Ab initio quantum theory of mass defect and time dilation in trapped-ion optical clocks

We derive a Hamiltonian for the external and internal dynamics of an electromagnetically bound, charged two-particle system in external electromagnetic and gravitational fields, including leading-order relativistic corrections. We apply this Hamiltonian to describe the relativistic coupling of the external and internal dynamics of cold ions in Paul traps, including the effects of micromotion, excess micromotion, and trap imperfections. This provides a systematic and fully quantum-mechanical treatment of relativistic frequency shifts in atomic clocks based on single trapped ions. Our approach reproduces well-known formulas for the second-order Doppler shift for thermal states, which were previously derived on the basis of semiclassical arguments. We complement and clarify recent discussions in the literature of the role of time dilation and mass defect in ion clocks.

Automated summary

This study derives a Hamiltonian to describe the dynamics of a bound, charged two-particle system in external electromagnetic and gravitational fields. The Hamiltonian is applied to investigate the relativistic coupling of cold ions in Paul traps, considering effects such as micromotion, excess micromotion, and trap imperfections. This provides a comprehensive and quantum-mechanical treatment for studying relativistic frequency shifts in atomic clocks based on single trapped ions.