Tailoring supercurrent confinement in graphene bilayer weak links

The Josephson effect is one of the most studied macroscopic quantum phenomena in condensed matter physics and has been an essential part of the quantum technologies development over the last decades. It is already used in many applications such as magnetometry, metrology, quantum computing, detectors or electronic refrigeration. However, developing devices in which the induced superconductivity can be monitored, both spatially and in its magnitude, remains a serious challenge. In this work, we have used local gates to control confinement, amplitude and density profile of the supercurrent induced in one-dimensional nanoscale constrictions, defined in bilayer graphene-hexagonal boron nitride van der Waals heterostructures. The combination of resistance gate maps, out-of-equilibrium transport, magnetic interferometry measurements, analytical and numerical modelling enables us to explore highly tunable superconducting weak links. Our study opens the path way to design more complex superconducting circuits based on this principle, such as electronic interferometers or transition-edge sensors.

in Ic as a function of carrier density and type in bilayer graphene (e.g. M.T. Allen et al., Nat. Phys. 12, 128-133 (2016)) and that a systematic study of the spatial variation of Ic is addressed, I think that it does not meet the elements of novelty and originality necessary for publication in Nature Communications and therefore I recommend its publication in a more specialist journal. Lastly, I think the authors may find useful to address the following minor points which would make the text of the manuscript clearer to the reader: 1. In paragraph on page 2 starting with "As a consequence, the bipolar region.." the text would be clearer if the authors explained all the labels used (NPnN, NPpN etc.) -which are actually clearly defined only in the caption of Figure 2. 2. The scale bar in Figure 2c does not look correct, as it seems that Figure 2c should contain some areas coloured in green in Figure 2b -which should correspond to a resistance value of 0.3-0.4 kΩ according to the scale bar in Figure 2b. 3. In the caption of Figure 3e, I think the authors mean 'differential resistance' other than just 'resistance'. 4. In the paragraph starting with "First, a beating pattern appears.." on page 5 the authors should clarify that the evolution from a Fraunhofer-like pattern into a bell-shaped Ic Vs. H curve happens for increasing gate voltages, as this only becomes clear to the reader after looking the captions of Figure 4a. 5. In the caption of Figure 4, the split-gate voltage values should be negative, in order to be consistent with what reported in Figures 4a and 4b as well as in Figure 1c. 6. At the end of page 6, the authors explain how Icnorm. is calculated, but they never define Icnorm. explicitly. This should be done, as Icnorm is used for the first time in the caption of Figure  5 but without any definition. 7. In the last section of the supplementary information, the authors state that the temperature used in the numerical simulation is higher than the actual measurement temperature, as the relevant scale is the thermal length. Using a lower temperature in the numerical code, however, should have an effect at least on the amplitude of the calculated Ic and perhaps on its spatial dependence as well. It would help if the authors could further clarify on this point and perhaps add to the SI file another Icnorm versus applied field contour plot, similar to that shown in Figure 5c, but calculated at a different temperature (e.g. kBT = ∆0/10).
Reviewer #3 (Remarks to the Author): The authors study encapsulated bilayer graphene contacted by superconducting titanium/aluminium electrodes. They fabricate a split gate on top of the sample (Figure 1), which eventually allows them to study superconducting current through a 1D constriction. They measure the normal state resistance and the critical current vs the back gate and the split gate in Figure 2. Different regimes are identified, depending on whether the regions of the channel and that under the split gate are P or N doped. In particular, the maps of Figure 2 allow them to find the range of parameters in which the region under the gate is gapped and depleted, but the channel is doped and conductive. This situation occurs when the back gate voltage is high enough (beyond about 6 V), and the split gate voltage is negative enough to shut off the conductance bypassing the channel formed by the spit gate. In this regime, the residual critical current of the order of 10-100 nA is observed (Figure 3d,e).
At this point the author make a rather ambitious claim that the critical current is quantized: "Importantly, despite the absence of signs of 1D subband formation while shrinking the constriction in the normal state, the critical current decreases in a step-wise fashion (see Fig. 3e) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]." I totally agree with them that without observing the quantized conductance, it is strange to expect the critical current quantization. To substantiate their claim, I believe that it is crucial to demonstrate this behavior in more than one sample. Also, it would be important to show explicitly how the map of Figure 3e is converted to a graph of critical current vs gate voltage. Finally, the normal state resistance in the range shown in Figure 3e appears to be about 500 Ohm, corresponding to conductance of about 50 e^2/h. It seems inconceivable that the critical current would be quantized and show only a few first few quantized steps as it may be suggested by the shape highlighted in Figure 3e.
