Reentrant superconductivity in a naturally occurring Josephson junction array tuned by radio-frequency power

Domain: neuroscience
Published: 2026-04-01
Editorial outcome: minor_revision

Abstract
Superconductivity, characterized by dissipationless current flow with flux expulsion or quantization, is usually suppressed when the magnetic field or the temperature is sufficiently high. However, in rare instances, superconductivity can reappear upon increasing the temperature or magnetic field, a phenomenon known as reentrant superconductivity. It usually emerges from competing orders in strongly correlated materials. Here we demonstrate reentrant superconductivity as a function of both temperature and magnetic field, tuned by radio-frequency power in a relatively simple system: granular aluminum, which exhibits the properties of a naturally occurring Josephson junction array. At low temperatures, giant Shapiro steps emerge, exhibiting characteristics of a single Josephson junction. Coherent phase locking across the array’s multiple junctions amplif ies the quantized voltage, enabling tunability at radio frequencies, as observed in artificially designed Josephson arrays. We show that our system can be tuned from a coherent superconducting (stiff-phase) to an insulating (phase-fluctuating) state using radio-frequency power. We propose that radio-frequency power modulates the Josephson coupling energy, 𝐸 . Remarkably, at elevated temperatures, the screening of the electron charge suppresses the charging energy, causing superconductivity to reappear. This many-body effect cannot be described within a single junction framework and involves many-body correlations. Our system can therefore be tuned to observe both the single-junction regime and many-body correlation effects, serving as a quantum simulator for complex phenomena in condensed matter physics.

Introduction
Complex behavior can arise in large networks governed by simple rules1, a principle observed across physics in phenomena such as spontaneous magnetization in Ising models, phase transitions in percolation theory, and synchronization in coupled oscillators2. One prominent example of a quantum network exhibiting such emergent behavior is the Josephson junction (JJ) array, which offers a rich platform for exploring collective quantum dynamics3. In this correlated quantum system, the relevant energy scales are the Josephson coupling across the junction 𝐸 = \hbar I_c / 2e, which determines the confinement of the Josephson phase, and the charging energy 𝐸 = e^2 / 2C, which determines the strength of Coulomb interactions, with I being the Junction’s critical current and C the capacitance. The interplay between these energy scales yields a superconductor-to-insulator quantum phase transition4, obtained for 𝐸 \simeq 𝐸 .

Reentrant superconductivity is a phenomenon in which the superconducting phase unexpectedly reappears from a dissipative state when increasing a tuning parameter of the system, such as temperature, magnetic field, and pressure. It has been observed in systems including superconductor-ferromagnet compounds5–8 and layered structures9,10, bismuthates 11, heavy-fermions12,13, Kagome materials14, topological insulator JJs15, and twisted moiré materials16. This phenomenon often arises from competing orders, such as ferromagnetism and superconductivity in ErRh4B4 and HoMo6S8 5,8, charge density waves and superconductivity in ZrTe 17, or different pairing mechanisms in UTe 13 and magic-angle graphene13,18. Reentrant superconductivity has also been ex plored through gauge-gravity duality, where black hole stability serves as a probe of holographic superconductivity in the dual non-relativistic field theory19.

Theoretically, it has been shown that within a specific regime of the quantum coupling ratio, 𝐸 /𝐸 , reentrant superconductivity can emerge in JJ arrays20. At moderate temperatures, quasiparticle screening reduces 𝐸 , enabling charge fluctuations and the reappearance of superconductivity. As the temperature decreases, this screening effect diminishes, restoring 𝐸 and driving the system back into a phase-fluctuating, dissipative state20,21. Despite its theoretical foundation, this reentrant behavior has remained experimentally elusive due to limitations in fabricating artificial JJ arrays with the required precision.

Various methods have been employed to tune 𝐸 /𝐸 in JJ arrays. For instance, varying the insulating barrier thickness to tune junction resistance22, tuning junction critical current with a magnetic field23, gating a two-dimensional electron gas placed in a proximity to the array to affect its screening24, and electron depletion in semiconductor-superconductor hybrid devices25. While these methods could effectively tune 𝐸 /𝐸 across the superconductor-to-insulator transition, only traces for quasi-reentrant behavior were observed experimentally22,26. A clear reappearance of a dissipationless superconducting state in a JJ array is still lacking.

We have employed a new approach for realizing a JJ array and for tuning its parameters. We utilize a naturally occurring JJ array made of granular aluminum (grAl). This material consists of superconducting islands coupled through insulating interlayers27 (See Figure 1c for illustration). The grain size and the barrier strength are controlled by the deposition conditions28,29. The system features nanoscale grain size, making the strong quantum fluctuations regimes of both charge and phase accessible30. Compared to artificial JJ arrays, the significantly smaller grain size results in a longer screening length, enhancing correlations between junctions across the array. Additionally, the electronic level spacing within each island exceeds both 𝐸 and the superconducting gap31–33. This strong separation makes the system less sensitive to microscopic details34, as disorder typically results in level broadening. Therefore, it is a promising building block for high kinetic inductance and low-loss quantum devices34. Finally, to control the Josephson parameters, we adapted a new approach by application of radio frequency (RF) power to tune the critical current 𝐼 35, thereby varying 𝐸 . The drive frequency should be much smaller compared to the characteristic frequency of the junction 𝜔 ~\sqrt{E_J E_C}, so it is mainly 𝐼 that depends on the RF amplitude36 (See Fig. 1c for illustration of the set-up and our interpretation for the effect of the RF power on junction properties and the quantum phase slip rate). Utilizing the unique properties of grAl and our new tuning approach, we can tune through the various regimes of the device. We realize a superconductor-to-insulator transition as a function of RF power with a strong insulating state. This insulator has ten times higher resistance compared with the device normal resistanceand the derivative of the device resistance with respect to temperature is negative (see Supplementary Information Fig. S5). In addition, in the insulator regime we observe a pronounced temperature dependent reentrant superconductivity. We explain our data using two regimes: at low temperatures, the system is described as a phase locked JJ array featuring giant Shapiro steps, obeying single-junction-type physics. At high temperatures, the reentrant behavior is described within the many-body correlated picture suggested by Efetov20, where reentrance is induced by the screening of long-range interactions. Our findings emphasize the potential of using granular superconductors as controlled platforms for developing novel quantum devices and exploring emergent quantum phenomena in strongly correlateNd systems.

Results
RF tuned superconductor-to-insulator transition

We study a JJ array implemented using a superconducting nanobridge fabricated from a grAl thin film (Fig. 1c). The nanobridge has a length of ℓ ≃ 120 nm, width 𝑤 ≃ 200 nm, and thickness 𝑡 ≃ 55 nm. These dimensions are significantly larger than the grain size of grAl, 𝑑 = 2±0.5 nm,37 the inter-grain spacing (𝑠 ≃ 1 nm), and the superconducting coherence length (𝜉 ≃ 10 nm) 38. Additionally, the bridge dimensions are much smaller than the in-plane magnetic penetration depth (𝜆 ≃ 1 μm) 39, resulting in a relatively uniform current distribution along the array40. Naively, the bridge would consist of approximately 𝑁 ×𝑁 ×𝑁 ≃ 40×67×18 inter-grain JJs. However, the effective number of junctions can differ significantly due to additional contributions coming from the contact region41.

