  What are the frequency, shape, kinematics, and luminosity of Ly\alpha\
envelopes surrounding radio-quiet quasars at high redshift, and is the
luminosity of these envelopes related to that of the quasar or not? As a first
step towards answering these questions, we have searched for Ly\alpha\
envelopes around six radio-quiet quasars at z~4.5, using deep spectra taken
with the FORS2 spectrograph attached to the UT1 of the Very Large Telescope
(VLT). Using the multi-slit mode allows us to observe several point spread
function stars simultaneously with the quasar, and to remove the point-like
emission from the quasar, unveiling the faint underlying Ly\alpha\ envelope
with unprecedented depth. An envelope is detected around four of the six
quasars, which suggests that these envelopes are very frequent. Their diameter
varies in the range 26<d<64 kpc, their surface brightness in the range
3x10^{-19}<\mu<2x10^{-17} erg/s/cm^2/arcsec^2, and their luminosity in the
range 10^{42}<L(Ly\alpha)<10^{44} erg/s. Their shape may be strongly
asymmetric. The Ly\alpha\ emission line full width at half maximum (FWHM) is
900<FWHM<2200 km/s and its luminosity correlates with that of the broad line
region (BLR) of the quasar, with the notable exception of BR2237-0607, the
brightest object in our sample. The same holds for the relation between the
envelope Ly\alpha\ luminosity and the ionizing luminosity of the quasar. While
the deep slit spectroscopy presented in this paper is very efficient at
detecting very faint Ly\alpha\ envelopes, narrow-band imaging is now needed to
measure accurately their spatial extent, radial luminosity profile, and total
luminosity. These observables are crucial to help us discriminate between the
three possible radiation processes responsible for the envelope emission: (i)
cold accretion, (ii) fluorescence induced by the quasar, and (iii) scattering
of the BLR photons by cool gas.
