to construct the invariants after using the standard derivation of the rate matrix. To automatically obtain the rates in dualityadapted form should be possible using the approach of Schulenborg, Saptsov et al presented in the supplement of B 93, 081411(R) (2016) which applies also to infinitegap superconducting leads. This is interesting since it extends beyond the leading order in the coupling to the reservoirs. It is interesting whether for finite gap a similar result can be found. This is now mentioned in the summary Section 8 (also in response to referee 1).
Response It means that the invariance of the rate variables ensures that the corresponding crossrelated terms of the kernel (as required by the duality relation (21)) are represented by one and the same rate, which enables a most compact representation.
Revision We have removed this sentence since it apparently confuses more than it clarifies at this point in the text.
Revision This is a good point: in fact in Eq. (43) one should transform $\gamma_s$ but with the + sign, $\gamma_s \to + \gamma_s$ [Eq. (30)]. This is now pointed out.