Gravitationally Induced Entanglement Between Particles in Harmonic Traps: Limits for Gaussian States

TL;DR

This work analyzes gravitationally induced entanglement between two masses in harmonic traps coupled solely by Newtonian gravity, focusing on Gaussian initial states and the time evolution of their covariance matrices. By deriving a quadratic effective Hamiltonian and employing symplectic evolution, it establishes a temperature threshold below which entanglement can arise for thermal states and shows an optimal trap frequency that maximizes entanglement for fixed parameters. Extending to two-mode squeezed and general Gaussian states, the study provides bounds on the achievable entanglement—saturating for thermal and squeezed cases—yet finds the effect to be extremely small under realistic conditions. The findings highlight the substantial experimental challenges in observing gravity-induced quantum effects and hint at interferometric approaches as more promising avenues.

Abstract

Gravitationally induced entanglement has been proposed as a probe of the quantum nature of gravity. This work analyzes a system of two particles in harmonic traps interacting only through gravity, considering thermal and two-mode squeezed initial states. For thermal states, a maximum temperature is identified above which entanglement cannot be generated, and for fixed system parameters an optimal trap frequency that maximizes the logarithmic negativity is found. Squeezing the initial state does not further enhance the entanglement generation, but increases the temperature range over which it can be observed. Extending the analysis to general Gaussian states, an upper bound on the achievable entanglement is derived and shown to be saturated, for example, by ground and squeezed states. The results show that the amount of entanglement generated in this setup is extremely small, highlighting the experimental challenges of observing gravitationally induced quantum effects.

Figures (2)

• Figure 1: Experimental setup: two particles of mass $m$ trapped in harmonic potentials with frequency $\omega$, interacting only via gravity.
• Figure 2: $E_{\mathcal{N}}$ as a function of $\omega$ for $m=10^{-15}$ kg and $T=10^{-10}$ K.