## 3 Mechanical Frequency Tuning  
The application of a positive voltage to a single membrane chip with a 200 \(\upmu\)m-thick frame, as depicted in Figure 1b, results in a reduction of the tensile stress of the SiN film. In this tensile stress-dominated regime the resonance (angular) frequencies of the mechanical modes are to a very good approximation given by  
\[\omega_{m,n}=\sqrt{\frac{\mathcal{T}}{\rho}}\frac{\pi}{a}\sqrt{m^{2}+n^{2}}, \tag{1}\]  
where \(\mathcal{T}\) is the tensile stress, \(\rho\) the density of SiN, \(a\) the membrane lateral dimension and \(m\), \(n\) strictly positive integers, and can thus be tuned by the application of a bias voltage \(V_{dc}\) to the piezoelectric element, as demonstrated in [56]. A linear reduction with \(V_{dc}\) of the resonance frequencies of both the fundamental (\(m=1,n=1\)) and a higher order (\(m=3,n=3\)) modes is observed (Figure 3a,b), with frequency shifts of \(-43\) Hz/V and \(-130\) Hz/V, respectively. Furthermore, as shown in Figure 3c, the mechanical quality factors are observed to be only weakly (slight increase for this sample) dependent on the bias voltage. We now turn to the piezoelectric actuation of double-membrane arrays as depicted in Figure 1c,d. The compression of the piezoelectric transducer (PZT) affects the whole Si frame/SiN films structure. In both geometries the lower (closer to the piezoelectric element) membrane sees its tensile stress decrease when a positive dc-voltage is applied to the PZT, whereas the upper (further away from the piezoelectric element) sees its stress increase. Figure 3d,e shows the variations with the bias voltage of the (1,1) and (3,3) modes of an array with a 200 \(\upmu\)m-thick frame. Shifts of \(-16\) Hz/V and \(-44\) Hz/V are respectively observed for the (1,1) and (3,3) modes of the lower membrane, and 9 Hz/V and 29 Hz/V for those of the upper membrane. The mechanical quality factors are observed to be essentially independent of the bias voltage (Figure 3f). Figure 2: Schematic experimental setup for the piezoelectric actuation and detection of the membrane vibrations (shown here for the array geometry of Figure 1d). ECDL: external cavity diode laser, BS: beamsplitter mirror, PZT: piezoelectric transducer, PD: photodiode, SA: spectrum analyzer. Piezoelectric tuning of the modes of a shorter (\(\sim 8.5\)\(\upmu\)m) array with thicker frame (500 \(\upmu\)m) and in the geometry of Figure 1d was also demonstrated and reported in [57]. Figure 4a,b shows the variations with \(V_{dc}\) of the resonance frequencies of the (1,1) and (2,2) modes of both membranes. The two membranes of this array exhibit fairly similar bare mechanical properties; for instance, the fundamental modes of the upper (A) and lower (B) membranes have resonance frequencies 721.05 kHz and 721.55 kHz, respectively, in absence of biasing. It is then possible to tune them to degeneracy by applying a bias voltage of approximately 56 V, as shown in Figure 4. The mechanical quality factors are also observed to be essentially independent of the bias voltage for this sample [57]. A careful analysis of the evolution of the thermal noise spectra allows furthermore for extracting the intermembrane coupling, as discussed in [57]. Figure 3: **(a,b)** Measured _single membrane_ (1,1) and (3,3) mode frequencies as a function of bias voltage \(V_{dc}\). **(c)** Corresponding resonance frequencies shifts versus \(V_{dc}\). **(d,e)** Measured (1,1) and (3,3) mode resonance frequencies of the A (red circles and empty blue squares) and B (red dots and filled blue squares) membranes in a _double-membrane_ array as in Figure 1c, as a function of \(V_{dc}\). **(f)** Corresponding resonance frequency shifts versus \(V_{dc}\). The error bars are smaller than the size of the data points. The insets in (a,b) and (d,e) show the variations of the mechanical quality factors with \(V_{dc}\).