Exploring the Dynamical Interplay between Mass-Energy Equivalence, Interactions, and Entanglement in an Optical Lattice Clock.

We propose protocols that probe manifestations of the mass-energy equivalence in an optical lattice clock interrogated with spin coherent and entangled quantum states. To tune and uniquely distinguish the mass-energy equivalence effects (gravitational redshift and second-order Doppler shift) in such a setting, we devise a dressing protocol using an additional nuclear spin state. We then analyze the dynamical interplay between photon-mediated interactions and gravitational redshift and show that such interplay can lead to entanglement generation and frequency synchronization dynamics. In the regime where all atomic spins synchronize, we show the synchronization time depends on the initial entanglement of the state and can be used as a proxy of its metrological gain compared to a classical state. Our work opens new possibilities for exploring the effects of general relativity on quantum coherence and entanglement in optical lattice clock experiments.