V. J. Martínez-Lahuerta

Dynamical decoupling techniques are a versatile tool for engineering quantum states with tailored properties. In trapped ions, nested layers of continuous dynamical decoupling by means of radio-frequency field dressing can cancel dominant magnetic and electric shifts and therefore provide highly prolonged coherence times of electronic states. Exploiting this enhancement for frequency metrology, quantum simulation or quantum computation, poses the challenge to combine the decoupling with laser-ion interactions for the quantum control of electronic and motional states of trapped ions. Ultimately, this will require running quantum gates on qubits from dressed decoupled states. We provide here a compact representation of nested continuous dynamical decoupling in trapped ions, and apply it to electronic $S$ and $D$ states and optical quadrupole transitions. Our treatment provides all effective transition frequencies and Rabi rates, as well as the effective selection rules of these transitions. On this basis, we discuss the possibility of combining continuous dynamical decoupling and Mølmer-Sørensen gates.

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.