We begin by demonstrating the tunability of the system at a low temperature of 𝑇 = 50 mK as shown in Fig. 1a. The differential resistance 𝑑𝑉/𝑑𝐼 is shown to be tunable by the RF power 𝑃 and the bias current 𝐼 . Below a critical power 𝑃 (𝑃 < 𝑃 = −13 dBm), the superconducting state (blue region) is observed for low bias currents, where a typical suppression of the critical current is observed. Upon increasing the power at zero bias (𝐼 ≃ 0), a peak in the differential resistance emerges. This peak reaches its maximum height and minimum width at 𝑃 = −10.5 dBm, which we define as the insulating state (I). We note that the zero-bias resistance 𝑅 is an order of magnitude larger than the resistance in the normal state 𝑅 , obtained at the highest DC currents and RF powers where superconductivity is completely quenched. Furthermore, when raising the temperature, 𝑅 decreases, a typical behavior for an insulating state. Previous studies on JJ arrays 42,43 and single junctions44 have reported power-induced transitions to a resistive state characterized by Shapiro step behavior. In contrast, the power-driven transition to an insulating state observed here is significantly more pronounced.

Upon increasing 𝐼 and 𝑃 further, clear oscillations of 𝑑𝑉/𝑑𝐼 are observed. We interpret these oscillations as giant Shapiro steps, as evidenced from our further analysis in Fig. 3. These Shapiro steps demonstrate that the low temperature dependence of the system can be described within an effective single JJ model as described in the theoretical analysis below and depicted in Fig. 1d.

Reentrant superconductivity

We now look at the 𝑃−𝑇 phase diagram of the system, shown in Fig. 1b. We observe reentrant superconductivity as a function of temperature; this is our main result. This phenomenon has been predicted by Efetov for an array of JJ’s as depicted in Fig. 1e. In our case we conjecture that a finite RF power is needed for carefully tuning 𝐸 for one to be able to observe the reentrance behavior as will be discussed further below.

For illustration, let us first follow the dashed line in Fig. 1b at 𝑃 = −14.25 dBm: at low temperatures, the system is superconducting with 𝑅 = 0. As the temperature increases above 1.5 K, 𝑅 goes through a peak: this is the insulating state. Upon increasing the temperature further, superconductivity surprisingly reappears, this being the reentrant superconducting phase SRC 2. Above 2.5 K, the normal state N of the system is observed. As noted above, the normal state resistance of the array is an order of magnitude smaller than the insulating state resistance.

The various regimes of the system, SC 1, I, SC 2, and N, are indicated on the phase diagram. Remarkably, we observe reentrant superconductivity as a function of temperature. For 𝑃 < 𝑃, the system undergoes from low 𝑇 superconductivity (SC 1) to high 𝑇 superconductivity (SC 2), while crossing through the insulating (I) regime. In the insulating regime, 𝑅(𝑇) is first rising with lowering 𝑇, then saturating at low 𝑇. This saturation at the lowest temperatures may not be an intrinsic property of the system but a result of the electronic heat generated by the RF power 45 (see Supplementary Information Sec. Ⅴ.B).

To investigate the effect of RF power and bias current, we fix the RF power at 𝑃 = −14.25 dBm, and plot the differential resistance 𝑑𝑉/𝑑𝐼 as a function of bias current 𝐼 and temperature 𝑇 (see Fig. 2a-b). As the temperature increases, the critical current 𝐼 of the superconducting state SC 1 decreases rapidly and vanishes completely at 𝑇 = 1.35 K. Upon increasing the temperature further, the resistance rises, reaching a peak in the zero-bias resistance around 𝑇 = 1.8 K.

Along the 𝐼 axis, (i.e., with increasing bias current), we observe oscillations in 𝑑𝑉/𝑑𝐼 (Shapiro steps). At 𝑇 = 2.16 K, the critical current 𝐼 reappears, signaling the onset of a reentrant superconducting phase (SC 2), which forms a dome-shaped region in the 𝐼 −𝑇 diagram at fixed RF power.

Fig. 2b, highlights this reentrant regime, where SC 2 is clearly visible. For comparison, Fig. 2c shows measurements taken without RF power, displaying a conventional 𝐼 (𝑇) dependence without reentrance. As discussed further below, we propose that the presence of RF power is essential for fine-tuning the effective coupling energy 𝐸 .

Giant Shapiro steps

The dependence of the system response on the RF frequency, 𝑓 , is explored in Fig. 3. We study the differential resistance 𝑑𝑉/𝑑𝐼 as a function of 𝐼 and 𝑓 , focusing on the insulating state, Fig. 3a-b. 𝑑𝑉/𝑑𝐼 exhibits damped oscillations as a function of 1/𝑓 , Fig. 3a. The oscillations are smeared at low frequencies, while the insulator zero-bias peak is enhanced (see inset), suggesting it has a different origin.

Looking at the 𝑑𝑉/𝑑𝐼 in the 𝑓 −𝑉 plane, Fig. 3b, we reveal the origin of the oscillations. They follow the giant Shapiro steps relation42, 𝑉 = 𝑛𝑁𝜙 𝑓 . Here 𝑛 is an integer, 𝜙 is the flux quantum, and 𝑁 is the number of phase locked junctions connected in series. Using the frequency dependence, we obtain contributions from 𝑁 = 430 junctions, which exceeds the number of junctions in the nanobridge (𝑁 ). This difference can be attributed to an extended phase drop across the device41, which comprises the nanobridge in series with long grAl contacts (see Supplementary Information Sec. Ⅴ.C and Fig. S7).

At a given frequency, the first consecutive Shapiro steps are missing. As the frequency decreases, more steps are hidden by the insulator peak, and only higher harmonic steps are visible. This behavior contrasts with the regular giant Shapiro steps that cross the origin in the frequency-voltage plane, as typically observed in JJ arrays43. Fig. 3c shows the giant Shapiro steps observed at 𝑓 = 67.5 MHz. They clearly illustrate the even-odd effect expected for the current step width46. They are characterized by a finite voltage width expected for arrays with minuscule disorder47 (see Supplementary Information Sec. Ⅴ.C).

We extend our investigation by examining the RF response of the system under an external magnetic field (Fig. 4). At low temperatures and at zero fields, we define 𝑃 = −13 dBm and 𝑃 = −10.5 dBm at the onset of SC 1 and the insulating (I) regimes, respectively. They exhibit non-monotonic modulation by the out-of-plane magnetic field 𝜇 𝐻 (Fig. 4a). The oscillations of the critical power 𝑃 are reminiscent of a Fraunhofer-type interference. The oscillatory, non-monotonic behavior of 𝑃 as a function of magnetic field, along with the strong suppression of 𝑅 at small 𝜇 𝐻 , suggests that the insulating phase emerges from the superconducting state.

