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In this paper, we investigate the potential of dark sirens by the space-borneatom interferometric gravitational-wave detectors to probe the Hubble constant.In the mid-frequency band, the sources live a long time. The motion of adetector around the Sun as well as in Earth orbit would induce large Dopplerand reorientation effects, providing a precise angular resolution. Such preciselocalization for the GW sources makes it possible to observe the dark sirenswith only one potential host galaxy, which are dubbed "golden dark sirens". Weconstruct the catalogs of golden dark sirens and estimate that there are around79 and 35 golden dark sirens of binary neutron stars (BNS) and binary blackholes (BBH) that would be pass the detection threshold of AEDGE in 5 years. Ourresults show that with 5, 10, and all 79 golden dark BNS tracked by AEDGE onecan constrain $H_0$ at 5.5\%, 4.1\%, and 1.8\% precision levels. With 5, 10,and all 35 golden dark BBH one can constrain $H_0$ at 2.2\%, 1.8\%, and 1.5\%precision levels, respectively. It suggests that only 5-10 golden dark BBH byAEDGE are sufficient to arbitrate the current tension between local andhigh-$z$ measurements of $H_0$.

Space-borne atom interferometric gravitational wave detections. Part II. Dark sirens and finding the one