Evolution of the Cool Gas in the Circumgalactic Medium (CGM) of Massive
  Halos -- A Keck Cosmic Web Imager (KCWI) Survey of Ly$\alpha$ Emission around
  QSOs at $z\approx2$
  Motivated by the recent discovery of the near-ubiquity of Ly$\alpha$ emission
around $z \gtrsim 3$ QSOs, we performed a systematic study of QSO
circumgalactic Ly$\alpha$ emission at $z\approx2$, utilizing the unique
capability of the Keck Cosmic Web Imager (KCWI) -- a new wide-field, blue
sensitive integral-field spectrograph (IFU). In this paper, we present KCWI
observations on a sample of 16 ultraluminous Type-I QSOs at $z=2.1-2.3$ with
ionizing luminosities of $L_{\rm{\nu_{\rm{LL}}}}=10^{31.1-32.3}$ erg s$^{-1}$
Hz$^{-1}$. We found that 14 out of 16 QSOs are associated with Ly$\alpha$
nebulae with projected linear-sizes larger than 50 physical kpc (pkpc). Among
them, four nebulae have enormous Ly$\alpha$ emission with the Ly$\alpha$
surface brightness $SB_{\rm{Ly\alpha}} >10^{-17}$ erg s$^{-1}$ cm$^{-2}$
arcsec$^{-2}$ on the $>100$ kpc scale, extending beyond the field of view of
KCWI. Our KCWI observations reveal that most $z\approx2$ QSO nebulae have a
more irregular morphology compared to those at $z\gtrsim3$. In turn, we measure
that the circularly-averaged surface brightness (SB) at $z\approx2$ is 0.4 dex
fainter than the redshift-corrected, median SB at $z\gtrsim 3$. The Ly$\alpha$
SB profile (SB$_{\rm{Ly\alpha}}$) of QSOs at $z\approx 2$ can be described by a
power law of SB$_{\rm{Ly\alpha},z\approx2.3}=
3.7\times10^{-17}\times(r/40)^{-1.8}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$,
with the slope similar to that at $z\gtrsim3$. The observed lower
redshift-corrected, circularly-averaged SB may be mainly due to the lower
covering factor of cool gas clouds in massive halos at $z\approx2$.
