## 2 Analysis and Simulations of SEB Arrays for Probing Using RF Reflectometry  
From the standpoint of circuit design, the SEB represents a two terminal voltage controlled variable admittance (\(\gamma\); Figure 1d). When Coulomb blockade prohibits electron transfer through the junction, its admittance reaches a minimum and approaches that of two capacitors \(C_{J}\) and \(C_{g}\) in series, while when the Coulomb blockade is lifted the SEB admittance is maximized. Near charge degeneracy points, two mechanisms associated with the underlying physics of single electron charge transfer are responsible for enhancement of admittance of the SEB. One is the so-called Sisyphus resistance [22]--excess dissipation at RF frequencies \(\omega\) approaching or exceeding tunneling rate \(\gamma\) through the junction with resistance \(R_{J}\), \(\omega\geq\gamma\). For a given temperature \(T\) the tunneling rate at the degeneracy point is given by  
\[\gamma=2k_{B}T/e^{2}R_{J}. \tag{1}\]  
The second effect is the enhancement of capacitance near charge transition points in metal island single-electron devices, where quantum capacitance effects [23; 24] are negligible--the so-called dynamic input capacitance [25] stems from the ability of an electron to travel through the junction leading to the enhancement of the perceived capacitance above the value of geometrical capacitance \(C_{g}\). These two components of total admittance change as the device goes in and out of the blockade, resulting in the phase and magnitude oscillations of the SEB admittance yielding and equivalent circuit of a parallel combination \(G_{\text{Sis}}(V_{g})\) and \(C_{Dyn}(V_{g})\)[26]:  
\[ \begin{array}{l}Y\ =\ G_{Sys}+j\omega C_{Dyn}\\ G_{Sys}\ =\ \frac{e^{2}\alpha^{2}\gamma}{4k_{B}T}(\frac{\gamma^{2}}{\omega^{2}}+1)^{ -1}\cosh^{-2}(\frac{-\alpha\alpha V_{g}}{2k_{B}T})\\ C_{Dyn}\ =\ \frac{e^{2}\alpha^{2}}{4k_{B}T}(\frac{\omega^{2}}{\gamma^{2}}+1)^{ -1}\cosh^{-2}(\frac{-\alpha\alpha V_{g}}{2k_{B}T})\end{array}  \tag{2}\]  
Here \(\alpha\ =\ C_{g}/(C_{g}+C_{j})\) is the lever arm factor of the gate and \(\Delta V_{g}\) is the gate voltage relative to an SEB population degeneracy point. It is straightforward to demonstrate that the magnitude of oscillations of both components of admittance for an SEB is much smaller than the respective magnitude of conductance oscillations in the SET with the same parameters \(C_{j}\), \(C_{g}\) and \(R_{j}\). Indeed, the SET source-drain admittance is predominantly real and for \(V_{ds}\approx 0\) oscillates between the constant value determined by the resistances of two SET junctions, \(R_{J1}\) and \(R_{J2}\): \(G_{max}=1/(2(R_{J1}+R_{J2}))\)[27] and at minima experience similar exponential suppression of conductance as in the SEB \(\sim\cosh^{-2}(-e\alpha\Delta V_{g}/2k_{B}T)\). By contrast, admittance in the SEB contains both real and imaginary parts whose relationship is a strong function of temperature and probing frequency. The simulated response of an SEB to the gate bias change calculated for \(f=\)198.8 MHz--the value chosen because of one of the MNs used in experiment--using (2) for two different temperatures 4.2 K and 50 mK, is presented in Figure 2. The temperatures are chosen to represent temperatures typical for the dilution and liquid helium refrigerators used in the experimental section of this paper. Figure 2a,b illustrate changes in admittance magnitude and phase, respectively, while Figure 2c,d show variations in active (Sisyphus conductance) and reactive (dynamic capacitance) parts of the SEB admittance, respectively. At the lower temperature of 50 mK (blue curves) both effects are contributing to admittance near transition points leading to oscillations in both magnitude and phase of the admittance. At the higher temperature of 4.2 K (red curves), however, the increase of the tunneling rate according to (1) leads to \(\gamma>\omega\) and charge transitions at the degeneracy points became quasiadiabatic. This minimized the dissipation and effectively eliminated the Sisyphus conductance part in (2).