In contrast, when an in-plane magnetic field 𝜇 𝐻 , aligned with the current direction, is applied, both 𝑃 and 𝑃 decrease monotonically, while 𝑅 is suppressed only at significantly higher fields (Fig. 4b). In both magnetic field orientations, and within the power range 𝑃 < 𝑃 < 𝑃, distinct resistance oscillations are observed. These oscillations correspond to the missing Shapiro steps, which reemerge once the zero-bias peak associated with the insulating state is suppressed by the magnetic field.

Remarkably, at elevated temperatures, we observe signatures of field-induced reentrant superconductivity, most prominently under in-plane field conditions (Fig. 4c-d). This reentrance may be consistent with Efetov’s picture: here, quasiparticles would be produced by pair-breaking48,49, enhancing screening, when the applied magnetic field is close to the upper critical field38.

Theoretical model

For simplicity, we conceptualize our system as a stack of many parallel one-dimensional chains of JJs. This model captures the Shapiro steps observed in Fig. 1a at low temperature and RF frequencies well below 𝜔 , using effective single-junction physics, as we now explain. The semiclassical equation of motion for each junction is given by the model of a resistively and capacitively shunted junction (RCSJ) under simultaneous DC and RF bias,

𝛽 \frac{\partial^2\phi}{\partial\tau^2} + \frac{\partial\phi}{\partial\tau} + \sin\phi = i_0 + i_1 \sin(\Omega\tau), (1)

where 𝛽 = 2𝜋𝑅^2𝐶𝐼 /𝜙 is set by the normal resistance 𝑅, the capacitance 𝐶, and the critical current 𝐼 of the junction. The time 𝑡 has been rescaled by 𝑡 = 𝜙 /(2𝜋𝑅𝐼 ) to define the dimensionless parameter 𝜏 = 𝑡/𝑡 , 𝛺 = 𝜔 𝑡 is the rescaled RF frequency, and 𝑖 = 𝐼 /𝐼 , 𝑖 = 𝐼 /𝐼 are the rescaled DC and RF currents, respectively (the latter being proportional to the square root of the RF power, 𝐼 ∼ \sqrt{P}).

The voltage drop across the junction is given by 𝑉 = 𝜙 \partial_t \phi/2𝜋. We assume that the DC and RF currents flowing through the sample are distributed equally between the one-dimensional chains; then, within each chain, the RCSJ equations decouple, and the total voltage drop across the chain is given by the voltage drop across a single junction multiplied by the number of junctions in each chain.

In Fig. 1d, we present calculations of 𝑑𝑣/𝑑𝑖 as a function of the rescaled DC current 𝑖 and RF power 𝑃, where 𝑣 = \partial_\tau \phi = 2𝜋𝑡 𝑉/𝜙 is the rescaled voltage, obtained for 𝛺 = 0.1 and 𝛽 = 1 (see Supplementary Information Sec. Ⅲ for extended discussion and other parameter regimes). While the model is simplified, it captures the Shapiro steps observed in Fig. 1a; this is in line with previous studies of JJ arrays42. Moreover, it accounts for the oscillatory behavior seen as a function of temperature in Fig. 2a: temperature reduces the critical current such that 𝑖 = 𝐼 /𝐼 (𝑇) in Eq. (1) become temperature-dependent, leading to the emergence of Shapiro steps.

Methods
Device fabrication

The naturally occurring JJ arrays have been implemented by granular aluminum nanobridge devices. The nanobridges were obtained using standard liftoff electron beam lithography. Device dimensions were examined by atomic force microscopy. Granular aluminum thin films were thermally evaporated on R-plane sapphire substrates held at liquid nitrogen temperature. Compared to films deposited on substrates held at room temperature, this approach reduces the mean grain size and narrows the grain size distribution29. For the nanobridge device presented in the main text, the film resistivity is 𝜌 = 2800 μΩcm, estimated by measuring an on-chip Hall bar device at room temperature. Measurements in additional devices are discussed in Supplementary Information Sec. Ⅴ.E. We employ a quasi-four-probe measurement configuration, where the granular aluminum nanobridge is connected in series with wide superconducting granular aluminum contacts (see Fig. S11a).

