Gravitational Forces Between Nonclassical Mechanical Oscillators

Interfacing quantum mechanics and gravity is one of the great open questions in natural science. Micromechanical oscillators have been suggested as a plausible platform to carry out these experiments. We present an experimental design aiming at goals, inspired by Schm\"ole et al., Class. Quantum Grav. 33, 125031 (2016). Gold spheres weighing on order milligram will be positioned large silicon nitride membranes, which are spaced submillimeter distances from each other. These mass-loaded membranes mechanical that vibrate $\sim 2$ kHz frequencies drum mode. They operated measured coupling microwave cavities. First, we show it possible measure gravitational force between deep cryogenic temperatures, where thermal noise strongly suppressed. investigate measurement when positions gravitating masses exhibit significant fluctuations, including preparation massive ground state, or squeezed state. also scheme realize experiment two prepared two-mode motional state exhibits nonlocal correlations same time. Although classical, pave way for testing true related arrangements. In proof-of-principle experiment, operate 1.7 mm diameter Si$_3$N$_4$ membrane loaded 1.3 mg gold sphere. At 10 mK temperature, observe mode with quality factor above half million kHz, showing strong promise Following implementation vibration isolation, positioning, phase filtering, foresee realizing experiments reach combining known pieces current technology.