Loss of coherence and coherence protection from a graviton bath

We consider a quantum harmonic oscillator coupled with a graviton bath. We discuss the loss of coherence in the matter sector due to the matter-graviton vertex interaction, which leads to a loss of coherence provided that the matter-wave interferometer can emit gravitons allowed by the kinematics, and the most dominant process up to $\sim\sqrt{G}$ and $\sim\mathcal{O}(c^{-2})$. Working in the quantum-field-theory framework, we obtain a master equation by tracing away the gravitational field at the leading order $\mathcal{\sim O}(G)$. We find that the decoherence rate is proportional to the cube of the harmonic trapping frequency and vanishes for a free particle, as expected for a system without a mass quadrupole. Furthermore, our quantum model of graviton emission recovers the known classical formula for gravitational radiation from a classical harmonic oscillator for coherent states with a large occupation number. In addition, we find that the quantum harmonic oscillator eventually settles in a steady state with a remnant coherence of the ground and first excited states. In particular, the superposition of number states $\frac{1}{\sqrt{2}}\left[\vert0\rangle+\vert1\rangle\right]$ never decoheres.