Inducing Strong Non-Linearities in a Phonon Trapping Quartz Bulk Acoustic Wave Resonator Coupled to a Superconducting Quantum Interference Device

A quartz Bulk Acoustic Wave resonator is designed to coherently trap phonons in a way that they are well confined and immune to suspension losses so they exhibit extremely high acoustic $Q$-factors at low temperature, with $Q\times f$ products of order $10^{18}$ Hz. In this work we couple such a resonator to a SQUID amplifier and investigate effects in the strong signal regime. Both parallel and series connection topologies of the system are investigated. The study reveals significant non-Duffing response that is associated with the nonlinear characteristics of Josephson junctions. The nonlinearity provides quasi-periodic structure of the spectrum in both incident power and frequency. The result gives an insight into the open loop behaviour of a future Cryogenic Quartz Oscillator in the strong signal regime.

Devices with nonlinear properties are very valuable in many areas of physics. This includes quantum metrological applications where nonlinearities are employed to perform quantum state preparation [1]. Another example is from the frequency control area, where nonlinear processes are used to to create a limit cycle in feedback frequency sources (oscillators) 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 Cryogenic Quartz Oscillators (CQO) [2]. Such oscillators are meant to employ ultra-high quality-factor Bulk Acoustic Wave (BAW) resonator [3][4][5] to achieve a new level of frequency stability for sources based on mechanical vibration.
Although, in general all devices are nonlinear in nature, the feasibility to achieve nonlinear regimes are usually limited for a variety of reasons. For example, these regimes usually require considerably high amounts of incident power that may also cause nonlinear effects in auxiliary components, induce noise processes such as flicker noise [6], as well as heating. So, in order to reduce this threshold of nonlinearity, it is required to reduce the system losses and increase nonlinear interactions. These requirements, in particular for BAW devices, usually contradict each other leading to trade offs. Another possible solutions is to utilise an ultra high Quality factor resonator and couple it to a nonlinear superconducting circuit with a low threshold power. The former can thus provide very narrow spectral lines, while the latter adds the strong nonlinearities required at low powers. In this work we utilize a Superconducting Quantum Interference Device (SQUID) circuit [7] coupled to a high Q BAW Cavity [8]. Similar SQUID-mechanical resonator systems have been used in the past as Gravity Wave detectors [9][10][11] and more recently as quantum hybrid systems [12] in the non-driven or weakly driven regimes.
The connection topologies involve a SQUID input coil, * maxim.goryachev@uwa.edu.au a two electrode BAW resonator and an external signal source. The parallel connection is realised when all three devices share the same voltage (see Fig. 1 (A)), and the series connection, when the three devices form a single current contour (see Fig. 1 (B)). These topologies may be understood as loaded feedback QCOs with an openloop. In both cases, the SQUID-BAW system is cooled down to 3.8K with a conventional pulse-tube cryocooler. The acoustic resonator is an SC-cut [13] BVA (electrodeless) [14] quartz BAW device, whereas the SQUID is a commercial Niobium amplifier. Both devices have been used for Nyquist noise measurements at liquid helium temperatures [8]. The signal is fed through a long coaxial line ending at a −60dB 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 − 10Hz 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 [15][16][17][18] arising from the nonlinear elastic terms of the constitutive equations or thermoelectroelasticity [19]. This holds true for the ultra-high Quality factor cryogenic BAW resonators under investigation in this work [5,20], with the exception of resonators which are not swept of impurities and as a result have a large amount [21].
The amplifier is a standard DC Niobium SQUID de- vice with an nominal input inductance of 400nH and the transfer coefficient of about 300µV/φ 0 . The experiment has been repeated with two identical amplifiers with similar results. For each experiment the SQUID bias has been kept in such a way that the system works on the linear part of its characteristic. Fig. 2 displays the transmission through the parallel system as a function of the incident power in the vicinity of the 3rd overtone of the C bulk acoustic mode (4.993027MHz) in terms of the absolute value and phase. The periodicity of the power dependence is apparent and is demonstrated further in Fig. 3 for three values of the detuning signal. Fig. 4 shows that the frequency response of this system for three values of the incident power. 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 electromechanical coupling). Fig. 5 shows dependence of the frequency response of series system connection on the incident power. The frequency is tuned around the 5th overtones of the A (quasilongitudinal) mode (15.732444MHz). The frequency and power responses of the system with the series connection of the components are demonstrated in Fig. 6 and    7 respectively. In addition to quasi-periodic structure of the spectrum, the system demonstrates symmetry around the resonance frequency in the linear regime: FIG. 6. Frequency response of the series system. Phase is unwraped mirror symmetry for the magnitude and diagonal symmetry for the phase. The same type of response is observed for other modes in the frequency range of the SQUID amplifier. It should be mention that such nonlinear response is often observed with Microstrip SQUID Amplifiers [22] where Quality factors are measured in 10-100 range. Also, similar to the parallel connection results, the intrinsic BAW nonlinearity is not observed.
The system Hamiltonian (in the units with = 1) can be written based on the equations of motion: where a (a † ) is an annihilation (creation) operator for an acoustic mode, ω m is an angular frequency of the mechanical mode, q i and φ i are conjugate variable for two SQUID branches, ω J = 1/ √ C J L J is the Josephson junction plasma frequency, φ + and φ − are properly scaled biasing current and flux, ξ = I0 φ0 L J . The charge terms containing q i may be removed since the effect of shunting capacitances is negligible at the working frequencies. Nevertheless, direct simulation of this system is associated with certain difficulties. For example, low dimensional nonlinear physical models are often treated asymptotically using perturbation techniques, which utilize only one higher harmonic [23,24]. Though these techniques are able to give adequate approximations for weakly nonlinear systems, systems with strong nonlinearities require complex numerical techniques such as the Harmonic Balance approach. The Harmonic Balance approach is a powerful method to simulate the steady-state response of nonlinear systems in the frequency domain [25,26]. Generally, this numerical approach splits a system into a linear and nonlinear parts and requires representation of each variable in a series of harmonics. Whereas the linear part represents the dynamical response at each frequency, the nonlinear part mixes the harmonic components. As a result the system is represented by a set of nonlinear algebraic equations that are solved numerically. The more advanced versions of this methods have been implemented for solving nonlinear circuit problems [27,28]. These types of software for designing nonlinear electrical circuits may be used for further numerical system analysis and design of future related devices.
Understanding of the nonlinearity of a SQUID amplifier coupled to a high-Q resonator is important for designing of a future CQOs. In fact, these experiments provide results for the open loop response of possible CQO topologies coupled to SQUID amplifiers. The nonlinearity is a necessary condition for any feedback oscillator to create a limit cycle, and the degree of nonlinearity controls the phase noise budget trade off. The strong low power nonlinearity keeps the oscillator circulating power low, which will typically reducing the flicker noise (the main limit in room temperature oscillators) but increasing the effect of thermal fluctuations due to the weaker signal to noise ratio of the acoustic resonance frequency determination. Weaker nonlinearity needs higher circulating power for saturation, thus minimising the thermal noise but inducing excess flicker noise [6]. For the CQO, thermal noise is naturally reduced by orders of magnitude by working at cryogenic temperatures, so the SQUID-BAW system gives a way to reduce the oscillator power significantly to keep flicker noise as low as possible. Thus, this work in an enabling step towards producing extraordinary frequency stability [2] beyond the limit achieved at room temperature devices [29] required for some test of fundamental physics e.g. the Lorentz invariance in the neutron sector [30].
This work was supported by the Australian Research Council Grant No. CE110001013.