Decoherence from general relativity

It is of great interest to explore matter in nontrivial quantum arrangements, including Schr\"odinger catlike states. Such states are sensitive to decoherence from their environment. Recently, in Allali and Hertzberg Gravitational decoherence of dark matter, J. Cosmol. Astropart. Phys. 07 (2020) 056 we computed the rate of decoherence of a piece of superposed matter that primarily interacts only gravitationally, a dark-matter-Schr\"odinger-cat state (DMSCS), within the nonrelativistic approximation. In this work we improve this to a general relativistic analysis. We first derive a single particle relativistic Schr\"odinger equation for a probe particle that passes through the DMSCS; the interaction is provided by the weak field metric of general relativity from the source. For a static DMSCS we find a neat generalization of our previous results. We then turn to the interesting new case of a time dependent DMSCS, which can be provided by a coherently oscillating axion field leading to superposed time dependent oscillations in the metric: a truly quantum-general relativistic phenomenon. We use scattering theory to derive the decoherence rate in all these cases. When the DMSCS is in a superposition of distinct density profiles, we find that the decoherence rate can be appreciable. We then consider the novel special case in which the density is not in a superposition, but the phase of its field oscillation is; this is a property that cannot be decohered within the nonrelativistic framework. We find that if the probe particle and/or the DMSCS's velocity dispersion is slow, then the rate of decoherence of the phase is exponentially suppressed. However, if both the probe and the DMSCS's velocity dispersion are relativistic, then the phase can decohere more rapidly. As applications, we find that diffuse galactic axions with superposed phases are robust against decoherence, while dense boson stars and regions near black hole horizons are not, and we discuss implications for experiment.