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We implement sympathetic cooling of highly charged ions (HCI) by fully
enclosing a linear Paul trap within a superconducting radio-frequency
resonator. A quantization magnetic field applied while cooling down into the
superconducting state remains present in the trap for centuries and external
electromagnetic fluctuations are greatly suppressed. A magnetic field decay
rate at the 10$^{-10}$ s$^{-1}$ level is found using trapped Doppler-cooled
Be$^+$ ions as hyperfine-structure (hfs) qubits. Ramsey interferometry and
spin-echo measurements on magnetically-sensitive hfs transitions yield
coherence times of >400 ms, showing excellent passive shielding at frequencies
down to DC. For sympathetic cooling of HCI, we extract them from an electron
beam ion trap (EBIT) and co-crystallize one together with Doppler-cooled Be$^+$
ions. By subsequently ejecting all but one Be$^+$ ions, we prepare single HCI
for quantum logic spectroscopy towards frequency metrology and qubit operations
with a great variety of HCI species.
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