Title: Determination of the dynamic Young's modulus of quantum materials in piezoactuator-driven uniaxial pressure cells using a low-frequency a.c. method

Abstract: We report on a new technique for measuring the dynamic Young's modulus, $E$, of quantum materials at low temperatures as a function of static tuning strain, $\epsilon$, in piezoactuator-driven pressure cells. In addition to a static tuning of stress and strain, we apply a small-amplitude, finite-frequency a.c. (1 Hz$ \lesssim \omega \lesssim $1000 Hz) uniaxial stress, $\sigma_{ac}$, to the sample and measure the resulting a.c. strain, $\epsilon_{ac}$, using a capacitive sensor to obtain the associated modulus $E$. We demonstrate the performance of the new technique through proof-of-principle experiments on the unconventional superconductor Sr$_2$RuO$_4$, which is known for its rich temperature-strain phase diagram. In particular, we show that the magnitude of $E$, measured using this a.c. technique at low frequencies, exhibits a pronounced nonlinear elasticity, which is in very good agreement with previous Young's modulus measurements on Sr$_2$RuO$_4$ under [100] strain using a d.c. method (Noad et al., Science 382, 447-450 (2023)). By combining the new a.c. Young's modulus measurements with a.c. elastocaloric measurements in a single measurement, we demonstrate that these a.c. techniques are powerful in detecting small anomalies in the elastic properties of quantum materials. Finally, using the case of Sr$_2$RuO$_4$ as an example, we demonstrate how the imaginary component of the modulus can provide additional information about the nature of ordered phases.