Measurement setup

Measurements were conducted in a dilution refrigerator with a base temperature of 20 mK and a (8 T,1 T,1 T) superconducting vector magnet. DC transport measurements were taken while simultaneously applying RF signals characterized by a power, 𝑃 (dBm), defined in the signal generator output. The measurement setup is illustrated in Supplementary Information Fig. S11b. The DC lines are filtered using a multi-stage RF and DC low-pass filter mounted on the mixing chamber plate, followed by additional DC low-pass filtering on the sample board, thereby reducing injected RF noise and ensuring that only the DC voltage is measured. The RF lines are attenuated at different stages of the dilution refrigerator and are coupled to the device through on-board RF bias-tees. The bias tees consist of a 50 kΩ DC bias resistor and a 22 nF RF capacitor. Differential resistance, 𝑑𝑉/𝑑𝐼 , data were taken by applying a DC current 𝐼 in addition to a small AC current 𝑑𝐼 = 10 nA at a frequency of 7.189 Hz, using a standard lock-in technique.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors report on the observation of reentrant superconductivity and a superconductor-to-insulator transition in granular aluminum, a material which acts as a natural Josephson junction (JJ) array.\nAs the authors correctly say in the introductory section of the manuscript, reentrant superconductivity in JJ arrays has been studied theoretically since a long time but the experimental situation remained elusive. In essence the effect should appear due to an interplay between the Josephson coupling energy E_J and the charging energy E_c, for specific values of the ratio E_J/E_c.\nIf E_J dominates one obtains a phase coherent state while for a dominating E_c the insulating state is obtained. If E_J/E_c decreases monotonically from values well above 1 to values well below 1, say, as a function of temperature one may observe the superconductor-to-insulator transition (SIT). However, at some moderate temperature quasiparticle screening may reduce E_c and thus increase E_J/E_c, enabling the reappearance of superconductivity. This effect has been observed by the authors. They use granular aluminum, having very small grain sizes of around 2 nm, as the experimental platform and apply an external radio frequency to tune E_J. With this system the SIT has been observed for specific values of the rf power, as summarized in Fig. 1 of the manuscript. The experimental results are accompanied by a detailed theoretical analysis.\nThe manuscript is well organized, the results are new and convincing. The paper is generally suitable for Nature Communications. The only drawback is that the authors never show current-voltage characteristics (IVCs) of their samples, which would help in understanding the properties of their system. In the past, such IVCs provided clear signature whether a system is Josephson coupled or in the flux-flow regime, if a Coulomb gap is present, if inhomogeneities play a big role, etc. It thus would be great to have comparable curves in the paper.\nAs I understand the authors primarily record the differential resistance and plot it as the color scale in various 2D plots. However, for example in Fig. S5b they integrate the signal to obtain the actual voltage V. Can the same procedure be used to obtain complete IVCs, i.e. plots of currents I_b vs. V? If so, it would be very helpful to see representative IVCs, say, along the vertical dashed lines in Fig. 1b. This could include a low-temperature IVC, obtained in region SC1, an IVC obtained in the insulating region, around 1.8 K, one in region SC2, and perhaps one or two more in the N regions. These plots could be presented in the Supplementary Information section. With these or a comparable modification I will be happy to recommend the manuscript for publication in Nature Communications.", "reviewer_2": "In “Reentrant superconductivity and superconductor-to-insulator transition in a naturally occurring Josephson junction array tuned by RF power” Avraham et al present a study on an RF-driven granular aluminum weak-link device. The most interesting observation is that, when driven, the device can evolve from superconducting, to insulating, and back to superconducting, and then to normal-conducting behavior as temperature is increased. The observation of re-entrant superconductivity is intriguing, and clearly goes beyond the physics that has been observed in granular superconductors so far. It is even more intriguing that the observation occurs in a driven context. Overall I find the study to be of high quality and that it is suitable for publication, subject to addressing some concerns below.\n\n(1) Is the phenomenon easily observed? Has it been observed in multiple devices?\n(2) The authors identify high-resistance regions as insulating (I). Presumably the insulating regime is due to phase quantum phase slips. On the other hand, there can be spikes in dV/dI at the edges of regions showing Shapiro steps, including in the RCSJ simulations shown by the authors, which know nothing about phase slips. Can the authors distinguish their insulating regime from the dV/dI ‘spikes’ that somehow come from RCSJ considerations?\n(3) Does the RF drive ever make the sample appear “more superconducting” compared to no RF drive? The comparison I have in mind is in particular near Tc (green region below):\nIt is difficult to tell from the figures, but it would be quite interesting if there is a hint that RF driving is promoting phase synchronization at a fixed temperature. Could the authors please comment?\n(4) In Fig. S3 the no RF and -18 dBm colors are almost imperceptibly different (at least for me). This caused quite some confusion. The issue is that in S3a one cannot completely tell that re-entrance is absent without RF drives. In the line cuts it is very difficult to see as well. I request the authors change the colors/plot layout to make the no RF data completely clear. Is it really true that there is no re-entrance without driving?", "reviewer_3": "Nature Communications - REFEREE RESPONSE\nManuscript Number: NCOMMS-25-63869-T\nReentrant superconductivity and superconductor-to-insulator transition in a naturally occurring Josephson junction array\ntuned by RF power, by S. Avraham, S. Sankar, S. Sandik, A. Burshtein, M. Goldstein, E. Sela, and Y. Dagan\nIn the manuscript «Reentrant superconductivity and superconductor-to-insulator transition in a naturally occurring\nJosephson junction array tuned by RF power» S. Avraham and co-authors claim that they «demonstrate reentrant\nsuperconductivity as a function of both temperature and magnetic field, tuned by radio frequency (RF) power in a relatively\nsimple system: granular aluminum (grAl), which exhibits the properties of a naturally occurring Josephson junction array».\nThe main manuscrit present color plots of experimental results measured on a single device in «quasi-four-probe\nconfiguration» and with resistance saturation that «is due to experimental limitations» as specified in the supplementary.\nThe measurements consist in resistance measurements at various current bias, temperature, RF-power, RF-frequency and\nmagnetic field. The measurements are discuss with a «theoretical model» that actually consist in two different models: a\nRCSJ model and a many-body effective model. They propose that the RF-current enables the reentrance of\nsuperconductivity.\nIf true, the results are definite worth being publish. However, the current presentation of the experimental results in the\nmanuscript casts some doubt on the veracity of the results and is very default to follow. I therefore ask the authors to make\nsignificant changes to the manuscript so that the experimental rigor of their work becomes clear and that readers can\nunderstand their work.\nFirst of all, I had a lot of difficulties to understand what superconducting re-entrance behavior(s) the authors is(are) referring to. Part of my difficulties are coming from the color plot presentation: raw data are needed both for clarity and to check for experimental rigor. Besides, the results were measured on only one device. As reproducibility is a key concept for science\nand as the variability between devices is interesting, in particular here for the discussion of the various energy scales at play\nand for the possible giant Shapiro steps: I ask for measurements on different devices.\nBelow, I detail some of my difficulties and questions.\nIf I understood properly with fig1a the authors intend to prouve re-entrance superconductivity with increasing RF-power.