  We propose a thought experiment, based on a mechanism that is reminiscent of
Cavendish's torsion balance, to investigate the possible quantum nature of the
gravitational field generated by the quantum superposition state of a massive
system. Our proposal makes use of the dynamics of a ultra-stable optically
levitated nanomechanical rotor endowed with a spin to generate a quantum
angular superposition that is then tested through standard Ramsey-like scheme.
Gravity manifests itself as an effective decoherence mechanism, whose strength
is different-and, as we show, appreciable-in the classical and quantum case. By
incorporating both the source for decoherence and the mechanism to probe it,
the experiment that we propose allows for a much reduced degree of control and
dynamical engineering.
