  We demonstrate cavity QED conditions in the Purcell regime for single quantum
emitters on the surface of an optical nanofiber. The cavity is formed by
combining an optical nanofiber and a nanofabricated grating to create a
composite photonic crystal cavity. Using this technique, significant
enhancement of the spontaneous emission rate into the nanofiber guided modes is
observed for single quantum dots. Our results pave the way for enhanced
on-fiber light-matter interfaces with clear applications to quantum networks.
