White paper: 1-10 Hz matter-wave interferometer to test the spin entanglement witness for quantum gravity

In this white paper, we highlight the importance of the ($1-10\mathrm{H}\mathrm{z}$) frequency range for laboratory tests of the quantum nature of gravity using the quantum gravity-induced entanglement of masses (QGEM) protocol. QGEM requires matter-wave interferometers with masses ($m\sim {10}^{-15}-{10}^{-14}\mathrm{k}\mathrm{g}$), brought within separations ($d\sim 30-50\mu \mathrm{m}$), while maintaining spatial superpositions of ($1-20\mu \mathrm{m}$) and coherence for ($\tau \sim 0.1-1\mathrm{s}$). These requirements make low-frequency environmental noise a central experimental challenge and place QGEM in a regime closely related to the low-frequency goals of the Einstein Telescope (ET) and the Cosmic Explorer (CE). In particular, QGEM is sensitive to relative acceleration noise (RAN) and to gravity-gradient noise (GGN) generated by seismic and other environmental mass-density fluctuations. For representative parameters $m={10}^{-14}\mathrm{k}\mathrm{g}$, $\mathrm{\Delta }x=10\mu \mathrm{m}$, and $\tau =1\mathrm{s}$, the differential acceleration-noise amplitude spectral density must be suppressed well below the ${10}^{-15}\mathrm{m}{\mathrm{s}}^{-2}\mathrm{/}\sqrt{\mathrm{H}\mathrm{z}}$ level to keep acceleration-induced dephasing below the relevant experimental scale. Achieving this level of low-frequency noise suppression is therefore a key requirement for QGEM and closely parallels the seismic and gravity-gradient noise challenges that ET and CE address.