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Light-matter interaction in optomechanical systems is the foundation forultra-sensitive detection schemes [1,2] as well as the generation of phononicand photonic quantum states [3-10]. Electromechanical systems realize thisoptomechanical interaction in the microwave regime. In this context, capacitivecoupling arrangements demonstrated interaction rates of up to 280 Hz [11].Complementary, early proposals [12-15] and experiments [16,17] suggest thatinductive coupling schemes are tunable and have the potential to reach thevacuum strong-coupling regime. Here, we follow the latter approach byintegrating a partly suspended superconducting quantum interference device(SQUID) into a microwave resonator. The mechanical displacement translates intoa time varying flux in the SQUID loop, thereby providing an inductiveelectromechanical coupling. We demonstrate a sideband-resolvedelectromechanical system with a tunable vacuum coupling rate of up to 1.62 kHz,realizing sub-aN Hz-1/2 force sensitivities. Moreover, we study the frequencysplitting of the microwave resonator for large mechanical amplitudes confirmingthe large coupling. The presented inductive coupling scheme shows the highpotential of SQUID-based electromechanics for targeting the full wealth of theintrinsically nonlinear optomechanics Hamiltonian.

Sideband-resolved resonator electromechanics on the single-photon level based on a nonlinear Josephson inductance