## 3 Experimental Setup  
The experimental apparatus used to electrically excite a QTF is sketched in Figure 3a. The analog output of a DAQ card was used to provide a sinusoidal electrical excitation for the QTF. The QTF was excited at its fundamental in-plane flexural mode. A LabVIEW-based software was used to vary the excitation frequency step-by-step around the resonance, with a peak-to-peak amplitude fixed to 0.5 mV. The piezoelectric current was converted into a voltage signal by a trans-impedance (TA) amplifier and acquired via the data acquisition (DAQ) card through LabVIEW-based software, which allowed retrieval of the QTF signal at the excitation frequency. The QTF was placed in stainless-steel housing (SSH) with an inlet and outlet to pump ambient air. Air passed through a first needle valve (NV1 in Figure 3a), the SSH, a second needle valve (NV2) and a vacuum pump. The valves were used to regulate the pressure inside the SSH. The pressure value was measured using a pressure display (DIGITRON 2025P). A pressure range from 10 Torr to 760 Torr was investigated. These measurements were repeated in three different configurations. Firstly, a bare T-shaped QTF was used. Then, a cylindrical V-groove (not shown in Figure 3a) was placed around the QTF to hold and fix two resonator tubes. The two tubes were mounted on both sides of the QTF (see Figure 3a) at a distance of \(d\) = 200 \(\upmu\)m, perpendicular to the QTF plane with the tube center 2 mm below the QTF top [10]. Lastly, the distance of 200 \(\upmu\)m was halved. A sketch of the T-QTF is reported in Figure 3b. The geometrical parameters of the T-QTF and the tubes are summarized in Table 1.  
Figure 3: (**a**) Experimental apparatus: SSH—stainless-steel housing; NV1—needle valve 1; NV2—needle valve 2; TA—transimpedance amplifier; DAQ—data acquisition card; WFG—waveform generator. (**b**) Sketch of T-shaped QTF. Taking into account the results obtained in Section 2.2, Equation (9) is used as fitting function in the form of:  
\[V_{out}=A\sqrt{\frac{(1-x)B\Big{(}C^{\prime 2}B(1-x)+\left(C^{\prime}x+1\right)^{2} \Big{)}}{x^{2}+(1-x)B}} \tag{10}\]  
where \(B=1/Q^{2}\) and \(x=1-f^{2}/{f_{0}}^{2}\). The \(f_{0}\) value corresponding to the peak is needed for the fitting procedure. Since its value is not modified by the stray capacitance thanks to the virtual ground imposed by the transimpedance amplifier, as is observed in Figure 2a, \(f_{0}\) firstly can be determined by imposing a Lorentzian fit. Then, Equation (10) can be used for a second iteration fit to estimate the \(Q\)-factor. The best fit by using Equation (10), for the highest (760 Torr) and lowest (10 Torr) pressures at which the bare QTF was investigated, is also shown in Figure 4a,b as solid red lines. From the preliminary Lorentzian fitting procedures, resonance frequencies \(f_{0}\) are extracted and plotted as a function of the air pressure in Figure 5 (datapoints) for the three configurations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(L_{1}\) & \(L_{2}\) & \(T_{1}\) & \(T_{2}\) & \(w\) \\ \hline T-QTF & 2.4 & 9.4 & 2.0 & 1.4 & 0.25 \\ \hline  & \(L\) & OD & ID & & \\ \hline Tubes & 12 & 1.83 & 1.59 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the T-QTF and of resonator tubes. \(L_{1}\) and \(T_{1}\) are the length and thickness of the T-shaped part of the QTF, \(L_{2}\) is the total prong length, \(T_{2}\) is the prong thickness, and \(w\) is the crystal width. OD and ID are the outer and internal tube diameters, respectively, and \(L\) is the tube length. All sizes are in millimeter units. Figure 4: Resonance curve (black datapoints) for T-QTF at (**a**) 10 Torr, and (**b**) 760 Torr. The solid red line is the best fit of experimental data obtained by using Equation (10). As expected from Equation (4), \(f_{0}\) decreases linearly as pressure increases. The intercept values of linear fits, i.e., the resonance frequencies in vacuum, are different for three configurations.