Double-Fourier engineering of Josephson energy-phase relationships applied to diodes

We thank both referees for their overall positive evaluation. We have implemented all the requested changes. We also provide the redlined manuscript with the changes highlighted at this URL: https://surfdrive.surf.nl/files/index.php/s/NwTnLoOVPoXTTGD

Ref 1:

One of the central predictions of this manuscript is, in fact, that the energy-phase relation (EPR) of two tunnel Josephson junctions connected in series is the same as that of a single junction but with high transparency. Hence, an effective transparency \tau can be defined, which, as explained by the authors, is 1 for identical junctions and 0 for highly asymmetric Josephson energies E_J1 >> E_J2. Cases of high \tau allow for obtaining higher-order harmonics in the EPR and realizing Josephson potentials of much more varied shapes than the simple cosine of a single tunnel junction, simply by assembling these basic units in parallel. I do not think this prediction is truly novel; after all, this building block is a Cooper pair transistor, as studied for many years. Nevertheless, it is employed in a new, intelligent and experimentally feasible strategy to design arbitrary Josephson potential.

We agree with the referee that the current-phase relation of two Josephson tunnel junctions is not truly novel, and indeed it has been studied for many years. We would like to clarify, though, that we consider a classical circuit $E_J \gg E_C$, and therefore not a Cooper pair transistor. While the current-phase relation of two tunnel junctions in series was studied in e.g. our Ref. 32, to the best of our knowledge, its correspondence to that of an SNS junction was not stated explicitly. The referee is also correct in saying that the main focus of our work is utilizing this current-phase relation to construct arbitrary ones.

For this reason, I believe it might be interesting for the readers to mention in the manuscript that their work is valid in the case of $E_J \gg E_C$, and maybe say a few short words on the role of $E_C$ in the DC properties of the device.

1) Discuss briefly the role of the charging energy for the DC properties of the device. Possibly explain that the method presented is only valid for E_J much larger than E_C.

Following the referee's suggestion we have now: - Explicitly stated that we neglect the charging energy when introducing the setup and stated that we expect that the residual charging energy will not qualitatively affect our conclusions. - Added a statement that also a Cooper pair transistor has the same current-phase relation to the discussion of current-phase relation correspondence.

2) Define $\phi_{min}$ and $\phi_{max}$ more explicitly.

We now explicitly define these in text.

3) The caption of figure 7 should start with "(a)" instead of "(Engineering a)". 4) In the caption of figure 8, (a) and (b) are not explicitly mentionned.

We thank the referee for pointing these out, and we have updated the captions accordingly.

Ref 2.

1) In fact, a recent preprint (ref. [26]) makes a rather similar observation and comes to similar conclusions. It would be fair to comment in the present manuscript on the relation to that work. E.g. the procedure to maximize the diode efficiency are almost the same.

We thank the referee for their remark. We have added a sentence to the introduction, clarifying the relation of our work to Ref. 26.

2) In Fig. 6 it is quite remarkable that disorder seems to have a strongly negative effect on the diode efficiency for large junction numbers. Maybe the authors can comment on this kind of counter-intuitive behavior

We now added a statement to the manuscript explaining that the increased disorder sensitivity for large junction numbers is due to unsuppressed root-mean-square fluctuations of the junction strengths.