\nFrom the color plot the re-entrance behavior is not obvious as there is no dissipationless superconducting state (dark blue),\nneither « Shapiro steps » (or at least a label indicating « Shapiro steps » or « SC2 »). Please, note, that I do not see any\nShapiro steps in the color plot. Please present the usual I-V curves for the Shapiro steps. If I have misunderstood and the\nauthors mean something else by “tuned by RF power,” please clarify here, as well as in the title and, when necessary, in the\ntext. I suggest that the authors mention in the manuscript that for the RF-tuning on the SC1-phase is pretty classic: an\nincreased RF-power, results in an increased RF-current, results in a decreased critical current/gap/EJ (because the total\neffective current biais equals I_b+I_RF), leading to the suppression of superconductivity.\nIf I understood properly with fig1b the authors intend to prouve re-entrance superconductivity with increasing temperature.\nFrom the raw data presented in the supplementary figS3 the dissipationless superconducting state reentrant is not\ndemonstrated: the resistance is far from being null (and the data are incomplete, points are missing). Probable reentrant\nsuperconductivity with temperature (with nonzero resistance) has already been reported in (highly) resistive granular\naluminum in the following publications:\n- H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M. Goldman, Phys. Rev. B 40, 182 (1989).\n- W. Wu and P. W. Adams, Phys. Rev. B 50, 13065(R) (1994).\n- F. Levy-Bertrand, T. Klein, T. Grenet, O. Dupre, A. Benooit, A. Bideaud, O. Bourrion, M. Calvo,\nA. Catalano, A. Gomez, J. Goupy, L. Grunhaupt, U. v. Luepke, N. Maleeva, F. Valenti, I. M. Pop,\nand A. Monfardini, Physical Review B 99, 094506 (2019).\nIn those publications non RF-signal was applied. Please compare your results with these papers and cite them. Can you\n(re)explain the novelty of your measurement compared to these papers? In particular can you show resistance versus\ntemperature measurements (raw data) with identical inputs but with and without RF-signal ? For now I do not agree with the\nsentence: «We reiterate that the reentrance appears only upon the application of RF current and does not occur using DC\ncurrent. We reiterate that the reentrance appears only upon the application of RF current and does not occur using DC\ncurrent.» A RF power P_RF=-20dBm is probably like not RF-signal and yet you observed (a non zero) re-entrance\nsuperconductivity with increasing temperature. The re-entrance seems wider in temperature (thus more robust) at low RF-\npower. Any comments ?\nI do not see the Shapiro steps in fig 3. Is fig3c a simulation ? I do not understand how the authors estimate the number of\ninter-grain JJs [for me Nx=120nm/(2x5nm+1nm)=24, Ny=200nm/(2x5nm+1nm)=40, Nz=55/(2x5nm+1)=11]. Can the authors\n(re)explain the estimation procedure?\nThe actual measurement set-up needs to be clarified. In particular the expression «quasi-four-probe configuration» p.5 of the\nsupplementary needs to be clarified in the main text. Can you provide an accurate scheme of the actual geometry ? Do you\nfilter out the ac-voltage (coming from the RF-current) from the dc-voltage? Do you mesure the resulting RF-current ?\nOther specific comments and suggestions are:\nI completely disagree with the last sentence of the introduction «Our system can therefore be tuned to observe both the\nsingle-junction regime and many-body correlation effects, serving as a quantum simulator for complex phenomena in\ncondensed matter physics. ». The behavior of granular aluminum is far from that of a single-junction as evidence (in the\nmanuscript itself!) by the «giant Shapiro-steps» relation between the voltage and the RF-frequency. Besides, I do not\nunderstand how the authors intend to control the many-body correlation effects, neither how a single experimental device\ncould be used for quantum simulations.\nI suggest you remove the following sentence that is cryptic to me: «Reentrant superconductivity has also been explored\nthrough gauge-gravity duality, where black hole stability serves as a probe of holographic superconductivity in the dual non-\nrelativistic field theory ».\nI suggest you elaborate more on the following sentence or you remove it: «Additionally, the electronic level spacing within\neach island exceeds both Ec and the superconducting gap, making the system less sensitive to microscopic details.»\nI hope you find my report helpful."}, "author_response": "December 14, 2025\nResponse to Reviewers for: NCOMMS-25-63869-T: Reentrant superconductivity and\nsuperconductor-to-insulator transition in a naturally occurring Josephson junction array\ntuned by RF power\nS. Avraham, S. Sankar, S. Sandik, A. Burshtein, M. Goldstein, E. Sela, and Y. Dagan\nSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel\nFirst, we would like to thank the Reviewers for their thorough reading and insightful remarks, which\nhave helped us improve our paper.\nReviewer #1 wrote: “The observation of re-entrant superconductivity is intriguing and clearly goes\nbeyond the physics that has been observed in granular superconductors so far.”\nReviewer #2 commented: “The results are new and convincing. The paper is generally suitable for\nNature Communications.”\nReviewer #3 stated: “The results are definitely worth being published.”\nThe reviewers provided specific comments, which we have addressed in detail in the response letter\nbelow. We hope that, after addressing all comments and criticisms, the Reviewers will find that the\nrevised version of our manuscript merits publication in Nature Communications.\nAll changes to the manuscript are highlighted in red. Our response is highlighted in red and the\nReviewer comments are black colored.\nBelow, we address the specific comments made by the Reviewers.\nResponse to Reviewer 1:\nIn “Reentrant superconductivity and superconductor-to-insulator transition in a naturally occurring Josephson\njunction array tuned by RF power” Avraham et al present a study on an RF-driven granular aluminum weak-link\ndevice. The most interesting observation is that, when driven, the device can evolve from superconducting, to\ninsulating, and back to superconducting, and then to normal-conducting behavior as temperature is increased. The\nobservation of re-entrant superconductivity is intriguing and clearly goes beyond the physics that has been\nobserved in granular superconductors so far. It is even more intriguing that the observation occurs in a driven\ncontext. Overall, I find the study to be of high quality and that it is suitable for publication, subject to addressing\nsome concerns below.\nWe thank the Reviewer for their comment and for appreciating our results.\n1. Is the phenomenon easily observed? Has it been observed in multiple devices?\nResponse\nWe have studied three different devices. In the revised Supplementary Information (Figs. S9-S10), we have\nincluded data obtained from the additional devices, as summarized in Table S1, and discussed in Section\nVE. The results are qualitatively similar.\nPlease note that the measurement setup for the two additional devices was not as fully optimized as the one\nused for the device presented in the main manuscript, for which comprehensive measurements were\nconducted. For example, the RF generator used had lower frequency resolution. As a result, additional noise\nand minor artifacts are present, as explained in the revised Supplementary Information Section VE.\n2. The authors identify high-resistance regions as insulating (I). Presumably the insulating regime is due to\nphase quantum phase slips. On the other hand, there can be spikes in dV/dI at the edges of regions showing\nShapiro steps, including in the RCSJ simulations shown by the authors, which know nothing about phase\nslips. Can the authors distinguish their insulating regime from the dV/dI ‘spikes’ that somehow come from\nRCSJ considerations?\nResponse\nThe origin of the high-resistance region can be distinguished from that of the traditional RCSJ-related Shapiro\nsteps (and spikes) based on the following arguments presented in the manuscript:\n• Magnitude of Resistance: The high resistance observed in the insulating state, 𝑅, in Fig. 1a is about\nI\nten times larger than both the normal-state resistance and the resistance observed in the rest of the\ngiant Shapiro steps. In contrast, in the RCSJ simulation, all spikes are of the same order of\nmagnitude. These simulation results are consistent with previous studies on single junctions (PRB\n67, 224501 (2003)) and Josephson junction arrays (PRL 64, 693 (1990); Commun. Phys. 3, 53\n(2020)).\n• Frequency Dependence: The insulating state exhibits an opposite frequency dependence compared\nto the Shapiro steps, as shown in Fig. 3a. 