## 3 Experimental Setup: Reflectometer and Matching Network Considerations  
### Hardware Configuration  
RF reflectometry is used in this paper to measure the response of the SEBs. The general approach for reflectometry measurements used in this work is presented in Figure 4a. Reflectometry measures the ratio of the reflected RF voltage wave \(V^{-}\), to the incident RF voltage wave \(V^{+}\), on a load, which is a function of the load impedance: \(\Gamma=V^{-}/V^{+}=(Z_{load}-Z_{o})/(Z_{load}+Z_{o})\). In this case, the load is an SEB coupled to a MN (red box in Figure 4a, see below for details). Therefore, the reflection coefficient, \(\Gamma\), will track the impedance of the SEB as a function of the gate voltage. We used a single port RF homodyne reflectometer where the probing RF signal is attenuated on the way to the sample (approximately 60 dB) by warm and cold attenuators and a directional coupler (ZFDC-20-50-S+ by Minicircuits). The attenuated signal is sent down the transmission line to the sample through the MN. A matching \(\Pi\) network, composed of input capacitor \(C_{in}\), surface mount inductor \(L\) with associated parasitic components, and pad capacitor \(C_{pad}\) (Figure 4a) is tuned to achieve a sensitive response to small variations in admittance schematically represented as \(\gamma(V_{g1},\,V_{g2})\). The photography of a typical printed circuit board used in experiments with a fabricated sample and surface mounted components is shown in Figure S1 (Supplementary Materials). Note that the input of the reflectometer is a DC ground because of the internal design of the directional coupler (R\({}_{\text{DC}}<\) 10 \(\Omega\)). The signal reflected from the devices is picked up by the cold preamplifier A1 ZX60-P33 ULN+ (Minicircuits) with a low noise figure (0.5 dB at 300 K) and gain about 20 dB are kept at \(T\approx\) 40 K and located at the 1 st stages of closed cycle refrigerators (CCRs). A directional coupler and cold attenuator are located at the second stage of the CCR. The PCB with the sample is bolted to the coldest part of each of the refrigerators used in the measurements: He3 pot (Janis He3); mixing chamber sample exchange platform (Bluefors) or 2 nd stage of He4 DE-210 CCR (ARS; the details on the board design are provided in Supplementary Materials). The noise floor of the setup is primarily determined by the cold amplifier. Our estimates show that the noise temperature of the amplifier is below 10 K [28]. Further amplification by about 40 dB is done at room temperature by amplifiers A2 and A3 (both ZX60-P103 LN+ by Minicircuits). The amplified signal is down converted to the base band by an ultra-high frequency lock-in amplifier by Zurich Instruments (UHF ZI) to produce a signal proportional to the reflection coefficient of the combination sample plus MN. Gate voltage biasing is achieved using computer controlled analog-to-digital converters with the signals lines going to the devices RF filtered and attenuated. Figure 4b shows a block diagram of an experiment with two gates wired to the SEBA, and Figure 4c shows a configuration designed to test relative sensitivity of the two SEBA with different coupling to the two gates using a single MN. Selection of the RF amplitude considers several factors. First, the amplitude must induce voltage swings less than the charging energy (\(E_{C}\)/e). Second, the heating caused by the RF signal must be less than the environmental heating (\(k_{B}\)/T/e). If these considerations are made, then the RF signal will not \"smear\" the response of the SEBs. Practically, this point is determined by ramping the RF voltage until smearing is observed. The measurement voltage is then chosen to be below this point while still retaining a sufficient signal-to-noise ratio. This usually results in RF amplitudes at the device on the order of tens to hundreds of microvolts.