We describe a thin-film superconducting Nb microwave resonator, tunable to within 0.3 ppm of the hyperfine splitting of $^{87}$Rb at $f_{Rb}=6.834683$ GHz. We coarsely tuned the resonator using electron-beam lithography, decreasing the resonance freq
uency from 6.8637 GHz to 6.8278 GHz. For emph{in situ} fine tuning at 15 mK, the resonator inductance was varied using a piezoelectric stage to move a superconducting pin above the resonator. We found a maximum frequency shift of about 8.7 kHz per 60-nm piezoelectric step and a tuning range of 18 MHz.
We have observed a few distinct anomalous avoided level crossings and voltage dependent transitions in the excited state spectrum of an Al/AlOx/Al Cooper-pair box (CPB). The device was measured at 40 mK in the 15 - 50 GHz frequency range. We find tha
t a given level crosses the CPB spectrum at two different gate voltages; the frequency and splitting size of the two crossings differ and the splitting size depends on the Josephson energy of the CPB. We show that this behavior is not only consistent with the CPB being coupled to discrete charged two-level quantum systems which move atomic distances in the CPB junctions but that the spectra provide new information about the fluctuators, which is not available from phase qubit spectra of anomalous avoided levels. In particular by fitting a model Hamiltonian to our data, we extract microscopic parameters for each fluctuator, including well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator. The tunneling rates range from less than 3.5 to 13 GHz, which represent values between 5% and 150% of the well asymmetry, and the dipole moments give a minimum hopping distance of 0.3 to 0.8 Anstrom. We have also found that these discrete two-level systems have a pronounced effect on the relaxation time (T1) of the quantum states of the CPB and hence can be a source of dissipation for superconducting quantum bits.