𝑅 continuously increases as the RF frequency decreases,\nI\nwhereas the spikes oscillate as a function of frequency and are gradually suppressed, becoming\ninvisible at sufficiently low RF frequencies. Furthermore, Fig. 3b shows that the Shapiro step regime\nis confined within a strip-like region in the voltage–frequency diagram, in contrast to the prediction\nof the RCSJ model (dashed lines). In particular, the expected low-voltage, low-frequency steps are\nhidden by the insulating state (bottom red region). This contrasts with regular giant Shapiro steps\nthat cross the origin, as observed in Josephson junction arrays (see, e.g., Panghotra et al. Commun.\nPhys. 3, 53 (2020), see comparison below). We have added clarification on this point in the revised\ndiscussion section and Supplementary Information Section VB.\n• Magnetic Field Dependence: By applying an external magnetic field (Fig. 4a), we demonstrate that\nfor out-of-plane fields as small as 10 mT, 𝑅 is significantly suppressed. This suppression uncovers\nI\na hidden region of giant Shapiro steps, similar to the one observed for finite DC current 𝑅 in Fig.\nI\n1a. At the same time, the revealed Shapiro spikes exhibit a shift as a function of power when the\nfield is further increased. The Shapiro steps remain visible up to high fields of about 1 T. We have\nadded an illustration of this point in the revised Supplementary Information Fig. S7.\nTo conclude: The zero-bias peak observed in the insulating state is an order of magnitude larger and\nexhibits contrasting behavior compared to Shapiro steps (and spikes) when varying parameters such as\nRF power, DC current, RF frequency, and magnetic field. These differences strongly suggest a distinct\nphysical origin. The above discussion has been added to the revised Supplementary Information Section\nVB4. We emphasize that the single-junction regime described by the RCSJ model does not connect in a\nsimple way to the insulating regime. We hope the revised discussion clarifies all these points.\n3. Does the RF drive ever make the sample appear “more superconducting” compared to no RF drive? The\ncomparison I have in mind is in particular near Tc (green region below):\nIt is difficult to tell from the figures, but it would be quite interesting if there is a hint that RF driving is\npromoting phase synchronization at a fixed temperature. Could the authors please comment?\nResponse\nIt is challenging to assess RF-induced enhancement of superconductivity in our setup. While we did not\nobserve any significant enhancement, our temperature resolution near T in the dilution refrigerator is limited,\nc\nmaking it difficult to determine whether a small effect might be present. The sample does not appear more\nsuperconducting under RF drive compared to the case without RF drive, but no conclusive statement can be\nmade. This point is mentioned in the revised Supplementary Information Section VA2.\n4. In Fig. S3 the no RF and -18 dBm colors are almost imperceptibly different (at least for me). This caused\nquite some confusion. The issue is that in S3a one cannot completely tell that re-entrance is absent without\nRF drives. In the line cuts it is very difficult to see as well. I request the authors change the colors/plot layout\nto make the no RF data completely clear. Is it really true that there is no re entrance without driving?\nResponse\nWe have tried to improve the color layout in the revised Supplementary Information Fig. S3. We hope that\nnow it is clear that there is no re-entrance without RF driving.\nResponse to Reviewer 2:\nThe authors report on the observation of reentrant superconductivity and a superconductor-to-insulator transition\nin granular aluminum, a material which acts as a natural Josephson junction (JJ) array.\nAs the authors correctly say in the introductory section of the manuscript, reentrant superconductivity in JJ\narrays has been studied theoretically since a long time but the experimental situation remained elusive. In\nessence the effect should appear due to an interplay between the Josephson coupling energy E_J and the\ncharging energy E_c, for specific values of the ratio E_J/E_c.\nIf E_J dominates one obtains a phase coherent state while for a dominating E_c the insulating state is obtained.\nIf E_J/E_c decreases monotonically from values well above 1 to values well below 1, say, as a function of\ntemperature one may observe the superconductor-to-insulator transition (SIT). However, at some moderate\ntemperature quasiparticle screening may reduce E_c and thus increase E_J/E_c, enabling the reappearance of\nsuperconductivity. This effect has been observed by the authors. They use granular aluminum, having very small\ngrain sizes of around 2 nm, as the experimental platform and apply an external radio frequency to tune E_J.\nWith this system the SIT has been observed for specific values of the rf power, as summarized in Fig. 1 of the\nmanuscript. The experimental results are accompanied by a detailed theoretical analysis.\nThe manuscript is well organized, the results are new and convincing. The paper is generally suitable for Nature\nCommunications.\nWe thank the Reviewer for their comment and for appreciating our paper.\nThe only drawback is that the authors never show current-voltage characteristics (IVCs) of their samples, which\nwould help in understanding the properties of their system. In the past, such IVCs provided clear signature\nwhether a system is Josephson coupled or in the flux-flow regime, if a Coulomb gap is present, if\ninhomogeneities play a big role, etc. It thus would be great to have comparable curves in the paper.\nAs I understand the authors primarily record the differential resistance and plot it as the color scale in various 2D\nplots. However, for example in Fig. S5b they integrate the signal to obtain the actual voltage V. Can the same\nprocedure be used to obtain complete IVCs, i.e. plots of currents I_b vs. V? If so, it would be very helpful to see\nrepresentative IVCs, say, along the vertical dashed lines in Fig. 1b. This could include a low-temperature IVC,\nobtained in region SC1, an IVC obtained in the insulating region, around 1.8 K, one in region SC2, and perhaps\none or two more in the N regions. These plots could be presented in the Supplementary Information section. With\nthese or a comparable modification I will be happy to recommend the manuscript for publication in Nature\nCommunications.\nResponse\nWe thank the Reviewer for suggesting that we present the integrated IVCs at different temperature regimes.\nWe have added this demonstration to the Supplementary Information Section VA2 and Fig. S4. As shown in the\ncomparison, regions SC1 and SC2 are characterized by a well-defined critical current, the insulating region\nexhibits a finite insulating gap, and the Shapiro step region displays visible steps in the IVC, supporting the\ndescription provided in the manuscript.\nAdditionally, we compared our data to previous studies, particularly Phys. Rev. B 99, 094522 (2019), where an\namorphous superconductor (TiN) is interpreted within the framework of emergent granularity-At B=0 the\nauthors suggest a phase coherent superconducting state, while applying a magnetic field suppresses E inducing\nJ\nan insulating state due to Quantum Phase Slips.\nResponse to Reviewer 3:\nIn the manuscript «Reentrant superconductivity and superconductor-to-insulator transition in a naturally\noccurring Josephson junction array tuned by RF power» S. Avraham and co-authors claim that they\n«demonstrate reentrant superconductivity as a function of both temperature and magnetic field, tuned by radio\nfrequency (RF) power in a relatively simple system: granular aluminum (grAl), which exhibits the properties of\na naturally occurring Josephson junction array». The main manuscript presents color plots of experimental\nresults measured on a single device in «quasi-four-probe configuration» and with resistance saturation that «is\ndue to experimental limitations» as specified in the supplementary. The measurements consist in resistance\nmeasurements at various current bias, temperature, RF-power, RF-frequency and magnetic field. The\nmeasurements are discussed with a «theoretical model» that actually consist in two different models: a\nRCSJ model and a many-body effective model. They propose that the RF-current enables the reentrance of\nsuperconductivity.\nIf true, the results are definite worth being publish.\nWe thank the Reviewer for finding our data worth publication.