  We report the detection of an ionized gas outflow from an $X$-ray active
galactic nucleus (AGN) hosted in a massive quiescent galaxy in a protocluster
at $z=3.09$ (J221737.29+001823.4). It is a type-2 QSO with broad ($W_{80}>1000$
km s$^{-1}$) and strong ($\log (L_{\rm [OIII]}$ / erg s$^{-1})\approx43.4$) [O
{\footnotesize III}]$\lambda\lambda$4959,5007 emission lines detected by slit
spectroscopy in three-position angles using Multi-Object Infra-Red Camera and
Spectrograph (MOIRCS) on the Subaru telescope and the Multi-Object Spectrometer
For Infra-Red Exploration (MOSFIRE) on the Keck-I telescope. In the all slit
directions, [O {\footnotesize III}] emission is extended to $\sim15$ physical
kpc and indicates a powerful outflow spreading over the host galaxy. The
inferred ionized gas mass outflow rate is $\rm 22\pm3~M_{\odot}~yr^{-1}$.
Although it is a radio source, according to the line diagnostics using
H$\beta$, [O {\footnotesize II}], and [O {\footnotesize III}], photoionization
by the central QSO is likely the dominant ionization mechanism rather than
shocks caused by radio jets. On the other hand, the spectral energy
distribution of the host galaxy is well characterized as a quiescent galaxy
that has shut down star formation by several hundred Myr ago. Our results
suggest a scenario that QSOs are powered after the shut-down of the star
formation and help to complete the quenching of massive quiescent galaxies at
high redshift.