The rest of the paper ( Figure 4) and most of the supplementary information describe the magnetic interference pattern. The similarity between the data and numerical simulations appears very conclusive.
PS. The title "Tailoring supercurrent confinement in graphene bilayer weak links" leads me to believe that multiple samples were measured. If so, could the authors clearly indicate which figures correspond to the additional samples?
Reviewer #1: "The authors investigated the supercurrent in bilayer graphene weak links using a QPC-like geometry. They used the split-top gates to create one-dimension constrictions. The ability to control the dimension of the weak links from two to one dimension allows them to tune critical currents as well as density profile of the supercurrent. The authors also studied the interference pattern of the supercurrent vs magnetic field as the weak link dimension changes from two to one dimension by varying split gate voltage and back gate voltage. The results agree well with their theoretical model." 1-"The ability to create 1D constriction in this experiment relies on an insulating state in BLG which can be induced by displacement field. However, such insulating state is very hard to achieve in BLG due to small band gap generated by displacement field and potential fluctuations from disorders. The authors should estimate the size of the band gap from displacement field (see Nature 459, 820-823 (2009)) and the potential fluctuation from a residue charge inhomogeneity and see if the band gap is larger than the potential fluctuation or not." We thank the reviewer for pointing out the importance of the comparison between the potential fluctuation and the band gap size. Indeed, we see that the expected values are in complete agreement with our experiment. We provide quantitative estimation of the gap induced by the displacement field that breaks the lattice symmetry underneath the split-gate using the formula provided in ref [31][32] (from the main text) and obtain a band gap of E g 8 5.4 meV at back and split-gate voltage values of 8 V and -7.6 V respectively which corresponds to a displacement field of D ~ 0.56 V/nm following Zhang et al. (ref. [36] from the main text). If we compare the band gap value to the potential fluctuation from our residual charge inhomogeneity n res ~ 2.6x10 10 cm -2 , we find that this energy value corresponds to an excitation of ~ 1 meV. This energy value is much smaller than the estimated band gap. Therefore, we have decided to integrate a short discussion in the supplementary information (page 4-5 and ref [8,9] have been added in the supplementary information too): " We provide quantitative estimation of the gap induced by the displacement field that breaks the lattice symmetry underneath the split-gate using the formula provided in [8,9]. The gap should be larger than the potential fluctuation coming from the residual charge carrier inhomogeneity n res ~ 2.6x10 10 cm -2 which corresponds to an excitation of E ~1 meV (a band gap of 1 meV can be obtained by applying back and top gate values of for example V BG = 0.14 V and V SG = -0.13 V respectively). At back and split gate values of V BG = 8 V and V SG = -7.6 V respectively, which correspond to a displacement field of D~ 0.56 V/nm following Zhang et al. [15], we obtain an energy band gap of E g ~ 85.4 meV, i.e. a value much larger than the potential fluctuation." 2-"From Fig. 3a, b in the supplementary information, Fabry-Perot oscillations exists in NP_nN regime and the period of these oscillations corresponds to the cavity underneath the split gates. Doesn't this mean that BLG under the split gates in the NP_nN regime is not completely insulating since electrons can still propagate through it. Therefore, the device does not exhibit 1D constriction in the NP_nN regime. " Fig. 3 in the supplementary information displays the differentiated gate map dG/dV BG (V BG ,V SG ) in the normal state which highlights fine Fabry-Pérot oscillations with periodicities corresponding to the cavities in which the carriers interfere. In the NP p N region interferences occur mainly in the cavities formed by the split-gate. The interferences are visible as parallel lines in the gate map which are tuned by both back-and top gate (shown in red in Fig. 3a). It is true that the NP n N region does show features which look like Fabry-Pérot interferences as well. While the almost horizontal parallel lines can be identified as oscillations between the contacts (cavity size of about 1 µm), they are nearly independent of V SG , indicating trajectories through the channel of the QPC which are only weakly tuned by the stray fields of the split-gate. If these interferences would take place by transport of charge carriers through the top-gated cavity, the interferences should be tuned in a stronger way by V SG as one can see in the PPP (and NP p N) region. Furthermore, we can identify non-parallel larger stripes in the NP n N region. When we extract the cavity size by assuming Fabry-Pérot interferences, we find < 150 nm at a back-gate value of 8 V which is much less than the size of the cavity corresponding to the split-gate width ≈ 300 nm. So we conclude that these features do not correspond to the cavity created by the split-gate. The sudden drop of the supercurrent together with the magneto-interference pattern transition combined with our analytical and numerical models confirms that the supercurrent is confined by the electrostatically defined constriction and that no current passes underneath the split-gate.
