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Solution of master equations by fermionic-duality: Time-dependent 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|
|Preprint Link:||https://arxiv.org/abs/2210.04973v1 (pdf)|
|Date submitted:||2022-10-12 17:53|
|Submitted by:||Wegewijs, Maarten|
|Submitted to:||SciPost Physics|
We analyze the time-dependent 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 non-local 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 large-gap superconductor which is weakly probed by charge and heat currents into a wide-band normal-metal electrode. We derive the complete time-dependent analytical solution of this problem involving non-equilibrium 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.
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:2210.04973v1, delivered 2022-11-25, doi: 10.21468/SciPost.Report.6194
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
1- (maybe contradictory with strength #1): the manuscript is very long
2- the introduction is difficult to read for non experts.
The manuscript "Solution of master equations by fermionic-duality: Time-dependent 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 single-ĺevel quantum dot (by being coupled to a infinite-gap 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. ). 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 "duality-dictated 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 duality-based 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 wide-band 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?
- 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?
- Cite as: Anonymous, Report on arXiv:2210.04973v1, delivered 2022-11-14, doi: 10.21468/SciPost.Report.6128
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/non-Markovianity.
The system used for the demonstration is an interacting spinfull single-level quantum dot weakly connected to a wide-band 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, non-degenerate 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