Detecting and influencing the motion of mechanical resonators has been a major topic in the study of fundamental physics and sensor technology. Optomechanical systems are particularly suitable for this due to their flexibility in design, causing them to be applicable in a wide range of parameter regimes. However, for all optomechanical systems there is a long-standing challenge to increase the single photon coupling strength. Whereas there are many ongoing developments in the field of linear optomechanics, there has been an increasing interest in nonlinear optomechanical systems. Namely, as this is a requirement for the creation of a massive superposition state in these platforms, treading the boundary between quantum mechanics and general relativity. In this thesis we couple a mesoscopic membrane to a superconducting microwave cavity in a flip-chip geometry. The silicon nitride membrane is embedded inside an in-substrate phononic shield. Its resonance mode has a large effective mass, while retaining a considerable zero-point fluctuation, making it an excellent candidate for gravitational quantum experiments. Developing this platform, we overcome multiple challenges, such as mitigating noise and increasing the single photon coupling rate. Furthermore, we include a nonlinearity by coupling a cavity to a superconducting qubit, taking a first step towards nonlinear optomechanical experiments in a flip-chip system.

Microwave cavities are commonly used in many experiments, including optomechanics, magnetic field sensing, magnomechanics, and circuit quantum electrodynamics. Noise, such as variations in the magnetic field or mechanical vibrations, can cause fluctuations of the natural frequency of the cavity, creating challenges in operating them in experiments. To overcome these challenges, we demonstrate a dynamic feedback system implemented by the locking of a microwave drive to the noisy cavity. A homodyne-interferometer scheme monitors the cavity resonance fluctuations due to low-frequency noise, which is mitigated by frequency modulating the microwave generator. The feedback has a bandwidth of 400 Hz, with a reduction of cavity fluctuations by 85% integrating up to a bandwidth of 2 kHz. Moreover, the cavity resonance frequency fluctuations are reduced by 73%. This scheme can be scaled to enable multitone experiments locked to the same feedback signal. As a demonstration, we apply the feedback to an optomechanical experiment and implement a cavity-locked multitone mechanical measurement. As low-frequency cavity frequency noise can be a limiting factor in many experiments, the multitone microwave locking technique presented here is expected to be relevant for a wide range of microwave-cavity experiments.