Effects of Newtonian gravitational self-interaction in harmonically trapped quantum systems

The Schr\"odinger-Newton equation has gained attention in the recent past as a nonlinear modification of the Schr\"odinger equation due to a gravitational self-interaction. Such a modification is expected from a fundamentally semi-classical theory of gravity, and can therefore be considered a test case for the necessity of the quantisation of the gravitational field. Here we provide a thorough study of the effects of the Schr\"odinger-Newton equation for a micron-sized sphere trapped in a harmonic oscillator potential. We discuss both the effect on the energy eigenstates and the dynamical behaviour of squeezed states, covering the experimentally relevant parameter regimes.