  We report on the design and performance of a cryogenic (300 mK) near-field
scanning microwave microscope. It uses a microwave resonator as the near-field
sensor, operating at a frequency of 6 GHz and microwave probing amplitudes down
to 100 uV, approaching low enough photon population (N~1000) of the resonator
such that coherent quantum manipulation becomes feasible. The resonator is made
out of a miniaturized distributed fractal superconducting circuit that is
integrated with the probing tip, micromachined to be compact enough such that
it can be mounted directly on a quartz tuning-fork, and used for parallel
operation as an atomic force microscope (AFM). The resonator is magnetically
coupled to a transmission line for readout, and to achieve enhanced sensitivity
we employ a Pound-Drever-Hall measurement scheme to lock to the resonance
frequency. We achieve a well localized near-field around the tip such that the
microwave resolution is comparable to the AFM resolution, and a capacitive
sensitivity down to 6.4x10^-20 F/rtHz, limited by mechanical noise. We believe
that the results presented here are a significant step towards probing quantum
systems at the nanoscale using near-field scanning microwave microscopy.
