Multiplet supercurrent in Josephson tunneling circuits

1) Possibly creating a link between the two communities studying Andreev physics and SIS based quantum circuits.
2) Clearly and pedagogically written.

1) Rather short, could go into more depth at times.

In this work, the authors revisit a three-terminal circuit based on standard SIS junctions under the new viewpoint of multiplet physics, typically studied in a very different regime of junctions with strong tunnel coupling (where Andreev bound states form). The authors consider in particular the dc current response to a commensurate voltage drive, focussing on the quartet line, which, according to the authors, cannot be distinguished (at least qualitatively) from the one studied in the Andreev regime of SNS junctions.

In spite of its brevity, the paper picks up on an interesting question, which has the potential of linking the community studying Andreev physics with the one investigating quantum circuits based on standard SIS junctions. Their work might also open up a number of follow-up questions, which could ignite further research into the transport physics of multi-terminal junctions. I therefore think, a publication can be recommended. However, it seems to me that in some places of the manuscript, there is some potential to go into more depth. I would kindly ask the authors to make corresponding changes (see requested changes below), and consider in addition the following basic thoughts.

1) It seems to me, that the main point of the authors is that in order to see a quartet line, one only needs is a mechanism which renders the trivial cosine dependence of the individual Josephson junction energies nontrivial, resulting in a total energy (or energy spectrum) which has Fourier components of the form cos(phi1+phi2). Of course, both Andreev physics, or the here considered charging energy term do the trick. This claim may therefore be true, but I think it needs to be worked out a little more explicitly, see subsequent thoughts. Also, as a little bit of a side point, I wonder if there is not an analogy to be made to classical sequential electron tunneling in NIN junctions? Here, a single tunnel junction will emit electrons in an uncorrelated Poissonian fashion, however two junctions in series (creating a charge island or a quantum dot in the middle) will create transport statistics deviating from the Poissonian distribution. It would not surprise me, if there were some existing works on classical multi-terminal transport, where nontrivial cross-correlations emerge, which could be considered as a classical analogy to the here considered effect.

2) The above leads me to wonder, whether it is not all essentially a question of creating an energy gap (either via the orbital interference effect in the Andreev physics, or via the charging energy term considered by the authors)? Because, in order to measure the quartet line along the lines proposed by the authors, it seems that there needs to be an adiabatic regime, requiring a finite energy spacing to avoid Landau-Zener transitions. Note that the EC->0 limit the authors discuss in the beginning, even though educational, does actually not fit the bill. For vanishing EC, the energy spacing will vanish, such that the adiabatic transport regime the authors discuss throughout the paper, cannot be reached. Another way of expressing the same is that I doubt that the system in the EC->0 regime would adiabatically change its state (the value of \varphi) to the time-dependent minimum value as the phases phi1,phi2 progress in time. Perhaps this is a caveat worth mentioning in the paper.

3) Ultimately, I expect the question is, whether one can map both kinds of circuits (multi-terminal SNS and multi-terminal SIS) to simply a Hamiltonian depending on two phases, H(phi1,phi2), which encompasses the current operator via the standard definition I_alpha=dH/dphi_alpha, such that there is no true distinguishing feature in the properties of the Hamiltonian in the most generic case (either with respect to symmetry/topology or other). It is however not clear to me, whether the authors claim that they have already shown the analogy to the most general extent, or whether this belongs on the "to-do-list" for subsequent work?

4) As a final thought, could it be, that the analogy between the two systems becomes increasingly nontrivial if one considers not only current expectation values, but in addition current-current correlations, or higher moments of the transport statistics?

I have the impression that some aspects of the above thoughts are already included in the manuscript, even though sometimes only in between the lines. Perhaps the authors could make such or similar thoughts more explicit.

1) I think the minimisation problem in the limit EC=0 has a very simple analytic solution, where \varphi = arctan(x) and x only depends on the junction energies and phi1,phi2, and the resulting current to ground should be of the form of sin(arctan(x))=x/sqrt{1+x^2}. Perhaps there could be some use in showing this result explicitly, since a subsequent expansion of the current in orders of x would much more explicitly show the appearance of the sought-after multiplet terms.

2) Also, as mentioned in the main report, in the limit EC->0, the system probably lacks a mechanism, which would allow for \varphi to "adjust" its state to the updated minimal value. The authors might want to comment on this caveat.

3) I am not sure 100% if I correctly understand the meaning of the quantity plotted in Fig. 3b, the so-called "maximum over offset phase of the average current". Could the authors please explain this in more detail, or with different words?

4) In particular with regard to the additional geometric contribution the authors discuss at the end, I think that an additonal plot of the actual dc quartet supercurrent (with and without the geometric correction) might help to get a feeling for the predicted effect.

5) The predicted indistinguishability of multiplet signals between SNS and SIS multi-terminal devices is a bold claim, and should be justified more deeply than it currently is. Perhaps, the thoughts laid out in the main report could help convey, where I see a possibility to improve the discussion and interpretation of the main result of this work.

