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

TL;DR

This paper develops a quantum-mechanical Hamiltonian model to accurately describe relativistic effects, including time dilation and mass defect, in trapped-ion optical clocks, improving understanding of frequency shifts.

Contribution

It introduces a systematic quantum approach to relativistic frequency shifts in ion clocks, incorporating micromotion and trap imperfections.

Findings

Reproduces known second-order Doppler shift formulas for thermal states.

Clarifies the role of time dilation and mass defect in ion clock accuracy.

Provides a comprehensive quantum framework for relativistic effects in trapped ions.

Abstract

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 on the role of time dilation and mass defect in ion clocks.