Fig. 1

A superconducting microwave circuit with magnetic-flux-mediated optomechanical coupling to a mechanical oscillator. a Circuit schematic of the device. The LC circuit is capacitively coupled to a microwave transmission line with characteristic impedance \({Z}_{0}\) by means of a coupling capacitor \({C}_{{\rm{c}}}\). In addition to the linear capacitors \(C\) and inductors \(L\), a superconducting quantum interference device (SQUID) is built into the circuit, consisting of two Josephson junctions with inductance \({L}_{{\rm{J}}}\) in a closed superconducting loop, of which a part is suspended and free to move perpendicular to the circuit plane. To bias the SQUID with magnetic flux \({\Phi }_{{\rm{b}}}\), a magnetic field can be applied perpendicular to the circuit plane. Motion of the mechanical element is transduced into modulations of the bias flux by a magnetic in-plane field \({B}_{| | }\). An optical micrograph of the circuit is shown in (b), light gray parts correspond to a \(20\)nm thick layer of aluminum, dark parts to silicon substrate. The black scale bar corresponds to 50 μm. The red dashed box shows the region, which is depicted in a tilted scanning electron micrograph in (c), showing the SQUID loop with the released aluminum beam. The bias flux through the SQUID loop \({\Phi }_{{\rm{b}}}\) can be changed by a bias current \({I}_{{\rm{b}}}\) sent through the on-chip flux bias line. The black scale bar corresponds to 3 μm. The inset shows a zoom into one of the constriction type Josephson junctions (JJs). In (d) the cavity resonance is shown, measured by sending a microwave tone to the microwave feedline and detecting the transmitted signal \({S}_{21}\). A fit to the data points (circles), shown as line, reveals a resonance frequency of \({\omega }_{0}=2\pi \times 5.221\) GHz and a linewidth \(\kappa =2 {\pi} \times 10.5\) MHz. Panel (e) shows color-coded the tuning of the cavity resonance absorption dip with magnetic bias flux in units of flux quanta \({\Phi }_{{\rm{b}}} / {\Phi }_{0}\), measured at \({B}_{| | }=1\) mT. Due to a large loop inductance of the SQUID, the arch exceeds a single flux quantum, for details see Supplementary Note 3