Lukas J. Spieß

Characterization of the trap rf induced a.c. Zeeman shift is essential for achieving high accuracy in optical ion clocks. In this work, we demonstrate the experimental characterization of this shift using highly charged $\mathrm{Ca}^{14+}$. The transverse component of the a.c. magnetic field is measured using the Autler-Townes splitting of the equally-spaced Zeeman components of the $^{3}\mathrm{P}_1$ when the Zeeman splitting is close to resonance with the trap rf drive frequency. We observe the resulting modulation by performing quantum logic spectroscopy using the co-trapped $\mathrm{Be}^{+}$. The longitudinal component is measured from probing the $\mathrm{Be}^{+}$ magnetic field-insensitive hyperfine transition $|F=2,m_F=0 \rangle \rightarrow | F=1,m_F=0 \rangle$. We confirm the small influence of the a.c. Zeeman shift in highly charged ions. The employed techniques can easily be transferred to other multi-level atomic systems.

In quantum logic spectroscopy, internal transitions of trapped ions and molecules can be probed by measuring the motional displacement caused by an applied light field of variable frequency. This provides a solution to ``needle in a haystack'' problems, such as the search for narrow clock transitions in highly charged ions, recently discussed by S. Chen et al. (Phys. Rev. Applied 22, 054059). The main bottleneck is the search speed over a frequency bandwidth, which can be increased by enhancing the sensitivity of displacement detection. In this work, we explore two complementary improvements: the use of squeezed motional states, explained using an analytical phase space model and optimal statistical postprocessing of data using a hypothesis testing framework. We demonstrate that each method independently provides a substantial boost to search speed. Their combination effectively mitigates state preparation and measurement errors, improving the search speed by an order of magnitude and fully leveraging the quantum enhancement offered by squeezing.

We measured the $g$-factor of the excited state $^3\text{P}_1$ in $\text{Ca}^{14+}$ ion to be $g = 1.499032(6)$ with a relative uncertainty of $4\times10^{-6}$. The magnetic field magnitude is derived from the Zeeman splitting of a $\text{Be}^+$ ion, co-trapped in the same linear Paul trap as the highly charged $\text{Ca}^{14+}$ ion. Furthermore, we experimentally determined the second-order Zeeman coefficient $C_2$ of the $^3\text{P}_0$ - $^3\text{P}_1$ clock transition. For the $m_J=0\rightarrow m_{J'}=0$ transition, we obtain $C_2 = 0.39\pm0.04\text{HzmT}^{-2}$, which is to our knowledge the smallest reported for any atomic transition to date. This confirms the predicted low sensitivity of highly charged ions to higher-order Zeeman effects, making them ideal candidates for high-precision optical clocks. Comparison of the experimental results with our state-of-the art electronic structure calculations shows good agreement, and demonstrates the significance of the frequency-dependent Breit contribution, negative energy states and QED effects on magnetic moments.

Recent advances in high-precision spectroscopy of highly charged ions necessitate an understanding of energy shifts of ionic levels caused by external electric and magnetic fields. Beyond the well-known Stark and linear Zeeman shifts, trapped ions may also exhibit quadratic Zeeman and electric quadrupole shifts. In this contribution, we present a systematic approach for the theoretical analysis of these shifts for arbitrary many-electron ions. Based on the derived expressions and making use of the multiconfigurational Dirac-Fock approach, we performed calculations of quadratic Zeeman shift coefficients and quadrupole moments for various ionic states in Ca$^{14+}$, Ni$^{12+}$ and Xe$^{q+}$ ions. These ions attract particular interest for ongoing and future experiments in optical clocks and tests of fundamental physics.

Most ions lack the fast, cycling transitions that are necessary for direct laser cooling. In most cases, they can still be cooled sympathetically through their Coulomb interaction with a second, coolable ion species confined in the same potential. If the charge-to-mass ratios of the two ion types are too mismatched, the cooling of certain motional degrees of freedom becomes difficult. This limits both the achievable fidelity of quantum gates and the spectroscopic accuracy. Here we introduce a novel algorithmic cooling protocol for transferring phonons from poorly- to efficiently-cooled modes. We demonstrate it experimentally by simultaneously bringing two motional modes of a Be$^{+}$-Ar$^{13+}$ mixed Coulomb crystal close to their zero-point energies, despite the weak coupling between the ions. We reach the lowest temperature reported for a highly charged ion, with a residual temperature of only $T\lesssim200~\mathrm{μK}$ in each of the two modes, corresponding to a residual mean motional phonon number of $\langle n \rangle \lesssim 0.4$. Combined with the lowest observed electric field noise in a radiofrequency ion trap, these values enable an optical clock based on a highly charged ion with fractional systematic uncertainty below the $10^{-18}$ level. Our scheme is also applicable to (anti-)protons, molecular ions, macroscopic charged particles, and other highly charged ion species, enabling reliable preparation of their motional quantum ground states in traps.