## 4 Parametric Amplification of the Thermal Fluctuations
### Theoretical Model
The variations with \(\epsilon\) of the HWHM obtained from the Lorentzian fits is also shown in Figure 6b. The solid line in Figure 6a shows the result of a fit according to Equation (15), while that in Figure 6b shows a linearly decreasing linewidth of the form \(\gamma_{B}(1-\epsilon)\), where \(\gamma_{B}\) is the HWHM in absence of modulation. As one approaches the threshold, effects due to the finite resolution bandwidth of the spectrum analyzer become visible and the measured linewidth is effectively limited by the RBW. Note that this does not affect the determination of the average energy, however, since the latter is related to the numerical integration of the noise spectrum and is independent of the RBW. Gains in energy of up to 125 (21 dB) were observed for this mode at this particular bias voltage, before the parametric oscillation threshold was reached. Similar behaviors were observed for both fundamental modes and different bias voltages (see Figure 8 and Supplementary Material), albeit with different gains and thresholds, as will be discussed below. The resonant nature of the parametric excitation was also verified by scanning the modulation voltage frequency around the second harmonic frequency. Figure 7 shows the variation of the energy gain--deduced from the noise spectra as previously--as a function of the pump detuning \(\delta\), for the fundamental mode of membrane B at a bias voltage such that \(\Delta/(2\pi)=250\) Hz and for two different parametric modulation amplitudes close to the threshold. The solid lines show the theoretical predictions of Equation (15), in which \(\epsilon\) is determined by the zero-detuning value and \(\gamma\) is the HWHM measured in absence of modulation. \[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]6 (**a**) Energy gain \(\bar{E}/\bar{E}_{0}\) as a function of the normalized parametric modulation amplitude \(\epsilon\) for membrane B’s fundamental mode at a bias voltage corresponding to \(\Delta/(2\pi)=30\) Hz. The solid line shows the result of a fit according to Equation (15). (**b**) HWHM of the noise spectra resulting from the Lorentzian fits, as a function of \(\epsilon\). The solid line shows a linearly decreasing function of the form \(\gamma_{B}(1-\epsilon)\). We finally investigated the influence of the bias voltage \(V_{dc}\) on the resonant parametric amplification of the fundamental modes of both membranes. The energy gains of both modes for different bias voltages are shown in Figure 8 as a function of the resonant modulation amplitude. The amplitude was arbitrarily normalized to the parametric amplification threshold amplitude, \(V_{ac}=448\) mV, of membrane A's fundamental mode at \(\Delta/(2\pi)=250\) Hz. One first observes that increasing the bias voltage leads to lower parametric oscillation thresholds for both membranes. Higher thresholds for membrane A than for membrane B at comparable \(V_{dc}\)s were also systematically observed. The reasons for this behavior were investigated in detail in [57]. In brief, the lower membrane (B) experiences a direct modulation of its tensile stress at the parametric resonance frequency through the compression of the lower frame. The parametric oscillation threshold can be reached by the application of a moderate modulation voltage. In contrast, such a direct modulation of the SiN film spring constant is much weaker for the upper membrane (A). However, the application of a static compressive force to the chips results in a modification of the nonlinear response/stress of membrane A, which accounts for the lowering of the parametric amplification threshold with the bias voltage. Let us also note that, due to thermal drifts during the measurements, variations in the highest achievable gains close to the threshold were typically observed, so that one cannot rule out that higher subthreshold gain values may be achievable by better temperature control.