## 5 Experimental Results and Discussion
### Characterization of SEB Arrays
#### 5.2.1 Characterization of a Small (\(N=3\)) SEBA
To illustrate this technique, an example of the reflectometry signal obtained from the same array, now coupled to two gates is presented in Figure 12. In Figure 12a, in the top and bottom subplots sharp lines represent respective derivatives that define sensitivity in magnitude and phase with respect to changing gate bias, \(d|\Gamma|dV_{g}\) (upper left plot) and \(d\Theta/dV_{g}\) (bottom left plot) of the reflected signal. Each SEB generates a set of lines with a distinct slope \(\partial V_{g\_tune}/\partial V_{g\_sens}~{}=~{}-C_{g\_tune}/C_{g\_tune}\). Constructive interference peaks appear at the points of line crossings. Note that the spacing between lines in each SEB along the \(V_{g\_tune}\) axis is almost the same, while along the \(V_{g\_tune}\) axis it is distinctly different, indicating dissimilar capacitance \(C_{g\_tune}\) for each SEB. To achieve weak and dissimilar coupling tuning gate can be simply positioned on the side of the array [33]. This combination ensures the appearance of line crossings within an easily accessible span of \(V_{g\_tune}\). Near these line crossings, one can find the points of maximum sensitivity where derivatives reach their extrema clearly visible at \"knots\" on the plot (there are eight such \"knots\" in Figure 12a). Clearly, the use of a second gate makes it easier to reach a high-sensitivity crossing point where the signals add up (and extrema of respective derivatives can be chosen as \"sweet spots\" for the sensing application) within a given \(V_{g}\) span, since to find such a point with a single gate would require a much broader span of \(V_{g\_sens}\). This example illustrates that the presence of the secondary gate greatly reduced the necessary \(V_{g\_sens}\) span in the search for a constructive interference peak as a point of highest sensitivity. Note, however, that in the case of very large ratios of \(E_{C}\)/\(k_{B}T\geq 100\) signals generated by each SEB start looking like periodic \"delta-functions like\" (Figure 8) with a very narrow voltage span for each peak leading to a collapse of overlapped regions and a disappearance of constructive interference peaks. Curiously, in the \(d\Theta/dV_{g}\) signal (lower panels in Figure 12) one of the lines, with the steepest slope, had a significantly stronger appearance compared to the other two while in \(d|\Gamma|dV_{g}\) plots the strength of the signal from each SEB are almost equal. We performed simulations of the response for this array using parameters of SEBs obtained from higher temperature measurements in Figure 11. However, to obtain a good fit to the experimental data in the simulated \(d\Theta/dV_{g}\) response, the resistance of the two junctions, corresponding to the two lines with poor contrast in this plot need to be increased by almost an order of magnitude. The reason for that likely stems from incomplete suppression of superconductivity in the two SEBs in the array at \(T<0.5\) K. Opening of the gap in the density of states at the metal-superconductor transition drastically reduces the tunneling rate at zero bias due to a lack of available quasiparticles. This is confirmed by the disappearance of the reflected signal from the SEB if no magnetic field is applied for temperatures \(T<T_{C}\approx 1.05\) K, where \(T_{C}\) is critical temperature for Al. Indeed, in our experimental setup with small permanent magnets the superconducting gap is not completely suppressed in about 50% of the samples. Due to tunnel rate reduction in the two SEB with developing the superconducting gap, the admittance there started to be dominated by Sisyphus resistance while significant reduction in dynamic capacitance term in (2), in turn, resulted in the greatly reduced phase swings at the charge degeneracy points. 2.2 Comparative Characterization of Larger (\({}^{\prime\prime}N=40^{\prime\prime}\) vs. \({}^{\prime\prime}N=200^{\prime\prime}\)) SEBAs  
As discussed in Section 2, utilization of \(N\) SEBs in an array is expected to enhance the \"collective\" oscillation magnitude by a factor of \(\sim\sqrt{N}\).