1) When reading the abstract and the manuscript, I did not understand
whether or not the authors want to deliver the message that Coulomb
interactions are at work in the three recent experiments reporting the
quartets. At present time, the message of the manuscript is ambiguous
in this respect and the paper could be interpreted as bringing
discredit on previous approaches based on Andreev processes, on the
basis of claims that are not scientifically demonstrated in the
paper. If the authors really want to make those claims, they should
work out their model in the light of a thorough comparison with the
available set of experimental data. They should evaluate all of the
relevant quantities (current and current-current cross-correlations)
that were measured in the experiments and determine how they vary as a
function of the bias voltage. They should also address within their
approach the experimentally relevant large or intermediate values of
the interface transparencies. But maybe, this is just a matter of
communication and choice of the words. Nevertheless, I ask the authors
to clarify in the revised version that their new proposal does not
apply under the same conditions of the contact transparencies as the
Andreev process which are relevant to different devices. In
particular, the authors should remove the third sentence of the
abstract.

Now that I have explained my opinion about those issues of
communication, I move to the following comments on the physics:

2) It seems to me that there is discrepancy between the claim that the
authors make about relevance to experiments (and even suggesting
"better relevance" than Andreev processes, see above) and the complete
absence of discussion of the experimental consequences of their own
proposal on topology. Could the authors deliver a schematic device and
protocol that can be used to probe what they say about topology ?
Should this experiment on topology be considered as being simple or
difficult ? This would be a great help regarding the experimental
audience.

3) I am confused by what the authors call "sequential tunneling". Do
they mean "sequential tunneling of single-electron states" or
"sequential tunneling of Cooper pairs" ? It seems to me that
tunneling one Cooper pair at a time should not produce quartet-like
signal, but maybe I am wrong on this point. A related question is
whether the authors' proposal relies on "breaking Cooper pairs" as in
Cooper pair splitting, or whether each Cooper pair remains bound
throughout the entire quantum transport process. I could well imagine
that Coulomb interaction is fully compatible with breaking two Cooper
pairs e.g. (e up 1, e down 1) and (e up 2, e down 2) which could
sequentially tunnel in the following order in the different leads: e
up 1 / e down 2 / e down 1 / e up 2. Then, this would not be very far
from the quartet-Andreev mechanism, wouldn't it ?

4) Also, in this sequential tunneling picture, I would like the
authors to tell whether there is an upper bound to the delay time
between two tunneling events. Arbitrarily large delay time between two
sequential tunneling events would sound like being paradoxical,
because I expect quantum coherence to be lost on long time
scale. Then, what is the upper bound to the delay time between two
tunneling events ? Is it set by the inverse of the superconducting gap
or by the inverse of the Coulomb interaction or by something else ?

5) Also, I would like the authors to comment about a simple intuitive
picture for entanglement in their proposal. Apparently, nontrivial
topology implies quantum effects and, likely, also entanglement. Are
quantum correlations and entanglement simple consequences of the
number-phase conjugation, as it would be there for two terminals
within the model of the authors ? Is entanglement a consequence of the
two-component Bogoliubov-de Gennes spinor acting like a spin-1/2, as
it is the case for Andreev molecules ? [see references A an B
below]. Is entanglement a consequence of four-fermion exchange as in
the quartet mechanism ?

6) It seems to me that the authors did not consider the following
consequence of their work. As far as I understand, there is the
following classification for nonlocality in three-terminal Josephson
junctions [made with S1-S2-S3 three-terminal junctions where S1 and S3
are laterally connected on S2 through any type of weak link].

I. Equilibrium + absence of Coulomb interactions: nonlocality is
limited by the superconducting coherence length, see the Andreev
molecules [References A-E below].

II. Finite bias voltage on the quartet line + absence of Coulomb
interactions: There is long range nonlocality due to the interplay
between Floquet and quasiparticle propagation [Reference F below], at
separations that exceed the superconducting coherence length by orders
of magnitude.

It seems to me that the authors could add and discuss or demonstrate
the following item for a double Josephson junction:

III. Equilibrium + electrostatics (i.e. the authors' proposal) imply
long range nonlocality in S1-S2-S3 at separation between the
interfaces that can exceed the superconducting coherence length by
orders of magnitude. Only electrostatic coupling between the S1-S2 and
the S2-S3 interfaces is needed and no bias voltage is required.

This discussion of the range of the effect is of interest towards
experimental audience, in a way that is more practical than topology.

In summary, the paper is interesting. I encourage the authors to take
the above comments into account, prepare a revised version and
resubmit their work.

[A] J.D. Pillet, V. Benzoni, J. Griesmar, J.-L. Smirr, and
\c{C}.\"{O}. Girit, Nonlocal Josephson Effect in Andreev Molecules,
Nano Lett. {\bf 19}, 7138 (2019).

[B] J.-D. Pillet, V. Benzoni, J. Griesmar, J.-L. Smirr, and \c{C} \"O
Girit, Scattering description of Andreev molecules, SciPost Phys. Core
{\bf 2}, 009 (2020).

[C] Z. Scher\"ubl, A. P\'alyl and S. Csonka, Transport signatures of
an Andreev molecule in a quantum dot-superconductor-quantum dot
setup, Beilstein J. Nanotechnol. {\bf 10}, 363 (2019).

[D] V. Kornich, H.S. Barakov, and Yu.V. Nazarov, Fine energy splitting
of overlapping Andreev bound states in multiterminal superconducting
nanostructures, Phys. Rev. Research {\bf 1}, 033004 (2019).

[E] V. Kornich, H. S. Barakov and Yu. V. Nazarov, Overlapping Andreev
states in semiconducting nanowires: competition of 1D and 3D
propagation, Phys. Rev. B {\bf 101}, 195430 (2020).

[F] R. M\'elin, Ultralong-distance quantum correlations in
three-terminal Josephson junctions, arXiv:2103.07971, Phys. Rev. B in
press.