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Description
Surface acoustic wave (SAW) resonators based on aluminum nitride (AlN) operating in the gigahertz frequency range are promising platforms for hybrid quantum systems . At a temperature of $10$ mK, the SAW resonator is governed by quantum noise, and the internal loss can be attributed to phonons coupling to two-level-systems (TLS), which arise from tunneling states linked to elastic and electric fields. One straightforward approach to determining TLS losses is to measure the resonance frequency shift as a function of temperature. However, the microscopic origin of TLS losses in AlN remains unresolved, particularly below $T = hf/2k_B$, where precise thermalization---and thus accurate temperature control---of the SAW resonator becomes challenging. Here, we present the resonance frequency shift of an AlN-based SAW resonator designed for $4$ GHz operation, measured across temperatures from $10$ to $700$ mK. Our analysis reveals a nonlinear shift in the resonance frequency $\Delta f_{r}=f_{r}(T)-f_{r}(T \to 0)$, ranging from $-0.040$ to $+0.187$ MHz around $f_{r}(T \to 0)=4274.794$ MHz, with a minimum near $T=100$ mK. This result is in good agreement with the standard theoretical model for TLS, which predicts a minimum at $T = hf_{r}/2k_B = 103$ mK. Furthermore, the extracted data points fit well to the TLS model when including a frequency offset of $-21$ kHz as the temperature approaches $0$ K ($T \to 0$). Finally, this analysis yields a TLS-related internal quality factor of $Q_{\text{i,TLS}} = 7237$. Our results demonstrate the presence of TLS-related losses in AlN-based SAW resonators operating at low-gigahertz frequencies and millikelvin temperatures. Moreover, precise temperature control of the SAW resonator enables accurate extraction of $Q_{\text{i,TLS}}$. This refinement in measurement precision is particularly significant, as TLS-related losses are highly relevant in the single-excitation limit and, consequently, for quantum applications.