Constraining the quasar radio-loud fraction at $z \sim 6$ with deep
  radio observations
  We carry out a series of deep Karl G. Jansky Very Large Array (VLA) S-band
observations of a sample of 21 quasars at $z\sim6$. The new observations expand
the searches of radio continuum emission to the optically faint quasar
population at the highest redshift with rest-frame $4400 \rm \AA$ luminosities
down to $3 \times10^{11} \ L_{\odot}$. We report the detections of two new
radio-loud quasars: CFHQS J2242+0334 (hereafter J2242+0334) at $z=5.88$ and
CFHQS J0227$-$0605 (hereafter J0227$-$0605) at $z=6.20$, detected with 3 GHz
flux densities of $87.0 \pm 6.3 \ \mu \rm Jy$ and $55.4 \pm 6.7 \ \mu \rm Jy$,
respectively. Their radio \replaced{loudness}{loudnesses} are estimated to be
$54.9 \pm 4.7$ and $16.5 \pm 3.2$, respectively. To better constrain the
radio-loud fraction (RLF), we combine the new measurements with the archival
VLA L-band data as well as available data from the literature, considering the
upper limits for non-detections and \deleted{and} possible selection effects.
The final derived RLF is $9.4 \pm 5.7\%$ for the optically selected quasars at
$z\sim6$. We also compare the RLF to that of the quasar samples at low redshift
and check the RLF in different quasar luminosity bins. The RLF for the
optically faint objects is still poorly constrained due to the limited sample
size. Our \replaced{result}{results} show no evidence of significant quasar RLF
evolution with redshift. There is also no clear trend of RLF evolution with
quasar UV/optical luminosity due to the limited sample size of optically faint
objects with deep radio observations.
