Optimal Form Factors for Experimental Proposals on Gravity-Induced Entanglement

The interface between quantum mechanics and gravity remains an unresolved issue. Recent advances in precision measurement suggest that detecting gravity-induced entanglement in oscillator systems could provide key evidence for the quantum nature of gravity. However, thermal decoherence imposes strict constraints on system parameters. For entanglement to occur, mechanical frequency ${\omega }_m$, dissipation rate ${\gamma }_m$, environmental temperature $T$, oscillator density $\rho$, and the form factor $\mathrm{\Lambda }$-determined by the geometry and arrangement of oscillators-must satisfy a specific constraint. This constraint, intrinsic to the noise model, is considered universal and cannot be improved by quantum control. Given the difficulty in further optimizing ${\omega }_m$, ${\gamma }_m$, $\rho$, and $T$, optimizing $\mathrm{\Lambda }$ can relax the constraints on these parameters. In this work, we prove that the form factor has a supremum of $2\pi$, revealing a fundamental limit of the oscillator system. We propose designs that approach this supremum, nearly an order of magnitude higher than typical spherical oscillators. This optimization could ease experimental constraints and bring quantum gravity validation based on gravity-induced entanglement closer to realization.