Mechanical mediation through shared apparatus in gravitational-entanglement experiments

Abstract

Can the support shared by two matter-wave interferometers entangle their path-recording spins, even when its final motion records neither path? It can. For prescribed forces coupled to a linear Gaussian apparatus, we bound the resulting spin channel by two force-weighted cross responses. Classical force replay measures the bound; local pulse changes can null both responses. For passive mounts, finite-time bounds on static compliances cover every mechanical mode, and a spectral gap gives a controlled elastic approximation. When gravity acts mainly during a hold, the permissible pulse-weighted compliance scales as the cube of the ramp time. A specified clamped horizontal link produces a phase comparable to a millisecond-ramp gravitational reference, but negligible for long ramps. These results provide a calibration criterion for excluding a mechanical entangling channel, rather than a contamination estimate for an uncharacterized instrument.