3-"The authors attribute the step-wise decrease of the critical current to ballistic supercurrents through QPC (Fig 3e in main text). However, they do not observe any signature of 1D subbands in the normal state. Do you have any explanation why 1D subbands would be more pronounced in the superconducting state? Can the "step-wise decrease" be the consequence of Fabry-Perot oscillations (It just happened to look like steps when in fact it's due to oscillations of resistance in the normal state). Figure 3d in the supplementary information also shows oscillations in critical current with similar periods as the one in Fig 3e of the main text." Here reviewer #1 (and to some extent reviewer #3) points out the fact that the step-like features observed in the NP n N area may appear due to the presence of Fabry-Pérot interferences. As aforementioned, these oscillations of the differentiated conductance which could be seen as Fabry-Pérot interferences do not correspond to any possible formed cavity within our device. As pointed out by referee #1, each of these features seen in the critical current correspond to one in the normal state, unlike what we mentioned in the main text. Therefore, we have decided to replace in the main text the following sentence: "Importantly, despite the absence of signs of 1D subband formation while shrinking the constriction in the normal state, the critical current decreases in a step-wise fashion (see Fig.  3e) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]. " by: "We note that, on the presented device, we observe features visible in the normal and the superconducting state which could be related to quantized conductance and supercurrent (see supplementary information) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]." We have measured these features in our smaller device (with a constriction defined by the split-gate of w ~ 60nm), but not in our larger device (with a constriction defined by the splitgate of w ~ 150nm). It sounded important to us to mention these features in the main text but since the data are not so clear, a conclusive explanation remains beyond the scope of our paper. Since it is not the real focus of our work, we have decided to remove Fig. 3e and place it together with normal and superconducting curves in the supplementary information (now Fig. 6) as well as a series of similar plots under various back gate conditions (in Fig. 7) in a new subsection entitled "C. Any signs of quantized supercurrent?" from part "iV. Additional superconductivity data" and reads: "In our experiments, we clearly observe Fabry-Pérot interferences highlighting ballistic transport of the charge carriers all across the length of the device. Therefore, one might expect to observe quantized conductance while inducing a constriction in a two-dimensional system [21,21]. This phenomenon has been extensively studied in particular in AlGaAs/GaAs heterostructures [23]. Until now, step-like features in the conductance have been observed in graphene (both single and bilayers) but none of them quantized in the expected value of 4e 2 /h [24-33] (the prefactor 4 refers to the spin and valley degeneracy). For our configuration using only one overall back-gate and a local split-gate, both the tuning of the Fermi energy and the opening of the gap cannot be fully independently controlled. At V BG and V SG sufficiently large to form the constriction and confine the supercurrent, the charge carrier density within the constriction which is mostly influenced by the back-gate compared to the stray field generated by the split-gate, might appear to large and the confinement not strong enough to clearly form 1D subbands and therefore both quantized conductance and supercurrent appear to be hard to observe. Therefore the picture drawn for more conventional semiconductors might not be applicable [34][35][36]. In our case we observe features in the normal and superconducting state conductance although non-quantized, here at V BG = 8.6 V (see Fig. 6a and b). At this back-gate voltage value and V SG ~ -8.5 V the stray field generated by the split-gate starts to balance the action of the back-gate on the charge carrier density. As we can see, the minimum conductance reaches 24e 2 /h corresponding to 6, four times degenerated, opened channels and a resistance of ~ 1 k. We note that these features are observable in a large back-gate range (see Fig. 7). What we define as signs of quantized supercurrent (together with signs of quantized conductance) was measured in our shorter devices with w ~ 65 nm split-gate distance but not in our longer device with a wider constriction (w ~ 150 nm split-gate distance, not shown here) which showed all properties of supercurrent confinement (in both amplitude and magneto-interferometric pattern) that we present in this work." A large set of new references has also been added.
