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Scattering description of Andreev molecules

by J. -D. Pillet, V. Benzoni, J. Griesmar, J. -L. Smirr, Ç. Ö. Girit

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Submission summary

Authors (as registered SciPost users): Çağlar Girit · Jean-Damien Pillet
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-02-26 01:00
Submitted by: Girit, Çağlar
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


An Andreev molecule is a system of closely spaced superconducting weak links accommodating overlapping Andreev Bound States. Recent theoretical proposals have considered one-dimensional Andreev molecules with a single conduction channel. Here we apply the scattering formalism and extend the analysis to multiple conduction channels, a situation encountered in epitaxial superconductor/semiconductor weak links. We obtain the multi-channel bound state energy spectrum and quantify the contribution of the microscopic non-local processes leading to the formation of Andreev molecules.

Current status:
Has been resubmitted

Reports on this Submission

Report 2 by Francois Lefloch on 2020-3-20 (Invited Report)

  • Cite as: Francois Lefloch, Report on arXiv:2002.10952v1, delivered 2020-03-20, doi: 10.21468/SciPost.Report.1588


This paper is a multi-channel extension of a previous paper published by the same authors to theoretically described the hybridization of ABS (Andreev Bound States) in three terminal Josephson junctions. Their approach is naturally based on the scattering formalism that includes both normal scattering in the junctions and Andreev reflections mechanisms at each superconducting interface.
In the context of quantum information, new quantum circuits including hybrid systems (here with superconducting and normal materials) and various geometries are widely investigated. In that sense, the results obtained by the authors, if not being ground-breaking’s, are of real interest for the community.
The paper is very well organized. In the limit of the assumptions made by the authors (see 1rst referee’s report) , the description of the model and the underlying physics phenomenon are very well described. This article a very pedagogical and clear.
Few comments that, in my opinion, limit the overall impact of this publication.
1 - The authors present results for a fixed number of channels (N = 20). It would have been interesting to discuss the effect of the number of channels in more details. This can be of real use for practical realizations with gated semiconducting nanowires.
2 - The origin of hybridization is due to (inverse) proximity effect in the central superconductor and explains the L/xi decay dependence. This inverse proximity effect is known to be strong when the transparency at the S/N interface is good (close to 1). But at the same time, the superconducting gap is locally reduced. On the contrary, when the transparency is smaller, the superconducting gap is restored but (inverse) proximity effect is less efficient.
It is not clear to me how the quality of the S/N interface will change the results but clearly the interface transparency is a real issue in experiments.
Note that the situation here seems different than in [22] as transport occurs necessarily though the central 3 D superconductor.

  • validity: good
  • significance: good
  • originality: ok
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2020-3-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.10952v1, delivered 2020-03-10, doi: 10.21468/SciPost.Report.1565


The authors consider a superconducting wire interrupted by two weak links. They study the spectrum of the central region, which they refer to as an "Andreev molecule". The manuscript builds on an earlier paper by the same group of authors (arXiv:1809.11011) in which they studied a single-mode wire, while here they consider the multi-mode case. The scattering description which they employ is routine in the field and the phenomenon of crossed Andreev reflection which is obtained is also very well studied and understood. This limits the novelty and significance of the work, but should not by itself prevent publication.

I have, however, also several concerns regarding the scientific validity, which do stand in the way of publication.

1. The central region of the superconducting wire (the "Andreev molecule") is grounded, and the authors say that this allows them to fix the superconducting phase of the central region at 0. Here they are confusing voltage bias and flux bias. "Grounding" means that the voltage is zero, it does not mean that the phase is zero. The phase difference phi_L-phi_R between the outer ends of the wire can be fixed by a flux bias, but the phase of the central region should then be determined selfconsistently, it is not fixed by the voltage ground.

2. The authors assume that the electron and hole propagate through the central region with longitudinal momentum k_e,h = k_F ± i/ξ. In a multi-mode junction the longitudinal momentum can vary between 0 and the maximal value of k_F. Setting it equal to k_F for all modes does not seem justified.

