Nuclear Heterodyne Interferometry for Gravitational Spectroscopy

Gravitational spectroscopy tests the coupling of gravity to matter by measuring gravitationally induced frequency shifts of quantum transitions. While modern optical clocks probe the gravitational response of electronic transitions with extraordinary precision, tests in the nuclear sector have not progressed since the M\"ossbauer measurements of the gravitational redshift by Pound and Rebka. Here we introduce a new approach to nuclear gravitational spectroscopy based on phase-sensitive heterodyne interferometry of time-resolved nuclear resonant scattering of synchrotron radiation. In this scheme the gravitational redshift appears as a slowly accumulating phase drift of a delayed heterodyne beat signal, converting nuclear gravitational spectroscopy from energy-domain detection to time-domain interferometry. A Fisher-information analysis supported by Monte Carlo simulations and experimentally confirmed photon count rates shows that the nuclear gravitational redshift of $^{57}$Fe can be detected within hours on a few-meter-scale vertical baseline, with percent-level precision on deviations from general relativity becoming accessible on day-scale timescales. The method thus establishes an experimentally realistic and scalable platform for precision tests of gravitational coupling to nuclear structure.