Scattering description of Andreev molecules

We have revised our manuscript in order to improve clarity and address the points raised in the two referee reports.
We thank both referees for their comments.

Report 1 suggests that the novelty and significance of our work was undermined by the prevalence in our field of the scattering approach and crossed Andreev reflection (CAR).
The referee should note that it is only recently that one finds an 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).
More importantly the major novelty of our work lies in the following two conclusions which are relevant for designing or understanding experiments:

a) In Figure 2 and accompanying discussion, we show that having multiple channels does not change qualitatively the signatures of hybridization of Andreev Bound States (ABS).
As mentioned in Report 2, "In the context of quantum information, new quantum circuits including hybrid systems (here with superconducting and normal materials) and various geometries are widely investigated."
Our results help direct experimental efforts using such systems.

b) In Figure 3 and accompanying discussion we show that the scattering amplitudes at maximal hybridization correspond to two specific microscopic mechanisms, double elastic cotunneling and double CAR (Fig. 3 and discussion).
We establish quantitatively the relationship between the well known phenomenon of CAR and elastic cotunneling (NSN systems) to the microscopic processes occurring in our SNSNS system, allowing a deeper, intuitive understanding of the Andreev molecule.

Our manuscript clearly states these two points, both in the abstract and in the conclusion.
We agree with Report 2 that although our results may not be ground-breaking, they "are of real interest for the community".
We hope that referee 1 will also be convinced of the novelty and significance of our manuscript.
We address the minor technical comments of both referees and the accompanying changes to the manuscript in detail below.

** Report 1, Issue 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./

** Response to Report 1, Issue 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 original 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 junction since there is a ground connection in the middle. As a consequence, phi_L and phi_R can be set independently.
We have changed 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 original 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 have modified this to:

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

** Report 1, Issues 2 and 3:

/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./

/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./

** Response to Report 1, Issues 2 and 3:
It is entirely correct that in general the longitudinal part of the wavevector can be different for each channel. In Fig. 2, we have chosen $k_F l\pmod{2\pi} = 0$ for simplicity and convenience. This implies that the phase acquired due to propagation only 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 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 already 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 have modified the text at the bottom of page 5 to state, "the longitudinal part of the wavevector can take any value between 0 and $k_F$." and added the following to the next paragraph:

"For convenience and visibility we set $k_Fl$ to constant values in the scattering coefficients while maintaining $k_F l \gg 1$.
In principle each channel may have a different phase factor resulting from interference but such offsets are already included via the random unitary scattering matrices $S_{L,R}$ and do not change the results qualitatively."

** Report 1, Issue 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./

** Response to Report 1, Issue 4

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.

To clarify how disorder at the superconductor or weak link-superconductor interface could be incorporated into S_L and S_R, we have added the following sentence at the bottom of page 2, "These matrices can in principle include additional scattering at the superconductor-weak link interface."

** Report 2, Issue 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./

** Response to Report 2, Issue 1

We agree that our results are of relevance to experiments with gated semiconducting nanowires and state this in the manuscript.
In experiments with such nanowires the number of highly transmitting channels is often in the single digits.
In our calculations with random scattering matrices we chose N = 20 in order to ensure that we would have a comparable number of highly transmitting channels, as shown in Figure 2.
In addition, for larger N, visibility in the spectra is reduced.
To clarify that there is no qualitative difference in the spectra with increasing N, we have added the following sentence at the end of section 3:
"Even though the spectra will become more dense as the number of channels is increased, this symmetry breaking will be relevant experimentally as long as $l\lesssim\xi_0$."

** Report 2, Issue 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./

** Response to Report 2, Issue 2

For semiconducting weak links in which the carrier density is much lower than in the metallic electrodes, the inverse proximity effect is negligible.
However it is true that the quality of the superconductor-semiconductor interface may play a strong role in hybridization.
In our formalism it is possible to add additional scattering at this interface by incorporating it into the matrices S_L and S_R, and as mentioned above, we have modified the text accordingly: "These matrices can in principle include additional scattering at the superconductor-weak link interface."
Although a more realistic 3D treatment of this interface is out of the scope of our manuscript, we provide references to recent preprints by Nazarov et al which address this question.