## 3 Experimental Setup
The trends show a strong dependence on air pressure, especially at pressures lower than 20 Torr, suggesting that the dominant loss mechanism is air damping. Moreover, working at pressures lower than 100 Torr, the \(Q\)-factor dramatically increases as pressure decreases, meaning that air pressure requires a higher efficient stabilization with respect to operating at pressures close to the atmospheric one. Negligible differences are observed when the QTF-tube distance is reduced from 200 \(\upmu\)m to 100 \(\upmu\)m: if the tube-QTF distance is reduced from 200 \(\upmu\)m to 100 \(\upmu\)m, the superchophene \(Q\)-factor is almost not affected. This \"relaxes\" the assembling procedure of the superchophene, avoiding the requirement of a superfine placement of tubes. This can be explained by noting that the prong spacing (800 \(\upmu\)m) is comparable with the tube ID (see Table 1); thus, the effective interacting surface can be neglected. Indeed, while prongs are vibrating, they could lose energy via interaction with the tube walls (squeeze damping). If this additional loss mechanism is present, it should be influenced by: (i) the tube-QTF distance: the lower the distance, the higher the contribution; (ii) the air pressure: the higher the pressure, the higher the squeeze damping. Thus, in Figure 6, a squeeze damping effect should result in a deviation at higher pressures of red circles (spectrophone with \(d=100\)\(\upmu\)m) towards lower values, with respect to blue triangles (spectrophone with \(d=200\)\(\upmu\)m). Conversely, red circles and blue triangles should overlap at lower pressures. Instead, both datasets are quite overlapped in the whole of the investigated pressure range, demonstrating that squeeze damping is negligible. In the whole pressure range, the coupling with tubes leads to a reduction in the spectrophone \(Q\)-factor with respect to the bare QTF. This reduction increases as pressure increases and is almost zero at pressures lower than 100 Torr. This behavior can be explained by considering that the effect of the acoustic coupling of the high \(Q\)-factor QTF with the low \(Q\)-factor tubes, which leads to a QTF loss of energy, is reduced at low pressures, because of the reduced number of surrounding molecules. Indeed, at 10 Torr,  
Figure 6: \(Q\)-factor values as a function of the air pressure (datapoints) for the three investigated configurations. Solid lines are best fits of experimental data by using Equation (12). the \(Q\)-factor of the spectrophone (45,454) is comparable with the one measured for the bare QTF (45,221). As a result, the surrounding air strongly affects the acoustic coupling of both resonators, namely, the tube and the QTF. It is worth noticing that datapoints in Figure 6 close to atmospheric pressure deviate from fitting with Equation (12). This is expected because [PERSON]'s model in Equation (5) was rewritten with the assumption that air density is proportional to pressure. This is valid only if the ideal gas law is assumed. Approaching atmospheric pressure, the ideal gas law is no longer valid, requiring a replacement with the van der Waals equation (real gas law). As the dissipation mechanisms described in Section 2.1 are independent of each other, the overall quality factor can be written as:  
\[\frac{1}{Q(P)}=\frac{1}{Q_{0}}+\frac{1}{Q_{air}(P)} \tag{11}\]  
where \(1/Q_{0}=1/Q_{TED}+1/Q_{sup}\). Using Equation (5) for \(Q_{air}\), \(Q(P)\) can be rewritten as:  
\[Q(P)=\frac{1}{C+D\sqrt{P}} \tag{12}\]  
where:  
\[\left\{ \begin{array}{c}C=\frac{1}{Q_{0}}+\frac{3\mu}{4\rho Tw_{0}}\\ \frac{3}{4\rho Tw_{0}}\\ D=\frac{1}{4\rho T_{0}}\end{array} \right. \tag{13}\]  
With this formulation, parameter \(C\) takes into account all the pressure-independent loss mechanisms. Equation (12) was used to fit the experimental data and the results are shown as solid lines in Figure 6. The obtained fit curves were used to evaluate the difference \(\Delta Q\) between the \(Q\)-factors of the bare QTF and the spectrophone with \(d=200\)\(\upmu\)m, as a function of the air pressure.