\nHowever, the current presentation of the experimental results in the manuscript casts some doubt on the veracity\nof the results and is very default to follow. I therefore ask the authors to make significant changes to the\nmanuscript so that the experimental rigor of their work becomes clear and that readers can understand their\nwork.\nWe appreciate the Reviewer’s comments and hope to demonstrate that our results are both solid and\nreproducible. We have also made a significant effort to improve the readability of the manuscript.\nFirst of all, I had a lot of difficulties to understand what superconducting re-entrance behavior(s) the authors\nis(are) referring to. Part of my difficulties are coming from the color plot presentation: raw data are needed both\nfor clarity and to check for experimental rigor.\nResponse\nWe regret the confusion resulting from our data presentation. We thank for the Reviewer’s suggestion to improve\nour presentation of the data. Below one can observe the reentrant behavior. For example, the green trace shows\nsuperconductivity up to Tc then an insulating regime and finally a second superconducting phase appears near\nBesides, the results were measured on only one device. As reproducibility is a key concept for science and as the\nvariability between devices is interesting, in particular here for the discussion of the various energy scales at play\nand for the possible giant Shapiro steps: I ask for measurements on different devices.\nResponse\nWe are grateful to the Reviewer for suggesting us to present measurements obtained for various devices. (See also\nour response to Reviewer #1). In the revised Supplementary Information, Fig. S9 and Fig. S10, we added some\nof the data obtained by the additional devices summarized in table S1. They exhibit the same qualitative effects\nas the device presented in the manuscript, as we discuss in Supplementary Information Section VE. We note that\nthe measurement setup used for one of the devices (device C) was not completely optimized as the one used for\nthe device presented in the manuscript (device A, for which comprehensive measurements have been conducted).\nFor example, the RF generator used had lower frequency resolution. Therefore, additional noise and minor\nartifacts can be seen, as we elaborate in the revised Supplementary Information Section VE. We also note that the\noverall behavior observed in the additional devices (device B and device C) is similar to the device presented in\nthe manuscript (device A); The quantized voltage of the Shapiro steps observed in device C is of the same order\nof magnitude as device A resulting from roughly 1000 junctions in the array.\nThe discussion regarding the various energy scales is now included in the revised Supplementary Information\nSection VF. First, we note that there are three distinct energy scales in our system: Δ,ℏ𝜔 ,ℏ𝜔 , which are the\np 𝑅𝐹\nsuperconducting gap, which is of the order of 100 GHz, the effective average plasma frequency, which we estimate\nto be of the order of 1 GHz, and the RF frequency, which is of the order of 10 MHz. The RF frequency and the\nsuperconducting gap do not significantly change from one device to another, and they are kept 4 orders of\nmagnitude apart. The plasma frequency may vary from one device to another. Moreover, it may change from one\njunction to another in our array. Nevertheless, we estimate the effective average plasma frequency within our\nsingle-junction picture to be of the order of 1 GHz. While this frequency depends on the junction resistance and\ncapacitance, we can state that the inequality\n𝜔 ≪𝜔 ≪Δ/ℏ\n𝑅𝐹 𝑝\nare still valid for all our devices. An exact determination of the plasma frequency for every junction in the array\nis, of course, impossible. Whereas the effective average 𝜔 is valid only within the simple description of single\np\njunction model.\nBelow, I detail some of my difficulties and questions.\nIf I understood properly with fig1a the authors intend to prouve re-entrance superconductivity with increasing\nRF-power. From the color plot the re-entrance behavior is not obvious as there is no dissipationless\nsuperconducting state (dark blue), neither « Shapiro steps » (or at least a label indicating « Shapiro steps » or «\nSC2 »). Please, note, that I do not see any Shapiro steps in the color plot. Please present the usual I-V curves for\nthe Shapiro steps. If I have misunderstood and the authors mean something else by “tuned by RF power,” please\nclarify here, as well as in the title and, when necessary, in the text. I suggest that the authors mention in the\nmanuscript that for the RF-tuning on the SC1-phase is pretty classic: an increased RF-power, results in an\nincreased RF-current, results in a decreased critical current/gap/EJ (because the total effective current bias\nequals I_b+I_RF), leading to the suppression of superconductivity.\nResponse\nWe thank the Referee for the remarks, and we will try to clarify accordingly. We clarify the explanation for Fig.\n1a according to the Reviewer’s suggestions. We have also modified the manuscript for clarity.\nAs mentioned in the figure caption, Fig. 1a aims to demonstrate tuning of system parameters by applying RF\npower. The reentrant behavior is shown later in figure 1b.\nWe demonstrate the giant Shapiro steps inferred from figure 1a including the requested labeling in the revised\nsupplementary Information Fig. S6. This supplementary figure also shows the usual I-V curve. The observed\ndifferential resistance oscillations in Fig. S6, which were taken at frequency of 24.5 MHz, are indicative of giant\nShapiro steps. They are characterized by rather small amplitudes. Consequently, it is difficult to see the giant\nShapiro steps in the I-V curve. Therefore, we refer to Fig. S7 where we demonstrate the giant Shapiro steps on\nthe I-V curves obtained at higher RF frequencies.\nBy writing “tuned by RF power” in the manuscript title, we mean that the effects observed in the manuscript,\nnamely, the Superconductor-to-Insulator transition, and the Reentrant Superconductivity, are both tuned by RF\npower, as we explicitly demonstrated in the manuscript in Figs. 1a and 1b, respectively. We also note that our\nsingle-junction model explains how the RF power is coupled to the array. We thank the Reviewer’s suggestion to\nbetter clarify the role of RF tuning in the SC1 phase and correspondingly we improved our description in the\nrevised manuscript in the results section.\nIf I understood properly with fig1b the authors intend to prouve re-entrance superconductivity with increasing\ntemperature. From the raw data presented in the supplementary figS3 the dissipationless superconducting state\nreentrant is not demonstrated: the resistance is far from being null (and the data are incomplete, points are\nmissing).\nResponse\nWe thank the Reviewer’s suggestion to improve the demonstration of reentrant superconductivity in Fig. S3. In\nthe revised Supplementary Information, Fig. S3c has been fixed to include all data points. The resistance values\ndemonstrating the dissipationless superconducting state are below the noise level, which is 4 orders of magnitude\nbelow the “insulating” state resistance and three orders of magnitude below the normal state one. Both SC1 and\nSC2 exhibit a dissipationless state which is equivalent in signal to superconductivity observed in the absence of\nRF power (see Fig. 2C and Fig. S3b in the revised supplementary information) where standard superconductivity\nin granular aluminum is observed.\nProbable reentrant superconductivity with temperature (with nonzero resistance) has already been reported in\n(highly) resistive granular aluminum in the following publications:\n- H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M. Goldman, Phys. Rev. B 40, 182 (1989).\n- W. Wu and P. W. Adams, Phys. Rev. B 50, 13065(R) (1994).\n- F. Levy-Bertrand, T. Klein, T. Grenet, O. Dupre, A. Benooit, A. Bideaud, O. Bourrion, M. Calvo,\nA. Catalano, A. Gomez, J. Goupy, L. Grunhaupt, U. v. Luepke, N. Maleeva, F. Valenti, I. M. Pop,\nand A. Monfardini, Physical Review B 99, 094506 (2019).\nIn those publications non RF-signal was applied. Please compare your results with these papers and cite them.\nCan you (re)explain the novelty of your measurement compared to these papers? In particular can you show\nresistance versus temperature measurements (raw data) with identical inputs but with and without RF-signal ?\nResponse\nWe thank the Reviewer for suggesting us to compare our results with these publications. We added such\ncomparison including citations to the discussion in the revised manuscript and in the Supplementary Information\nSection VA3.