This paper points out the importance of the quantum nature of the gravitational interaction with matter in a linearized theory of quantum gravity induced entanglement of masses. We will show how the quantum interaction entangles the steady states of a closed system (eigenstates) of two test masses placed in the harmonic traps, and how such a quantum matter-matter interaction emerges from an underlying quantum gravitational field. We will rely upon quantum perturbation theory highlighting the critical assumptions for generating a quantum matter-matter interaction and showing that a classical gravitational field does not render such an entanglement. We will consider two distinct examples: one where the two harmonic oscillators are static, and the other where the harmonic oscillators are nonstatic. In both cases it is the quantum nature of the gravitons interacting with the harmonic oscillators that are responsible for creating an entangled state with the ground and the excited states of harmonic oscillators as the Schmidt basis. We will compute the concurrence as a criterion for the above entanglement and compare the two ways of entangling the two harmonic oscillators.

• Received 14 February 2022
• Accepted 16 May 2022

DOI:https://doi.org/10.1103/PhysRevD.105.106028