3. On page 6 the authors write "we ignore fast phase oscillations in t_S and r_S arising from the small Fermi wavelength by fixing k_F l arbitrarily and independent of l". This assumption makes no sense to me. In a phase coherent treatment these phase oscillations should play a crucial role.

4. The plots are calculated by "randomly generated symmetric unitary matrices S_L and S_R". These matrices should represent the weak link, which I presume is a tunnel junction. Since no disorder is included in the superconductor, the randomness needed for this assumption is not present and I do not understand the justification for this choice.

  • validity: low
  • significance: low
  • originality: low
  • clarity: ok
  • formatting: good
  • grammar: good

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Author:  Çağlar Girit  on 2020-03-19  [id 771]

answer to question

Thank you for reading our manuscript and providing feedback.

It is correct that the scattering description we employ is widely used to model Josephson junctions and does not represent a technical challenge. Obtaining expressions of our scattering matrices is relatively straightforward but it is not the main message of our manuscript. The novelty of our work lies in reporting what a scattering description has to tell us about Andreev molecules:

a) Having multiple channels (Fig. 2) does not change qualitatively the spectrum of hybridized Andreev Bound States (ABS) compared to what was reported in our previous work for a single channel (arXiv:1809.11011),

b) The scattering amplitudes (Fig. 3) obtained for configuration of maximal hybridization, phi_L = - phi_R and phi_L = phi_R, demonstrates that the formation of bound states is almost exclusively dominated by a specific microscopic mechanism, dEC and dCAR respectively.

These conclusions are important for the experimental realization of Andreev molecules since it provides the appropriate conditions for observation of novel phenomena such as avoided crossings between ABS in the energy spectrum and non-local Josephson Effect. On top of that, it gives more intuition about the nature of the electronic states in an Andreev molecule.

We also agree that the Crossed Andreev Reflection (CAR) phenomenon has been extensively studied (we give [17] as a reference since it is one of the pioneering works but we are happy to add more references if necessary). We simply mention CAR in the introduction and believe that our subsequent description in terms of successive individual microscopic scattering events is enlightening and complementary.

We also would like to mention that, though the result is simple to obtain, it is only recently that one can find explicit and complete expression of the scattering matrix for a superconductor of finite thickness (see for example an initial version of our previous work arxiv:1809.11011v1, or, in 2019, PRR 1 033212 and arxiv:1912.10307). Our work only tackles specific questions about Andreev molecule, but we believe it brings novel insights and tools to understand this physics.

We have carefully read your 4 concerns and we believe that none of them call into question the scientific validity of our work. However in order to improve the clarity of the manuscript we propose to make the following changes which address your comments:

  1. The ground has nothing to do with fixing the superconducting phase of the central superconductor. The central superconducting part is grounded in order to allow the left and right junction to carry different supercurrent while preserving current conservation. In practice, this allows to independently current bias the left and right junction in order to measure their respective critical currents, or alternatively to current bias one of the junction while the other is closed by a superconducting loop such that it can be phase biased with a magnetic flux. These two setups allow to detect the non-local nature of ABS in the Andreev molecule (see our first manuscript arXiv:1809.11011).

In our work, the phase of the central superconductor is arbitrarily set to zero. By gauge invariance, this does not imply any loss of generality. What matters physically are the phase differences between superconductors. Since the central phase is chosen to be zero, these phase differences are given by phi_L and phi_R.

We believe the confusion could come from the phrasing in the caption of Fig. 1:

“The ground connection allows applying the phase differences phi_L,R independently”

What we meant here is that the current can be different in each junctions since there is a ground in the middle. As a consequence, phi_L and phi_R can be set independently. We propose changing this sentence to:

“The ground connection allows applying the phase differences phi_L,R independently by flowing different current through each junctions”

The confusion could also come from a similar sentence in the text page 5:

“Because of the ground connection, there are effectively two loops connecting the left and right superconductors which allow tuning phi_L,R independently with external magnetic fields”

This is indeed confusing since the ground is actually not necessary to form loops between superconductors. We propose modifying it with:

“Each junction can be shorted by a superconducting loop which allow tuning phi_L,R independently with external magnetic fields.”

