Thin-film superconducting resonator tunable to the ground-state   hyperfine splitting of $^{87}$Rb
  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 frequency 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. 
Z. Kim, C. P. Vlahacos, J. E. Hoffman, J. A. Grover, K. D. Voigt, B.   K. Cooper, C. J. Ballard, B. S. Palmer, M. Hafezi, J....
quant-ph