  We investigate the nature of a gravitational two-state system (G2S) in the
simplest setup in Newtonian gravity. In a quantum description of matter a
single motionless massive particle can in principle be in a superposition state
of two spatially-separated locations. This superposition state in gravity, or
gravitational cat state, would lead to fluctuations in the Newtonian force
exerted on a nearby test particle. The central quantity of importance for this
inquiry is the energy density correlation. This corresponds to the noise kernel
in stochastic gravity theory, evaluated in the weak field nonrelativistic
limit. In this limit, quantum fluctuations of the stress energy tensor manifest
as the fluctuations of the Newtonian force. We describe the properties of such
a G2S system and present two ways of measuring the cat state for the Newtonian
force, one by way of a classical probe, the other a quantum harmonic
oscillator. Our findings include: (i) mass density fluctuations persist even in
single particle systems, and they are of the same order of magnitude as the
mean; (ii) a classical probe generically records a non-Markovian fluctuating
force; (iii) a quantum probe interacting with the G2S system may undergo Rabi
oscillations in a strong coupling regime. This simple prototypical
gravitational quantum system could provide a robust testing ground to compare
predictions from alternative quantum theories, since the results reported here
are based on standard quantum mechanics and classical gravity.
