Gravitationally Induced Entanglement of Matter in Quadratic Curvature Gravity and Constraints on Ghost Mass

Abstract

We investigate gravitationally generated entanglement in two quantum harmonic oscillators induced by quadratic (Stelle) gravity, including corrections up to $1.5$ post-Newtonian order. Starting from the quadratic action, we derive the effective two-body Hamiltonian for two harmonically trapped masses, incorporating the contributions of the massive spin-$2$ ghost ($m_2$) and massive spin-$0$ ($m_0$) degrees of freedom of the gravitational field. For two quantised oscillators prepared in their ground state, we compute the von Neumann and Rényi entropies of the reduced state and identify a frequency at which the gravitationally-induced entanglement vanishes due to cancellation between relativistic momentum squeezing and quantum-delocalisation-induced position squeezing. We further analyze the cancellation frequency and derive the approximate constraint $m_0 < \sqrt[3]{4}\, m_2$ for the spin-$2$ and spin-$0$ modes. This relation follows from demanding a stable harmonic oscillator description. Finally, we study gravitationally-induced concurrence in a non-Gaussian setup and show how quadratic gravity modifies the entanglement generated between two spatial superpositions. The concurrence can approach $\mathcal{O}(1)$ for certain choices of mass, spatial superposition, particle distance, and spin-$2$ and spin-$0$ modes. The concurrence will deviate from Newtonian gravity at certain particle separations, depending on the energy of the spin-$2$ and spin-$0$ modes. For example, spin modes as low as $0.0197$ eV become distinguishable from Newtonian gravity at a distance $d \sim 40 μ$m. This allows us to constrain the spin-$2$ and spin-$0$ masses in experiments.