This thesis reports on frequency comparisons with optical clocks based on trapped ions, and their application to probing variations of fundamental constants, with a focus on searches for ultralight dark matter (UDM). The work was carried out at the Physikalisch-Technische Bundesanstalt (PTB).

An existing optical clock based on a single ¹⁷¹Yb⁺ ion is operated with high availability over several years. Two atomic transitions are probed in an interleaved fashion: An electric octupole (E3) transition, which is highly sensitive to possible variations of the fine-structure constant, and an electric quadrupole (E2) transition.
Including the resulting measurement data with that previously obtained in our group further improves the best constraints on a linear drift of the fine-structure constant α to
(1/α)(dα/dt)=1.8(2.5)×10⁻¹⁹/yr,
and its coupling to the gravitational potential Φ of the sun to
(c²/α)(dα/dΦ)=−2.4(3.0)×10⁻⁹,
reducing uncertainties by about a factor of four. The same data, along with that from a comparison between the ¹⁷¹Yb⁺ E3 clock and a Sr lattice clock, is analysed for the oscillatory signatures associated with small hypothetical couplings between UDM and normal matter.
Leveraging the high α-sensitivity of both frequency ratios, more than an order-of-magnitude improvement in limits on a scalar UDM coupling to photons is achieved over a wide range of dark matter masses. UDM couplings to the nuclear sector are investigated with optical clocks for the first time, by considering the effect an oscillating nuclear charge radius would have on the E3/E2 frequency ratio. It is further demonstrated that space- and time-separated clock comparisons provide sensitivity to dark matter even for oscillators with identical sensitivities to fundamental constants, while also offering access to unique dark matter signatures and complementary UDM couplings.

A new dual-species apparatus for combining ¹⁷¹Yb⁺ and ⁸⁸Sr⁺ ions enables the realization of a multi-ion clock based on the ²S_1/2→²D_5/2 transition in ⁸⁸Sr⁺ with up to ten ions. A fractional systematic uncertainty of 5.3×10⁻¹⁹ is obtained for the multi-ion clock, comparable to the current best single-ion clocks. A frequency comparison between the new optical clock based on ⁸⁸Sr⁺ and the single-ion ¹⁷¹Yb⁺ clock operating on the ²S_1/2 (F=0)→²F_7/2 (F=3) E3 transition yields an unperturbed frequency ratio of 0.6926711632159660399(20).

With a fractional uncertainty of 2.9×10⁻¹⁸, this is the most accurate determination of any physical constant to date. The data obtained with multiple ⁸⁸Sr⁺ ions is compatible with that obtained using a single ⁸⁸Sr⁺ ion, but offers a significantly improved instability. This establishes multi-ion clocks as a robust platform for next-generation tests of fundamental physics.