## 2 Results: Towards a Cryogenic Quartz Oscillator
Figure 2 shows the amplitude and phase response of the parallel system as a function of the incident power and frequency of excitation in the vicinity of the 3 rd overtone of the C bulk acoustic mode (4.993027 MHz), with the periodicity of the power dependence apparent and demonstrated further for three values of the detuning signal. These dependencies can not be described by the Duffing model used to describe the response of bare mechanical resonators and requires full trigonometric function representation. It is seen that in both frequency and power the system response is best described by chirp functions. The applied powers are too low to induce the resonator own nonlinearity, and the same type of response is observed for all low frequency BAW modes with the nonlinearity becoming apparent at different power levels depending on the mode Quality factor and motional resistance (which describes the electromechanical coupling). For the 3 rd overtone of the C bulk acoustic mode, the Quality factor is \(4.9\times 10^{7}\) as found from [PERSON] noise spectrum [27]. Figure 3 shows the amplitude and phase response of the series system as a function of the incident power and frequency of excitation in the vicinity of the 5 th overtone of the A (quasi-longitudinal) mode  
Figure 2: Top: Amplitude (**A**) and Phase (**B**) response of the parallel system as a function of the incident power in the vicinity of the 3 rd overtone of the C bulk acoustic mode (4.993027 MHz). Bottom Left: Amplitude (**C**) and Phase (**D**) response as a function of power for three specific values of the detuning frequency from the data represented on the left. Bottom Right: Amplitude (**E**) and Phase (**F**) response for three specific values of the incident power from the data represented on the left. (15.732444 MHz). In addition to quasi-periodic structure of the spectrum, the system demonstrates symmetry around the resonance frequency in the linear regime: mirror symmetry for the magnitude and diagonal symmetry for the phase. Also, similar to the parallel connection results, the intrinsic BAW nonlinearity is not observed. For this mode, the Quality factor is \(27\times 10^{7}\) obtained from Nyqvist noise spectrum [27]. The system Hamiltonian (in the units with \(\hbar=1\)) can be written based on the equations of motion:  
\[H = \omega_{m}a^{\dagger}a+\omega_{I}\sum_{i}\left(\frac{q_{i}^{2}}{2}+ \frac{\delta_{i}^{2}}{2}-\xi\cos\delta_{i}\right)\,-\omega_{I}\delta_{1}\delta _{2}-\omega_{I}(\phi_{+}\delta_{1}+\phi_{-}\delta_{2})\,-g(a^{\dagger}+a)( \delta_{1}-\delta_{2}) \tag{1}\]  
where \(a\) (\(a^{\dagger}\)) is an annihilation (creation) operator for an acoustic mode, \(\omega_{m}\) is an angular frequency of the mechanical mode, \(q_{i}\) and \(\phi_{i}\) are conjugate variables for two SQUID branches, \(\omega_{I}=1/\sqrt{C_{I}L_{I}}\) is the Josephson junction plasma frequency, \(\phi_{+}\) and \(\phi_{-}\) are properly scaled biasing current and flux, \(\xi=\frac{\hbar}{\phi_{0}}L_{I}\). The charge terms containing \(q_{i}\) may be removed since the effect of shunting capacitances is negligible at the working frequencies. It is also worth noting that none of the higher order modes of the resonator is an integer or rational multiple of the modes of interest. The deviation from being integer multiples of the fundamental mode for quartz resonators is due to the piezoelectric effect. Figure 3: Top: Amplitude (**A**) and Phase (**B**) response of the series system as a function of the incident power in the vicinity of the 5 th overtones of the A (quasi-longitudinal) mode (15.732444 MHz). Bottom Left: Amplitude (**C**) and Phase (**D**) response as a function of power for three specific values of the detuning frequency from the data represented on the left. Bottom Right: Amplitude (**E**) and Phase (**F**) response for three specific values of the incident power from the data represented on the left. Nevertheless, direct simulation of this system is associated with certain difficulties.