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 Hz) 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 about10-15-10-14 kg), brought within separations (d about30-50 μ m), while maintaining spatial superpositions of (1-20 μ m) and coherence for (τ about 0.1 - 1 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 kg, Δx=10 μ m, and τ=1 s, the differential acceleration-noise amplitude spectral density must be suppressed well below the 10-15 m,s-2/sqrt Hz 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.