Relativistic corrections and dynamic decoupling in trapped ion optical atomic clocks

Abstract

This thesis describes relativistic corrections and the use of dynamical decoupling in the context of trapped ion optical atomic clocks. The shifts that contribute to trapped ion optical atomic clocks can be divided into relativistic effects and environmental perturbations, which correspond to the two parts of this thesis.

In the first part, we investigate how to properly account for relativistic corrections from an ab initio derivation. For that purpose, we start from a charged two-particle system in external electromagnetic and gravitational fields described by the classical Lagrangian for two particles interacting with the electromagnetic field. From this starting point, we derive a quantum Hamiltonian including leading order relativistic corrections in a systematic way. We then apply this Hamiltonian to describe the relativistic coupling of external and internal dynamics of cold ions in Paul traps, including the effects of micromotion, excess micromotion and trap imperfections. This approach provides a systematic and fully quantum mechanical treatment of relativistic frequency shifts in atomic clocks based on single trapped ions. Additionally, we reproduce well-known formulae 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. Furthermore, we also study the problem of an ion in a Penning trap for the case of a transition between manifolds with spin 0. Our Hamiltonian gives a basis for properly treating the relativistic effects of an ion that can be applied to an extensive variety of experiments after the proper implementation.

The second part considers the mechanism of continuous dynamical decoupling, focusing on gaining insensitivity to some environmental perturbations, such as magnetic field fluctuations and quadrupole shifts. This mechanism consists in the application of a radio-frequency magnetic field orthogonal to the quantization axis of a given spin manifold. We show how this is achieved for one manifold and then extend the treatment to two manifolds. In that process, we make some approximations, consisting of rotating wave approximations and neglecting the effect that the off-resonant radio-frequency magnetic field of one manifold has on the other manifold and vice versa. Nevertheless, we account for those approximations perturbatively by using the so-called Magnus expansion, showing that they can be considered as an effective shift of the Zeeman splitting of the manifolds. Afterwards, we can apply our formalism to properly describe a quadrupole transition between two manifolds, where the particular case of a transition between S = 1/2 and D = 5/2 of 40Ca+ is studied; a comparison with experimental data will be presented elsewhere. We compare our approximate treatment with the true solution of the periodic Hamiltonian presented in the introductory part of the thesis, finding that the corrected dressed basis corresponds to the time-independent part of the Floquet states. We finish this part by considering the implementation of a Mølmer-Sørensen gate within the framework of continuous dynamical decoupling.