New Step-like features in the supercurrent under various back-gate conditions. Critical current I c and conductance G (top) and differential resistance dV/dI(V SG ,I) maps (bottom) showing step-like features in the supercurrent for various back-gate voltages, from left to right V BG =8.6, 9, 9.1, 9.2 and 9.3V respectively.  )) and that a systematic study of the spatial variation of Ic is addressed, I think that it does not meet the elements of novelty and originality necessary for publication in Nature Communications and therefore I recommend its publication in a more specialist journal."

4-"The authors mention
We disagree with reviewer #2 on the critics regarding the novelty of our work presented in our manuscript. The reviewer first cites P. Rickhaus et al., Nano Lett. 15 (9) [42] in our manuscript). In these works, the supercurrent is tuned solely by an overall back-gate. None of the references cited by reviewer #2 do involve local gate control of supercurrent in its amplitude and spatial distribution.
Here we have used a quantum point contact geometry to show that supercurrent could be confined which has not been studied anywhere else to our knowledge. In addition, we demonstrate that gapped bilayer graphene may not be disturbed by any edge current.
"Lastly, I think the authors may find useful to address the following minor points which would make the text of the manuscript clearer to the reader: 1-In paragraph on page 2 starting with "As a consequence, the bipolar region.." the text would be clearer if the authors explained all the labels used (NPnN, NPpN etc.) -which are actually clearly defined only in the caption of Figure 2." We follow the request of reviewer #2 and insert the text which was in the caption of Fig. 2 in the paragraph which describes the different doping regions in the QPC geometry. Now the first sentence of the last paragraph of page 2 reads: "The schematics in Fig.2d summarize the different scenarios which govern the behaviour of such an electrostatically induced constriction, i.e. the formed 1D constriction area NP n N, the unipolar regime NNN and the non-uniform NP p N junction." 2-"The scale bar in Figure 2c does not look correct, as it seems that Figure 2c should contain some areas coloured in green in Figure  3-"In the caption of Figure 3e, I think the authors mean 'differential resistance' other than just 'resistance'." Reviewer #2 is correct. We have changed the caption text with "differential resistance" instead of "resistance".
4-"In the paragraph starting with "First, a beating pattern appears.." on page 5 the authors should clarify that the evolution from a Fraunhofer-like pattern into a bell-shaped Ic Vs. H curve happens for increasing gate voltages, as this only becomes clear to the reader after looking the captions of Figure 4a." To clarify our explanation, we have revised the sentence which describes the evolution of the magneto-interference pattern as follows: "By increasing V SG , i.e. the confinement, a progressive change of the interference pattern is observed as the split-gate is tuned and the 1D constriction forms. First, a beating pattern appears, resembling Fraunhofer-like interference (upper panel) when the system remains two-dimensional." 5-"In the caption of Figure 4, the split-gate voltage values should be negative, in order to be consistent with what reported in Figures 4a and 4b as well as in Figure 1c." We have changed the split-gate voltage values to negative.
6-"At the end of page 6, the authors explain how Icnorm. is calculated, but they never define Icnorm. explicitly. This should be done, as Icnorm is used for the first time in the caption of 7-"In the last section of the supplementary information, the authors state that the temperature used in the numerical simulation is higher than the actual measurement temperature, as the relevant scale is the thermal length. Using a lower temperature in the numerical code, however, should have an effect at least on the amplitude of the calculated Ic and perhaps on its spatial dependence as well. It would help if the authors could further clarify on this point and perhaps add to the SI file another Icnorm versus applied field contour plot, similar to that shown in Figure 5c, but calculated at a different temperature (e.g. kBT = ∆0/10)." We have repeated the simulations, this time at k B T=∆ 0 /20. No strong variations have been seen in particular on the interference pattern transition corresponding to a change in spatial distribution of the supercurrent. At lower temperature, small features are slightly more pronounced. We have changed Fig.5c in the main text as well Fig.10  Reviewer #3: The authors study encapsulated bilayer graphene contacted by superconducting titanium/aluminium electrodes. They fabricate a split gate on top of the sample (Figure 1), which eventually allows them to study superconducting current through a 1D constriction. They measure the normal state resistance and the critical current vs the back gate and the split gate in Figure 2. Different regimes are identified, depending on whether the regions of the channel and that under the split gate are P or N doped. In particular, the maps of Figure 2 allow them to find the range of parameters in which the region under the gate is gapped and depleted, but the channel is doped and conductive. This situation occurs when the back gate voltage is high enough (beyond about 6 V), and the split gate voltage is negative enough to shut off the conductance bypassing the channel formed by the spit gate. In this regime, the residual critical current of the order of 10-100 nA is observed (Figure 3d,e).
