Optically coherent nitrogen-vacancy centers in {\mu}m-thin etched diamond membranes

Diamond membrane devices containing optically coherent nitrogen-vacancy (NV) centers are key to enable novel cryogenic experiments such as optical ground-state cooling of hybrid spin-mechanical systems and efficient entanglement distribution in quantum networks. Here, we report on the fabrication of a (3.4 $\pm$ 0.2) {\mu}m thin, smooth (surface roughness r$_q$ < 0.4 nm over an area of 20 {\mu}m by 30 {\mu}m diamond membrane containing individually resolvable, narrow linewidth (< 100 MHz) NV centers. We fabricate this sample via a combination of high energy electron irradiation, high temperature annealing, and an optimized etching sequence found via a systematic study of the diamond surface evolution on the microscopic level in different etch chemistries. While our particular device dimensions are optimized for cavity-enhanced entanglement generation between distant NV centers in open, tuneable micro-cavities, our results have implications for a broad range of quantum experiments that require the combination of narrow optical transitions and {\mu}m-scale device geometry.