  The atmosphere of a terrestrial planet that is replenished with secondary
gases should have accumulated hydrogen-rich gas from its protoplanetary disk.
Although a giant impact blows off a large fraction of the primordial atmosphere
of a terrestrial planet in the late formation stage, the remaining atmosphere
can become water-rich via chemical reactions between hydrogen and vaporized
core material. We find that a water-rich post-impact atmosphere forms when a
basaltic or CI chondrite core is assumed. In contrast, little post-impact water
is generated for an enstatite chondrite core. We investigate the X-ray- and
UV-driven mass loss from an Earth-mass planet with an impact-induced
multi-component H$_2$$-$He$-$H$_2$O atmosphere for Gyrs. We show that water is
left in the atmosphere of an Earth-mass planet when the low flux of escaping
hydrogen cannot drag water upward via collisions. For a water-dominated
atmosphere to form, the atmospheric mass fraction of an Earth-mass planet with
an oxidizing core after a giant impact must be less than a few times 0.1%. We
also find that Earth-mass planets with water-dominated atmospheres can exist at
semimajor axes ranging from a few times 0.1 au to a few au around a Sun-like
star depending on the mass loss efficiency. Such planets are important targets
for atmospheric characterization in the era of JWST. Our results indicate that
efficient mixing between hydrogen and rocky components during giant impacts can
play a role in the production of water in an Earth-mass planet.