At this point the author make a rather ambitious claim that the critical current is quantized: "Importantly, despite the absence of signs of 1D subband formation while shrinking the constriction in the normal state, the critical current decreases in a step-wise fashion (see Fig. 3e) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]." I totally agree with them that without observing the quantized conductance, it is strange to expect the critical current quantization. To substantiate their claim, I believe that it is crucial to demonstrate this behavior in more than one sample. Also, it would be important to show explicitly how the map of Figure 3e is converted to a graph of critical current vs gate voltage. Finally, the normal state resistance in the range shown in Figure 3e appears to be about 500 Ohm, corresponding to conductance of about 50 e^2/h. It seems inconceivable that the critical current would be quantized and show only a few first few quantized steps as it may be suggested by the shape highlighted in Figure 3e.
The rest of the paper (Figure 4) and most of the supplementary information describe the magnetic interference pattern. The similarity between the data and numerical simulations appears very conclusive.
PS. The title "Tailoring supercurrent confinement in graphene bilayer weak links" leads me to believe that multiple samples were measured. If so, could the authors clearly indicate which figures correspond to the additional samples?
Reviewer #3 appears to have similar remarks on the quantized supercurrent as reviewer #1. In order to address the reviewer's concerns we the added part on the signs of quantized supercurrent in the supplementary info. We note that reviewer #3 thinks that Fig. 3e, which was in the original draft and now appears in the supplementary information, reaches the normal state at the high bias values. However, since the data here is shown as a function of current and not voltage, the biasing is not large enough to reach the normal state resistance (in terms of energy the biasing is still within twice the superconducting gap, i.e. below ~200 eV, in the scale of the current displayed in this color map). So the normal state differential resistance above twice the superconducting gap corresponds to the normal state resistance when applying a small magnetic field (~20 mT) to fully break the superconductivity. Therefore, the normal state conductance, as now shown if Fig. 6b of the supplementary information, drops to 24e 2 /h. Finally, as we mention in the new part on the sign of quantized supercurrent, we have seen these features in our shorter device with w ~ 65 nm split-gate distance. Our longer device with a wider constriction w ~ 150 nm split-gate distance (not shown here) showed all properties of confinement that we present in this work (we also have recently measured devices with other confinement configurations which fully confirms our findings presented in this paper), but the signs of quantization. Additionally:  We have added under brackets the thickness of the two (bottom and top) hBN multilayers in the caption of Fig.1. Now it reads: "…(with a bottom and top hBN multilayer of ~35 and ~38 nm thick) on a pre-patterned overall back-gate (BG) covered with a 20 nm thick Al 2 O 3 and a split-gate (SG) on top of the heterostructure. The superconducting leads are edge connected to the mesa. The width W = 3.2m and length L = 950 nm while the distance between the two fingers of the split-gate w ~ 65 nm (and d' ~ 38 nm and d" ~ 55 nm). c AFM image of the device. Scale bar is 1 m."  We have found that our extracted contact resistance was not properly calculated. We have changed the text of the part "II. Normal state characterization" as follows: " Fig.1a shows both normal state resistance R and conductance G as a function of back-gate voltage V BG and charge carrier density n, while Fig.1b displays the electron conductivity (minus the estimated contact resistance calculated below) vs charge carrier density. The contact resistance per contact is estimated as R C = (R-R Q )/2, where the quantum resistance R Q is subtracted from the measured resistance R. The quantum resistance R Q = (h/ge 2 )(1/M) is defined as the resistance set by the ballistic limit of all contributing conductance modes M = W/ F /2, where  F = 2/k F = 2/(n) 1/2 is the Fermi wavelength at charge carrier density n and g = 4 accounts for the spin and valley degeneracy. At high charge carrier density n~4x10 12 cm -2 (M = 361 and R Q = 18) a resistance of R = 90 is measured, yielding the contact resistance R C = 36 and contact resistivity  C = R C W = 115m, comparable to the values given by Wang et al. [1]." As a consequence, the Fig. 1b has been replotted and corrected. The extracted value for the residual carriers is now 2.6x10 10 cm -2 .
