## 2 Results: Towards a Cryogenic Quartz Oscillator  
One of the major requirements for feedback oscillators is an existence of a nonlinear element used to to create a limit cycle by transferring the main tone energy to higher harmonics. Despite the long history of classical feedback oscillators, the latter problem has recently emerged again for designing of CQO. Typically a CQO operating around 7 K has an extremely high power dependence with a power coefficient of frequency of order 1 Hz \(\mu\)W at \(-\)30 dBm of input power, about 1000 times higher than a room temperature oscillator [37]. A solution is to utilise an ultra high Quality factor resonator coupled to a nonlinear superconducting circuit to lower the threshold power. The BAW resonator can thus provide very narrow spectral lines, while the superconducting circuit adds the strong nonlinearities required at low powers. In this work we utilizes a Superconducting Quantum Interference Device (SQUID) circuit [31] coupled to a high \(Q\) BAW Cavity [27]. The connection topologies involve a SQUID input coil, a two electrode BAW resonator and an external signal source. The parallel connection is realised when all three devices share the same voltage (see Figure 1A), and the series connection, when the three devices form a single current contour (see Figure 1B). These topologies may be understood as loaded feedback QCO with an open-loop. In both cases, the SQUID-BAW system is cooled down to 3.8 K with a conventional pulse-tube cryocooler. The acoustic resonator is an SC-cut [38] BVA (electrodeless) [39] quartz BAW device (1 mm thick, 30 mm diameter), whereas the SQUID is a commercial Niobium amplifier for which the current bias is applied across the loop. The SQUID amplifier does not have additional noise cancellation, and its flux bias is static. Bias parameters are tuned to maximise gain in the small signal case that is estimated to be 1.2 MOhm in terms of transimpedance relating input current and the output voltage. Both devices have been used for Nyquist noise measurements at liquid helium temperatures [27]. The signal is fed through a long coaxial line ending at a \(-\)60 dB cold attenuator. The output signal of the cold part of the DC SQUID amplifier is retrieved via a micro-coaxial line, which is connected to and read out by a room temperature amplifier. All the data is acquired by a Vector Network Analyser locked to a Hydrogen maser providing extra frequency stability over long averaging times. Long averaging times are required to keep the measurement bandwidth as low as 3-10 Hz in order to avoid system ringing due to very high Quality factors. The BAW resonator equivalent model may be represented by an equivalent circuit comprising a number of motional branches and a shunt capacitance. Each motional branch is a series connection of resistive, capacitive and inductive components. In this work we limit the investigation to a few low order modes that fall into the frequency range of the SQUID amplifier. The nonlinear response of mechanical resonators can be usually approximated by the Duffing model [40; 41; 42; 43] arising from the nonlinear elastic terms of the constitutive equations or thermoelectroelasticity [44]. This holds true for the ultra-high Quality factor cryogenic BAW resonators under investigation in this work [24; 45], with the exception of resonators which are not swept of impurities and as a result have a large amount [46]. Figure 1: Experimental setups representing two ways to connect a Bulk Acoustic Wave (BAW) resonator and a Superconducting Quantum Interference Device (SQUID) amplifier: (**A**) parallel connection, (**B**) series connection. The amplifier is a standard DC Niobium SQUID device with an nominal input inductance of 400 nH and the transfer coefficient of about 300 \(\mu\)V/\(\phi_{0}\). The experiment was repeated with two identical amplifiers with similar results. For each experiment the SQUID bias was kept in such a way that the system works on the linear part of its characteristic.