## 3 Experimental Setup: Reflectometer and Matching Network Considerations
### Matching Network Design  
To use an SEB as a detector using reflectometry one needs to detect impedance changes in the devices in response to a gate voltage that yields the change in the reflection coefficient \(\Gamma=(Z_{Load}-Z_{0})/(Z_{Load}+Z_{0})\). As shown above, the impedance of a single SEB with parameters from Figure 3 probed with the RF signal in the range of 100-1000 MHz is on the order of 100-10 M\(\Omega\). The impedance for SEBA scales down as \(\sim\)1/\(N^{1/2}\), yielding a minimal expected impedance \(\geq\)0.7 M\(\Omega\) for N \(\leq\) 200 studied here. Figure 4: **(a)** Simplified circuit diagram of the single-port reflectometry setup. Circuit delineated by a dashed line is the sample attached to the PCB board with the MN and cooled to a temperature defined by the setup; the MN circuitry is outlined in a red box with port 1 facing the sample and port 2 facing the directional coupler. Blue rectangle delineates the cryogenic part of the measurement apparatus. Amplifier at 40 K, ZX60-P33 ULN+ is thermally anchored to the 1 st stage of the cryocooler, \(T\approx\) 40 K; two stage amplifier at room temperature uses ZX60-P103 LN+; both types of amplifiers are by Mini-Circuits. Directional coupler ZFDC-20-50-S+ and cold attenuator are located at the second stage of cryocooler @2.7 K (Janis He3, Bluefors) or 3.5 K (ARS). The PCB with the sample is bolted to the He3 pot (Janis He3); mixing chamber sample exchange platform (Bluefors) or 2 nd stage of cryocooler (ARS). **(b)** Block diagram of the experiment with a SEB array coupled to two gates with capacitances C\({}_{11}\)... C\({}_{1n}\) to \(V_{g1}\) (“sensing gate”) and C\({}_{21}\)... C\({}_{2n}\) to \(V_{g2}\) (“tuning gate”). **(c)** Block diagram of the experiment for comparative measurements of two arrays. The junction sides of the SEBs for both arrays are connected to the same MN. For simplicity each array is shown as a single SEB. The devices are spatially separated on the chip to minimize capacitive crosstalk. Direct measurements of large impedances using reflectometry are challenging because the impedance of transmission lines is low, typically \(Z_{0}\) = 50 \(\Omega\). Without an impedance transformation, the changes in the reflected signal induced by changing device impedances are extremely small. Therefore, a carefully designed MN becomes crucially important for the signal extraction. The main purpose of the MN that performs this impedance transformation is therefore to convert the load impedance to the value close to a characteristic impedance of the transmission line, \(Z_{0}\), so that changes in the load impedance \(Z_{load}\) will result in noticeable changes of the reflected signal. Note that strictly speaking all of the following considerations are only applicable if the measurement system is calibrated against known standards. This means, for example, that at a calibrated reference plane, both open and short standards are expected to produce \(|I|\) = 1 and a matched load should produce \(|I|\) = 0. Experimental setups never produce these results without the use of error correction because of a wide range of non-idealities in the signal paths (standing wave resonances in the transmission lines, deviations from exact \(Z_{0}\), matched values in components used, frequency dependent transfer characteristics and phase shifts in the amplifiers and couplers, etc.). If calibration cannot be performed, the experimental results can only qualitatively be compared with the theoretical predictions. The optimization of MN that uses a critically coupled resonator is discussed in [29]. The secondary role of MN is to act as a band-pass filter to enable propagation of the carrier and sidebands while attenuating the out-of-band signals and thus improving the signal to noise ratio. This bandwidth consideration needs to be taken into account for proper design of the MN. As mentioned above, the MN used in this work is a \(\Pi\) network (\(C_{pud}\)-\(L\)-\(C_{IN}\)) shown in red box Figure 4a.