New Fig.1b of the supplementary information:  Change in Fig.3 of the supplementary information: numbers on the gray-scale map bar were wrong and have been corrected.
 We have refined the analytical model that was based in the original version of the manuscript on the approach from Barzykin, V. & Zagoskin, A.N. Coherent transport and nonlocality in mesoscopic SNS junctions: anomalous magnetic interference patterns. Superlatt. Microstruc. 25, 797 (1999) (ref. [37] of the supplementary information), see subsection entitled "A. Analytical model: Long junction" from the part V entitled "Analytical and numerical model" of the supplementary information. In the new version, we have recalculated the critical current in the long-junction limit by going beyond the approximation of that reference. We have also corrected the phase difference in Eq. (9) of supplementary information. The modifications concern the geometrical weight of the Andreev trajectories (cf. for example Eq. (2) in the previous and new versions) and the overall scale of the interference pattern as a function of magnetic field (the latter was anyway used as a fitting parameter). The fit using our refined formula is presented in new Fig. 5b: Old figure 5b Importantly, the modifications in our analytical expressions do not produce any qualitative change of the behavior of the interference pattern for the QPC setup. Moreover, as one can see from the comparison of the new version of Fig. 5b with the previous version, the two fits are almost indistinguishable. We have also re-written some parts of the corresponding subsection of supplementary information to improve the presentation.
I find the changes that the authors made satisfy the concerns I raise in my review. I now believe that this work can be published in Nature Communications. I have some minor comments the authors should address prior to publication: 1) In Figure 2 in the main text, the authors should give a value of current bias used in the resistance map measurement (fig 2a, b, and c).
2) In Fig. 5c, the definition of t of the label on the y-axis \psi_{SG}/t is not given in the main text.
3) Is it possible to relate \psi_{SG}/t to the measured V_{SG} in order to check if the transition from beating to non-beating patterns agrees with the theory? 4) In the supplementary Information on page 2, one of the sentences reads "In a subsequent step, split-gates are fabricated in the fashion". Do you mean "in the similar fashion"?
Reviewer #2 (Remarks to the Author): The authors have made a significant effort in addressing all the questions and remarks of the referees, and inserted additional data which make the manuscript and its discussion more solid and sound from a scientific point of view. However, it must be noted, that reviewer #3 pointed out the need for showing evidence for a quantised supercurrent in more than a single device, which I agree it is a very important point. In response to this remark, the authors simply clarify that features of quantised conductance were only observed for a single device with a split-gate distance of ~ 65 nm (they state "as we mention in the new part on the sign of quantized supercurrent, we have seen these features in our shorter device with w ~ 65 nm split-gate distance" in the rebuttal letter), therefore leaving the issue of reproducibility in more than a single device raised by reviewer #3 still open. Apart from this question that has to be still addressed, I think the manuscript can now be considered more suitable for publication in Nature Communication.
Reviewer #3 (Remarks to the Author): I have read the authors' replies to all referees. I am glad that the incorrect claim of a quantized critical current (criticized by refs. 1 &3) has been mostly moved to the supplementary. I am surprised that they kept the following remnant of that claim: "We note that, on the presented device, we observe features visible in the normal and the superconducting state which could be related to quantized conductance and supercurrent (see supplementary information) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]." On that level, it is up to the authors to decide whether they want to cling to that statement. At least most of it will not be presented in the text of the paper. It would be lovely if they explained why their sample appears to show no edge conductance, while similar samples from the Manchester group (ref. [50]) do show pronounced edge supercurrent.

Dear Editor,
We wish to thank the reviewers once more for their constructive remarks and comments. Here, we address each of the reviewer's questions and issues point by point (modification-additional text in red).
