  Light-matter interaction in optomechanical systems is the foundation for
ultra-sensitive detection schemes [1,2] as well as the generation of phononic
and photonic quantum states [3-10]. Electromechanical systems realize this
optomechanical interaction in the microwave regime. In this context, capacitive
coupling arrangements demonstrated interaction rates of up to 280 Hz [11].
Complementary, early proposals [12-15] and experiments [16,17] suggest that
inductive coupling schemes are tunable and have the potential to reach the
vacuum strong-coupling regime. Here, we follow the latter approach by
integrating a partly suspended superconducting quantum interference device
(SQUID) into a microwave resonator. The mechanical displacement translates into
a time varying flux in the SQUID loop, thereby providing an inductive
electromechanical coupling. We demonstrate a sideband-resolved
electromechanical 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 frequency
splitting of the microwave resonator for large mechanical amplitudes confirming
the large coupling. The presented inductive coupling scheme shows the high
potential of SQUID-based electromechanics for targeting the full wealth of the
intrinsically nonlinear optomechanics Hamiltonian.
