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.