## 3 Experimental Setup: Reflectometer and Matching Network Considerations
### Matching Network Design
In our experiment a typical value of \(C_{pud}\approx\) 0.5 pF is determined by the size of the bond pads and stray capacitance of the bond wire to ground. We use off the shelf surface mount size 0805 ceramic core inductors (220-820 nH) by Coilcraft and ceramic surface mount capacitors by Johanson Technology. The lower frequency limit of the reflectometry setup (\(\approx\)100 MHz) is chosen to increase the SEBA admittance in accordance with equation (2); the upper frequency limit is determined by the homodyne detector used for carrier demodulation (600 MHz in our experiment, set by a frequency limit of ZI UHF). By changing a value of the input capacitance, \(C_{IN}\), the MN can be adjusted to convert \(Z_{SEBA}\) to \(Z_{load}\approx Z_{0}=50\)\(\Omega\). To determine \(C_{IN}\) that ensures the operation near a match point, the response of the MN is simulated using a realistic 5-element model of the inductor provided by the manufacturer (Coilcraft). However, to accurately simulate the MN these parameters need to be adjusted for low ambient temperature; failing to do so leads to gross errors and non-functioning MN. Note that due to large values of device impedance (\(>\)10 M\(\Omega\)), the RF power is dissipated almost entirely in the parasitic resistances of the inductor coil rather than the SEBA components. This factor ultimately limits the sensitivity of the MN [15]. Experiment shows that DC resistance of inductors used in this work drops by approximately 70 times when it is cooled from room temperature down to about 20 K, and changes insignificantly at lower temperatures (Figure 5, red curve), which is consistent with the resistivity of a copper coil. This drop in resistivity also impacts the skin-effect resistance (\(R_{SE}\) in MN circuit, inside the red box in Figure 4a) as the pre-factor scales down proportional to the resistivity. Experiment shows that by contrast, the capacitance of 0805 size Johansson Technology chip capacitors used in experiments changes only by about 1% from 300 K down to low temperature (Figure 5, black curve). Using this information, the response of a single SEB connected to a MN (Figure 6a) is simulated using Matlab code and Keysight Advanced Design Systems (ADS) @ software. To illustrate the significance of proper MN tuning three examples of \(C_{IN}\) are considered. The blue curve in Figure 6b corresponded to a slightly overcoupled case, for which \(Z(f_{res})<50\)\(\Omega\); the red curve corresponded to a slightly undercoupled case, \(Z(f_{res})>50\)\(\Omega\) and the green curve corresponded to the critically coupled case, \(Z(f_{res})\approx 50\)\(\Omega\). Next, the oscillations of admittance of a single SEB connected to the MN in Figure 6a in response to sweeping \(V_{g}\) are simulated (Figure 6(c1),(c2)). Note that both magnitude and phase of admittance are simulated for each resonant frequency corresponding to the respective minima in Figure 6b (462.309 MHz for \(C_{IN}\) = 20 pF, 461.288 MHz for \(C_{IN}\) = 28.2 pF and 460.439 MHz for \(C_{IN}\) = 40 pF) but they are indistinguishable in Figure 6(c1),(c2). By contrast, the oscillations in the reflected signal resulting from SEB admittance oscillations differed drastically for closely matched (\(C_{IN}\) = 28.2 pF) vs. the other two (\(C_{IN}\) = 20 pF and \(C_{IN}\) = 40 pF) cases. In Figure 6(c3),(c4) we plotted respective derivatives \(d|\Gamma|/dV_{S}\) and \(d\Theta/dV_{S}\) of the reflected signals as they represent sensitivity as is explained in Section 2. It is straightforward to see that the MN with \(C_{IN}\) = 28.2 pF yields a much stronger signal and therefore it offered a significant improvement in both the magnitude and phase response. For the chosen parameters of the devices and temperature the resulting enhancement of oscillations is far more prominent in the \(d\Theta/dV_{S}\) signal than in \(d|\Gamma|/dV_{S}\) for the reasons discussed below in the experimental section.