  1. It is entirely correct that k_F can be different for each channel and, in general, this will be the case. In Fig. 2, we have chosen, for simplicity, that k_F*l = 0 (mod 2pi). This does not imply that k_F is the same for each channel, it simply implies that the phase acquired due to propagation is the same for each channel. There is no physical reason to make this particular choice. In an actual experiment, these phases would actually take random values between 0 and 2pi for each channel. This is something we can easily introduce in our calculations but it does not change qualitatively the global shape of our spectra (the main difference would be a variation of the amplitude of the ABS avoided crossings). Moreover, since the normal scattering matrices S_L and S_R are chosen randomly, they introduce different scattering phases, which prevents peculiar situations where all channels would host constructive or destructive interference maximizing or minimizing the magnitude of each avoided crossings.

We propose adding the following sentence to the caption of Fig. 2:

“Note that k_F might be different in each conduction channel.”

  1. We agree that the sentence "we ignore fast phase oscillations in t_S and r_S arising from the small Fermi wavelength by fixing k_F l arbitrarily and independent of l" (pg 6) is unclear and might lead to some confusion.

Indeed interference plays an important role in the construction of bound states. If one could gradually increase the length l of the central superconducting part while keeping k_F constant, this would lead to very fast oscillations for most quantities. For example, the width of avoided crossings in the spectra of ABS would oscillate to their full magnitude for the dCAR at phi_L = phi_R and partially for the dEC at phi_L = -phi_R. For the dCAR, when the avoided crossing fully closes, it means there is no hybridization, which leads to the cancellation of certain coefficients.

We have attached two plots (dEC and dCAR) with k_F = 10/xi_0 in our reply to show the effect of interference. In practice, it would be much bigger but the oscillations would be too fast to be visible on the plots.

In our manuscript, we made the deliberate choice to maintain k_F*l = constant while varying l in Fig. 3 in order to show how bringing two Josephson junctions close to each other, within a few superconducting coherence lengths, leads to the transition from localized ABS to hybridized ABS. This also shows how dEC and dCAR gradually take over local microscopic mechanisms. We believe this transition is more visible without the oscillations caused by interference, but that is purely a pedagogical choice.

In order to make this clearer, we propose changing the following sentence:

“In addition we ignore fast phase oscillations in t_S and r_S arising from the small Fermi wavelength by fixing k_Fl arbitrarily and independent of l while maintaining k_F*l>>1.”


“Moreover, in order to improve visibility, we remove fast oscillations of the scattering coefficients arising from interference by fixing k_F*l to constant values. This corresponds to plotting the k_F-independent envelope of the coefficients."

  1. The randomly generated scattering matrices S_L and S_R indeed represent the weak link. The weak link is not necessarily tunnel junctions (for which the non-local Josephson Effect is weak) but could be any quantum conductor. Since our approach is 1D, our model preferably describes one-dimensional quantum conductors, such as carbon nanotubes or semiconducting nanowires, but in principle S_L and S_R could also describe any quantum conductors in 2 or 3 dimensions.

The reason we choose random S_L and S_R for quantum conductors with many conduction channels (Fig. 2) is not to include disorder but rather to keep a very generic approach with no assumption made on the nature of the conduction channel. For example, they have no reason to be the same than in the central superconductor.

The disorder of the superconducting region is not included in the matrix S_S. It could perfectly be taken into account in our approach by adding in series with S_S two normal scattering matrices S_S^L and S_S^R, on its left and on its right. They would describe scattering at the left and right interfaces of the superconductor and disorder in between. Nevertheless, it is possible to combine S_S^L and S_L (or S_S^R and S_R) into a single scattering matrix describing the left (resp. right) quantum conductor, which would be formally equivalent to what is done in our manuscript.

A better description of disorder could be done using a tight-binding approach as was done in the initial version of our previous work (arxiv:1809.11011v1) or using a semi-classical approach in more recent works by the Nazarov group. These results shows that there is no qualitative change in the formation of an Andreev molecule when one introduces disorder. Disorder mainly leads to a weakening of the ABS hybridization because of a reduction of the superconducting length xi (xi is replace by sqrt[xi*mfp] where mfp is the mean free path). However, it should remain measurable as long as the junction are separated by a distance comparable to this new xi.