\nGoldmann et al. are focused on the superconductor to insulator transition, and their devices are tuned by the film\nthickness, which is not possible for in-situ application. They report the emergence of low-resistance state at high\ntemperature that has sheet resistance of the same order of magnitude as the normal state resistance. There are no\ntwo distinct superconducting states as in our experiment and for example in field tuned UTe (Nat. Phys. 15, 1250\n2\n(2019)).\nWu and Adams study granular aluminum, however their work is focused on the SIT and tune the insulating state\nby DC current. They do not show resistance that is significantly lower than the normal state one. In addition, there\nare no two distinct superconducting states as in our experiment and for example in field tuned UTe .\ndeposition, which is not possible for in-situ application. Their work is also focused on the SIT. They report\nemergence of low-resistance state at high temperatures. However, the sheet resistance at this emergent state is of\nthe same order of magnitude as, or even greater than the normal state resistance. Furthermore, there are no two\ndistinct superconducting states as in our experiment and for example in field tuned UTe .\nsystem parameters by using RF power and temperature, which leads to the simultaneous observation of reentrant\nsuperconductivity and giant Shapiro steps in a single experimental platform. Our measurement methods are\ndedicated to mesoscopic nanobridge devices that include a relatively small number of Josephson junctions, thus\nenabling our clear experimental observations. Therefore, they are different from those presented in the\npublications mentioned by the Reviewer, which study much larger macroscopic granular samples. In addition, our\ngranular samples are characterized by a smaller grain size that has been shown to improve their uniformity and\ntheir superconducting properties. As suggested by the Reviewer we have added a discussion about the proposed\nreferences with the similarities and differences. A comparison between the Resistance versus temperature curves\nmeasured with RF signal and without RF signal (raw data) is given in Fig. S3b.\nFor now I do not agree with the sentence: «We reiterate that the reentrance appears only upon the application of\nRF current and does not occur using DC current.» A RF power P_RF=-20dBm is probably like not RF-signal\nand yet you observed (a non zero) re-entrance superconductivity with increasing temperature.\nResponse\nWe are sorry that this point was not clear enough. In the revised manuscript we refer to figure 2c when stating\nthat «We reiterate that the reentrance appears only upon the application of RF current and does not occur using\nDC current.» This figure clearly shows that upon application of DC current superconductivity is suppressed as\nexpected from simple Ginzburg Landau theory without reentrance or superconductor-to-insulator behavior. This\ndoes not contradict nor cast any doubt on the paper by Wu and Adams or on that of Levy-Bertrand et al. where a\nmuch more resistive and/or with larger grains samples have been studied.\nThe re-entrance seems wider in temperature (thus more robust) at low RF-power. Any comments ?\nResponse\nThe observation that the reentrant superconducting phase SC2 is wider in temperature for low RF powers is\nconsistent with our theoretical interpretation elaborated in the manuscript. For higher E_J/E_C values (low RF\npower), the degree of screening needed to preserve superconductivity is smaller, thus the reentrant phase will\nappear at lower temperatures. We added these clarifications to the Supplementary Information Section VA1.\nI do not see the Shapiro steps in fig 3. Is fig3c a simulation ?\nResponse\nFig. 3c is showing measurement results. It is indeed more convenient to observe the Shapiro steps in the revised\nsupplementary Information, Figure S7.\nI do not understand how the authors estimate the number of inter-grain JJs [for me\nNx=120nm/(2x5nm+1nm)=24, Ny=200nm/(2x5nm+1nm)=40, Nz=55/(2x5nm+1)=11]. Can the authors\n(re)explain the estimation procedure?\nResponse\nThere are two ways to estimate the number of JJs in the array. The first is from geometrical considerations which\nyield roughly 40×67×18 junctions. The second method is by fitting the differential resistance minima points\nto the giant Shapiro steps relation 𝑉 =𝑛×𝑁 ×Φ ×𝑓 . This fit yields 430 Junctions. We associate the\nn 𝐽 0 𝑅𝐹\napparent discrepancy with a contribution from grains in the contact region that also contribute to the overall signal.\nDemonstrations of the giant Shapiro steps and their fitting procedure have been added to the revised\nSupplementary Information (Fig. S7) and are now referenced in the revised manuscript in the Results section.\nThe actual measurement set-up needs to be clarified. In particular the expression «quasi-four-probe\nconfiguration» p.5 of the supplementary needs to be clarified in the manuscript. Can you provide an accurate\nscheme of the actual geometry ? Do you filter out the ac-voltage (coming from the RF-current) from the dc-\nvoltage? Do you measure the resulting RF-current ?\nResponse\nWe added a scheme for the quasi-four probe measurement configuration to the revised Supplementary\nInformation, Fig. S11a. The RF-voltage goes through two filtering steps that eliminate it. First, the low pass filters\nlocated in each DC line as demonstrated in the experimental setup scheme Fig. S11b. Then, the lock-in technique\nis sensitive only to frequencies close to DC (a few Hertz). We therefore do not measure the RF voltage. We added\nthese clarifications to the revised manuscript in the methods section. As for the measurement of the RF current\ninduced to our device, we do not directly measure this current, and we only observe its effects on the JJ array\nproperties.\nOther specific comments and suggestions are:\nI completely disagree with the last sentence of the introduction «Our system can therefore be tuned to observe\nboth the single-junction regime and many-body correlation effects, serving as a quantum simulator for complex\nphenomena in condensed matter physics. ». The behavior of granular aluminum is far from that of a single-\njunction as evidence (in the manuscript itself!) by the «giant Shapiro-steps» relation between the voltage and the\nRF-frequency. Besides, I do not understand how the authors intend to control the many-body correlation effects,\nneither how a single experimental device could be used for quantum simulations.\nResponse\nOur system can be described within a single-junction like model in the low power low temperature regime (The\nRCSJ model) on the other hand it is showing the behavior expected for many-body correlation system as predicted\nby Efetov. In that sense a single system can be tuned to show both regimes.\nIt is common to present a vision of the future in many papers. In our case the use of our system as a quantum\nsimulator is merely a vision that has not yet been realized.\nI suggest you remove the following sentence that is cryptic to me: «Reentrant superconductivity has also been\nexplored through gauge-gravity duality, where black hole stability serves as a probe of holographic\nsuperconductivity in the dual non-relativistic field theory ».\nResponse\nWe thank the Reviewer for the suggestion. We are interested in keeping this sentence which aims to illustrate the\nimportance of reentrant phenomenon to the broader physics community.\nI suggest you elaborate more on the following sentence or you remove it: «Additionally, the electronic level\nspacing within each island exceeds both Ec and the superconducting gap, making the system less sensitive to\nmicroscopic details.»\nResponse\nThis sentence is intended to provide intuition to why despite the random packing our system exhibits coherent\neffects such as the giant Shapiro steps. Disorder usually results in smearing of the energy levels, however when\nthey are strongly separated apart the disorder is less effective. We elaborated on this sentence in the revised\nmanuscript in the introduction section.\nI hope you find my report helpful.\nWe appreciate the time of the Reviewer and the effort they made to improve our manuscript.\n8"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The replies are completely convincing. I thank the authors for their work, and look forward to seeing this published.", "reviewer_2": "In their reply and the revised manuscript the authors addressed all my comments and criticisms. Particularly, they provided the requested plots of current voltage characteristics and the corresponding discussion in the text.\nFrom my side the manuscript is ready for publication."}, "author_response": ""}]

Decision letter