  We cooled ultrathin tapered fibers to cryogenic temperatures and controllably
coupled them with high-Q microsphere resonators at a wavelength close to the
optical transition of diamond nitrogen vacancy centers. The 310-nm-diameter
tapered fibers were stably nanopositioned close to the microspheres with a
positioning stability of approximately 10 nm over a temperature range of 7-28
K. A cavity-induced phase shift was observed in this temperature range,
demonstrating a discrete transition from undercoupling to overcoupling.
