Measure quantum time dilation of a clock in a momentum superposition

For a clock of rest mass m with internal Hamiltonian $H_{\mathrm{clock}}$ whose center of mass is in a coherent superposition of momenta, the mass-energy coupling $H = \sqrt{p^{2} c^{2} + (m c^{2} + H_{\mathrm{clock}})^{2}}$ gives a proper-time distribution that differs from that of the corresponding classical mixture of momenta by a coherence-dependent term (Smith and Ahmadi, Nature Communications 2020). Determine whether atomic or ion clocks can resolve this quantum contribution, the momentum separation and fractional frequency resolution required, and measure it.

A clock comparison in which a momentum-superposed clock and a matched classical mixture show the predicted difference in frequency shift.