## 4 Parametric Amplification of the Thermal Fluctuations
### Theoretical Model
If, like in the experiment, one does not keep track of the quadrature angle \(\theta\), one observes an average noise spectrum  
\[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]0  
given by  
\[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]1  
At the parametric resonance (\(\delta=0\)), the average noise spectrum in Equation (12) can be simply written as  
\[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]2  
i.e., the sum of two Lorentzians with HHHMs (half-width at half-maximum) \(\gamma(1-\epsilon)\) and \(\gamma(1+\epsilon)\), corresponding to the noise spectrum of the amplified and deamplified quadratures, respectively. While in general not Lorentzian, the average noise spectrum is, to a good approximation, Lorentzian either at low gains (\(\epsilon\ll 1\)) or close to the parametric threshold \(\epsilon\to 1\), where the contribution of the amplified quadrature noise dominates. At the parametric threshold \(\epsilon=1\)-or, equivalently, \(\bar{\varsigma}=2/Q\)-the linewidth of the noise spectrum diverges (in absence of nonlinearities). The variance of the amplitude, \(\Delta x^{2}=\langle x^{2}\rangle\), is obtained by integrating this one-sided noise spectrum over the positive frequencies  
\[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]3  
to give the average energy in the mode \(\bar{E}=\frac{1}{2}m\omega_{m}^{2}\langle x^{2}\rangle\) as  
\[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]4  
where \(\bar{E}_{0}=\frac{1}{2}k_{B}T\) is the average energy in absence of parametric modulation. In presence of parametric modulation modulation at the second harmonic frequency of the fundamental mode of one of the membranes. The evolution of the noise spectrum of the fundamental modes was then monitored as a function of the modulation voltage amplitude \(V_{ac}\). The bias voltage \(V_{dc}\), which determined the frequency separation between the fundamental mode frequencies \(\Delta=\omega_{1,1}^{B}-\omega_{1,1}^{A}\) (difference between the filled and open circles in Figure 4a), was chosen in such a way that the second harmonic of the excited fundamental mode frequency, \(2\omega_{1,1}^{A}\) or \(2\omega_{1,1}^{B}\), did not coincide with a (2,2) mode resonance frequency. We verified experimentally that neither the non-resonantly driven (1,1) mode nor the (2,2) modes were excited in presence of the parametric drive. \[x(t) =\left[A(t)e^{-i\omega_{m}t}+A^{*}(t)e^{i\omega_{m}t}\right]/2 \tag{3}\]5 Average noise spectra of membrane B’s fundamental mode at a bias voltage corresponding to \(\Delta/(2\pi)=30\) Hz and for different resonant modulation amplitudes corresponding to (**a**) \(\epsilon=0.069\), (**b**) \(\epsilon=0.809\), (**c**) \(\epsilon=0.908\) and (**d**) \(\epsilon=0.996\). The solid lines are the results of Lorentzian fits to the data. The \(y\)-axis scale, in arbitrary units, is the same for all figures. The same RBW of 0.5 Hz is used for recording all the spectra. Figure 5 shows examples of noise spectra of membrane B's fundamental mode, obtained for a bias voltage corresponding to a frequency separation \(\Delta/(2\pi)=30\) Hz, and for increasing modulation amplitudes (see Supplementary Material for the complete spectrum dataset). The modulation amplitudes were normalized to the parametric threshold voltage and the same resolution bandwidth of 0.5 Hz is used for all spectra. The results of Lorentzian fits to the data are also shown. It can be seen that the amplified noise spectra are generally well-approximated by Lorentzians with increasing peak value/area and reduced linewidth, as the parametric modulation amplitude is increased. The average energy is obtained by numerical integration of the spectrum and the energy gain--i.e., the ratio \(\bar{E}/\bar{E}_{0}\) of the average energy for a given \(\epsilon\) to the average energy for \(\epsilon=0\)--is shown in Figure 6a as a function of \(\epsilon\).