AC evidence of a field tuned 2D superconductor-metal transition in a low-disorder InO$_x$ film

Employing microwave spectroscopy, we investigated the field tuned quantum phase transition between the superconducting and the resistive states in a low-disorder amorphous InO$_x$ film in the frequency range of 0.05 to 16 GHz. Our AC measurements are explicitly sensitive to the critical slowing down of the characteristic frequency scales approaching a transition. The relevant frequency scale of superconducting fluctuations approaches zero at a field $B_{sm}$ far below the field $B_{cross}$ where different isotherms of resistance as a function of magnetic field cross each other. The phase stiffness at the lowest frequency vanishes from the superconducting side at B $\approx B_{sm}$, while the high frequency limit extrapolates to zero near $B_{cross}$. Our data are consistent with a scenario where $B_{sm}$ is the true quantum critical point for a transition from a superconductor to an anomalous metal, while $B_{cross}$ only signifies a crossover to a regime where superconducting correlations make a vanishing contribution to both AC and DC transport measurements in the low-disorder limit.