  Context: High contrast imaging is a powerful technique to detect and
characterize planetary companions at large orbital separations from their
parent stars. Aims: We aim at studying the limiting magnitude of the VLT/SPHERE
Adaptive Optics system and the corresponding instrument performance for faint
targets (G $\ge$ 11.0 mag). Methods: We computed coronagraphic H-band raw
contrast at 300 [mas] and FWHM of the non-coronagraphic PSF, for a total of 111
different stars observed between 2016 and 2022 with IRDIS. For this, we
processed a large number of individual frames that were obtained under
different atmospheric conditions. We then compared the resulting raw contrast
and the PSF shape as a function of the visible wave front sensor instant flux
which scales with the G-band stellar magnitude. We repeated this analysis for
the top 10\% and 30\% best turbulence conditions in Cerro Paranal. Results: We
found a strong decrease in the coronagraphic achievable contrast for star
fainter than G $\sim$ 12.5 mag, even under the best atmospheric conditions. In
this regime, the AO correction is dominated by the read-out noise of the WFS
detector. In particular we found roughly a factor ten decrease in the raw
contrast ratio between stars with G $\sim$ 12.5 and G $\sim$ 14.0 mag.
Similarly, we observe a sharp increase in the FWHM of the non-coronagraphic PSF
beyond G $\sim$ 12.5 mag, and a corresponding decrease in the strehl ratio from
$\sim$ 50\% to $\sim$ 20\% for the faintest stars. Although these trend are
observed for the two turbulence categories, the decrease in the contrast ratio
and PSF sharpness is more pronounced for poorer conditions.
