Nano-electromechanical systems implement the opto-mechanical interaction combining electromagnetic circuits and mechanical elements. We investigate an inductively coupled nano-electromechanicalsystem, where a superconducting quantum interference device (SQUID) realizes the coupling. We show that the resonance frequency of the mechanically compliant string embedded into the SQUID loop can be controlled in two different ways: (i) the bias magnetic flux applied perpendicular to the SQUID loop, (ii) the magnitude of the in-plane bias magnetic field contributing to the nano-electromechanical coupling. These findings are quantitatively explained by the inductive interaction contributing to the effective spring constant of the mechanical resonator. In addition, we observe a residual field dependent shift of the mechanical resonance frequency, which we attribute to the finite flux pinning of vortices trapped in the magnetic field biased nanostring.

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.

We present a hybrid system consisting of a superconducting coplanar waveguide resonator coupled to a nanomechanical string and a transmon qubit acting as nonlinear circuit element.We perform spectroscopy for both the transmon qubit and the nanomechanical string. Measuring the ac-Stark shift on the transmon qubit as well as the electromechanically induced absorption on the string allows us to determine the average photon number in the microwave resonator in both the low and high power regimes. In this way, we measure photon numbers that are up to nine orders of magnitude apart. We find a quantitative agreement between the calibration of photon numbers in the microwave resonator using the two methods. Our experiments demonstrate the successful combination of superconducting circuit quantum electrodynamics and nano-electromechanics on a single chip.