Reviewer #1 (Remarks to the Author): I find the changes that the authors made satisfy the concerns I raise in my review. I now believe that this work can be published in Nature Communications. I have some minor comments the authors should address prior to publication: 1) In Figure 2 in the main text, the authors should give a value of current bias used in the resistance map measurement (fig 2a, b, and c).
As we have mentioned in the method section, we have used ac excitation for the lock-in detection between 1 and 10 V. For Fig 2a (normal state data), within this resistance map the largest voltage drop at the sample was 2.5 V while the maximum current was 2.6 nA. For Fig. 2b (and c) (i.e. superconducting state) the voltage drop at the sample was at a maximum of 2 V while the maximum current was 2.5 nA. We added the detailed values of the maximum measured ac current in the caption of figure 2.
2) In Fig. 5c, the definition of t of the label on the y-axis \psi_{SG}/t is not given in the main text.
 SG /t represents the strengths of the on-site potentials introduced on the split-gate in units of the intralayer hopping constant t. We have added this information in the text (in the last sentence before the conclusion section) as follows: "We finally show tight-binding simulations using Kwant package [53] of I c as a function of magnetic field B and split-gate strength  SG (in units of the intralayer hopping constant t) in Fig. 5c (see supplementary information for details) which are in good qualitative agreement with our experimental data of Fig. 4b." 3) Is it possible to relate \psi_{SG}/t to the measured V_{SG} in order to check if the transition from beating to non-beating patterns agrees with the theory?
Unfortunately realistic and quantitative modeling of electrostatic potentials caused by a constriction in similar geometries is beyond the current state-of-the-art tight binding simulations. We therefore limit our consideration to a qualitative comparison.

4)
In the supplementary Information on page 2, one of the sentences reads "In a subsequent step, split-gates are fabricated in the fashion". Do you mean "in the similar fashion"?
We thank the reviewer for pointing this out. We have corrected it with "in the similar fashion" in the text.
Reviewer #2 (Remarks to the Author): The authors have made a significant effort in addressing all the questions and remarks of the referees, and inserted additional data which make the manuscript and its discussion more solid and sound from a scientific point of view. However, it must be noted, that reviewer #3 pointed out the need for showing evidence for a quantised supercurrent in more than a single device, which I agree it is a very important point. In response to this remark, the authors simply clarify that features of quantised conductance were only observed for a single device with a split-gate distance of ~ 65 nm (they state "as we mention in the new part on the sign of quantized supercurrent, we have seen these features in our shorter device with w ~ 65 nm split-gate distance" in the rebuttal letter), therefore leaving the issue of reproducibility in more than a single device raised by reviewer #3 still open. Apart from this question that has to be still addressed, I think the manuscript can now be considered more suitable for publication in Nature Communication.
We thank Reviewer #2 for these positive comments. As we have mentioned in the text, we have measured two devices with similar geometry (QPC, one shorter and with small distance between the split-gates, shown here and a longer device) and both showed the similar behavior while confining the supercurrent. Only the signs of quantized supercurrent were seen in the smaller device presented in this article as we mention in our reply to reviewers and described in section IV.C. We have also observed the effect of confinement in another geometry (with side-top-gates, i.e. forming a 1D channel the entire length of the device) which ends to a very similar magneto-interferometric pattern. We will describe these experiments in another paper.
Reviewer #3 (Remarks to the Author): I have read the authors' replies to all referees. I am glad that the incorrect claim of a quantized critical current (criticized by refs. 1 &3) has been mostly moved to the supplementary. I am surprised that they kept the following remnant of that claim: "We note that, on the presented device, we observe features visible in the normal and the superconducting state which could be related to quantized conductance and supercurrent (see supplementary information) as predicted for ballistic supercurrents in quantum point contacts [38][39][40]." On that level, it is up to the authors to decide whether they want to cling to that statement. At least most of it will not be presented in the text of the paper. It would be lovely if they explained why their sample appears to show no edge conductance, while similar samples from the Manchester group (ref. [50]) do show pronounced edge supercurrent.
We thank reviewer #3 to raise the issue of edge currents. We address it in a new section (section VI entitled "Effect of the edge currents on the magneto-interferometric pattern" and added 8 references) in the supplementary information. First by implementing our analytical model with edge channels (figure 12 presenting the geometry of the system), we introduce a correction factor which